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 Showing 1 - 82 of 82 Journals sorted alphabetically Advances in Applied Mathematics       (Followers: 15) Advances in Applied Mathematics and Mechanics       (Followers: 6) Advances in Applied Mechanics       (Followers: 15) AKCE International Journal of Graphs and Combinatorics American Journal of Applied Mathematics and Statistics       (Followers: 11) American Journal of Applied Sciences       (Followers: 22) American Journal of Modeling and Optimization       (Followers: 3) Annals of Actuarial Science       (Followers: 2) Applied Mathematical Modelling       (Followers: 22) Applied Mathematics and Computation       (Followers: 31) Applied Mathematics and Mechanics       (Followers: 4) Applied Mathematics and Nonlinear Sciences Applied Mathematics and Physics       (Followers: 2) Biometrical Letters British Actuarial Journal       (Followers: 2) Bulletin of Mathematical Sciences and Applications Communication in Biomathematical Sciences       (Followers: 2) Communications in Applied and Industrial Mathematics       (Followers: 1) Communications on Applied Mathematics and Computation       (Followers: 1) Differential Geometry and its Applications       (Followers: 4) Discrete and Continuous Models and Applied Computational Science Discrete Applied Mathematics       (Followers: 10) Doğuş Üniversitesi Dergisi e-Journal of Analysis and Applied Mathematics Engineering Mathematics Letters       (Followers: 1) European Actuarial Journal Foundations and Trends® in Optimization       (Followers: 3) Frontiers in Applied Mathematics and Statistics       (Followers: 1) Fundamental Journal of Mathematics and Applications International Journal of Advances in Applied Mathematics and Modeling       (Followers: 1) International Journal of Applied Mathematics and Statistics       (Followers: 3) International Journal of Computer Mathematics : Computer Systems Theory International Journal of Data Mining, Modelling and Management       (Followers: 10) International Journal of Engineering Mathematics       (Followers: 7) International Journal of Fuzzy Systems International Journal of Swarm Intelligence       (Followers: 2) International Journal of Theoretical and Mathematical Physics       (Followers: 13) International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems       (Followers: 3) Journal of Advanced Mathematics and Applications       (Followers: 1) Journal of Advances in Mathematics and Computer Science Journal of Applied & Computational Mathematics Journal of Applied Intelligent System Journal of Applied Mathematics & Bioinformatics       (Followers: 6) Journal of Applied Mathematics and Physics       (Followers: 9) Journal of Computational Geometry       (Followers: 3) Journal of Innovative Applied Mathematics and Computational Sciences       (Followers: 6) Journal of Mathematical Sciences and Applications       (Followers: 2) Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance       (Followers: 12) Journal of Mathematics and Statistics Studies Journal of Physical Mathematics       (Followers: 2) Journal of Symbolic Logic       (Followers: 2) Letters in Biomathematics       (Followers: 1) Mathematical and Computational Applications       (Followers: 3) Mathematical Models and Computer Simulations       (Followers: 3) Mathematics and Computers in Simulation       (Followers: 3) Modeling Earth Systems and Environment       (Followers: 1) Moscow University Computational Mathematics and Cybernetics Multiscale Modeling and Simulation       (Followers: 2) Pacific Journal of Mathematics for Industry Partial Differential Equations in Applied Mathematics       (Followers: 1) Ratio Mathematica Results in Applied Mathematics       (Followers: 1) Scandinavian Actuarial Journal       (Followers: 2) SIAM Journal on Applied Dynamical Systems       (Followers: 3) SIAM Journal on Applied Mathematics       (Followers: 11) SIAM Journal on Computing       (Followers: 11) SIAM Journal on Control and Optimization       (Followers: 18) SIAM Journal on Discrete Mathematics       (Followers: 8) SIAM Journal on Financial Mathematics       (Followers: 3) SIAM Journal on Imaging Sciences       (Followers: 7) SIAM Journal on Mathematical Analysis       (Followers: 4) SIAM Journal on Matrix Analysis and Applications       (Followers: 3) SIAM Journal on Numerical Analysis       (Followers: 7) SIAM Journal on Optimization       (Followers: 12) SIAM Journal on Scientific Computing       (Followers: 16) SIAM Review       (Followers: 9) SIAM/ASA Journal on Uncertainty Quantification       (Followers: 2) Swarm Intelligence       (Followers: 3) Theory of Probability and its Applications       (Followers: 2) Uniform Distribution Theory Universal Journal of Applied Mathematics       (Followers: 2) Universal Journal of Computational Mathematics       (Followers: 3)
Similar Journals
 SIAM Journal on Scientific ComputingJournal Prestige (SJR): 1.973 Citation Impact (citeScore): 3Number of Followers: 16      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1064-8275 - ISSN (Online) 1095-7197 Published by Society for Industrial and Applied Mathematics  [17 journals]
• Multirate Partially Explicit Scheme for Multiscale Flow Problems

Authors: Wing Tat Leung, Yating Wang
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1775-A1806, June 2022.
For time-dependent problems with high-contrast multiscale coefficients, the time step size for explicit methods is affected by the magnitude of the coefficient parameter. With a suitable construction of multiscale space, one can achieve a stable temporal splitting scheme where the time step size is independent of the contrast [E. T. Chung et al., J. Comput. Phys., 445 (2021), 110578]. Considering the parabolic equation with a heterogeneous diffusion parameter, the flow rates vary significantly in different regions due to the high-contrast features of the diffusivity. In this work, we aim to introduce a multirate partially explicit splitting scheme to achieve efficient simulation with the desired accuracy. We first design multiscale subspaces to handle flow with different speeds. For the fast flow, we obtain a low-dimensional subspace for the high-diffusive component and adopt an implicit time discretization scheme. The other multiscale subspace will take care of the slow flow, and the corresponding degrees of freedom are treated explicitly. Then a multirate time stepping is introduced for the two parts. The stability of the multirate methods is analyzed for the partially explicit scheme. Moreover, we derive local error estimators corresponding to the two components of the solutions and provide an upper bound of the errors. An adaptive local temporal refinement framework is then proposed to achieve higher computational efficiency. Several numerical tests are presented to demonstrate the performance of the proposed method.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-30T07:00:00Z
DOI: 10.1137/21M1440293
Issue No: Vol. 44, No. 3 (2022)

• On Thermodynamically Compatible Finite Volume Schemes for Continuum
Mechanics

Authors: Saray Busto, Michael Dumbser, Ilya Peshkov, Evgeniy Romenski
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1723-A1751, June 2022.
In this paper we present a new family of semidiscrete and fully discrete finite volume schemes for overdetermined, hyperbolic, and thermodynamically compatible PDE systems. In the following, we will denote these methods as HTC schemes. In particular, we consider the Euler equations of compressible gasdynamics as well as the more complex Godunov--Peshkov--Romenski (GPR) model of continuum mechanics, which, with the aid of suitable relaxation source terms, is able to describe nonlinear elasto-plastic solids at large deformations as well as viscous fluids as two special cases of a more general first-order hyperbolic model of continuum mechanics. The main novelty of the schemes presented in this paper lies in the fact that we solve the entropy inequality as a primary evolution equation rather than the usual total energy conservation law. Instead, total energy conservation is achieved as a mere consequence of a thermodynamically compatible discretization of all the other equations. For this, we first construct a discrete framework for the compressible Euler equations that mimics the continuous framework of Godunov's seminal paper [Dokl. Akad. Nauk SSSR, 139(1961), pp. 521--523] exactly at the discrete level. All other terms in the governing equations of the more general GPR model, including nonconservative products, are judiciously discretized in order to achieve discrete thermodynamic compatibility, with the exact conservation of total energy density as a direct consequence of all the other equations. As a result, the HTC schemes proposed in this paper are provably marginally stable in the energy norm and satisfy a discrete entropy inequality by construction. We show some computational results obtained with HTC schemes in one and two space dimensions, considering both the fluid limit as well as the solid limit of the governing PDEs.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-28T07:00:00Z
DOI: 10.1137/21M1417508
Issue No: Vol. 44, No. 3 (2022)

• An Efficient MultiModes Monte Carlo Homogenization Method for Random
Materials

Authors: Zihao Yang, Jizu Huang, Xiaobing Feng, Xiaofei Guan
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1752-A1774, June 2022.
In this paper, we propose and analyze a new stochastic homogenization method for diffusion equations with random and fast oscillatory coefficients. In the proposed method, the homogenized solutions are sought through a two-stage procedure. In the first stage, the original oscillatory diffusion equation is approximated, for each fixed random sample $\omega$, by a spatially homogenized diffusion equation with piecewise constant coefficients, resulting in a random diffusion equation. In the second stage, the resulting random diffusion equation is approximated and computed by using an efficient multimodes Monte Carlo method which only requires solving a diffusion equation with a constant diffusion coefficient and a random right-hand side. The main advantage of the proposed method is that it separates the computational difficulty caused by the spatial fast oscillation of the solution and caused by the randomness of the solution, so they can be overcome separately using different strategies. The convergence of the solution of the spatially homogenized equation (from the first stage) to the solution of the original random diffusion equation is established, and the optimal rate of convergence is also obtained for the proposed multimodes Monte Carlo method. Numerical experiments on some benchmark test problems for random composite materials are also presented to gauge the efficiency and accuracy of the proposed two-stage stochastic homogenization method.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-28T07:00:00Z
DOI: 10.1137/21M1454237
Issue No: Vol. 44, No. 3 (2022)

• On Port-Hamiltonian Approximation of a Nonlinear Flow Problem on Networks

Authors: Björn Liljegren-Sailer, Nicole Marheineke
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B834-B859, June 2022.
This paper deals with the systematic development of structure-preserving approximations for a class of nonlinear partial differential equations on networks. The class includes, for example, gas pipe network systems described by barotropic Euler equations. Our approach is guided throughout by energy-based modeling concepts (port-Hamiltonian formalism, theory of Legendre transformation), which provide a convenient and general line of reasoning. Under mild assumptions on the approximation, local conservation of mass, an energy bound, and the inheritance of the port-Hamiltonian structure can be shown. Our approach is not limited to conventional space discretization but also covers complexity reduction of the nonlinearities by inexact integration. Thus, it can serve as a basis for structure-preserving model reduction. Combined with an energy-stable time integration, we demonstrate the applicability and good stability properties using the example of the Euler equations on networks.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-28T07:00:00Z
DOI: 10.1137/21M1443480
Issue No: Vol. 44, No. 3 (2022)

• Numerical Reparametrization of Periodic Planar Curves Via Curvature
Interpolation

Authors: Kazuki Koga
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1703-A1722, June 2022.
A novel static algorithm is proposed for numerical reparametrization of periodic planar curves. The method identifies a monitor function of the arclength variable with the true curvature of an open planar curve and considers a simple interpolation between the object and the unit circle at the curvature level. Since a convenient formula is known for tangential velocity that maintains the equidistribution rule with curvature-type monitor functions, the strategy enables one to compute the correspondence between the arclength and another spatial variable by evolving the interpolated curve. With a certain normalization, velocity information in the motion is obtained with spectral accuracy while the resulting parametrization remains unchanged. Then, the algorithm extracts a refined representation of the input curve by sampling its arclength parametrization, whose Fourier coefficients are directly accessed through a simple change of variables. As a validation, improvements to spatial resolution are evaluated by approximating the invariant coefficients from downsampled data and observing faster global convergence to the original shape.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-27T07:00:00Z
DOI: 10.1137/21M145197X
Issue No: Vol. 44, No. 3 (2022)

• State Estimation with Model Reduction and Shape Variability. Application
to biomedical problems

Authors: Felipe Galarce, Damiano Lombardi, Olga Mula
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B805-B833, June 2022.
We develop a mathematical and numerical framework to solve state estimation problems for applications that present variations in the shape of the spatial domain. This situation arises typically in a biomedical context where inverse problems are posed on certain organs or portions of the body which inevitably involve morphological variations. If one wants to provide fast reconstruction methods, the algorithms must take into account the geometric variability. We develop and analyze a method which allows us to take this variability into account without needing any a priori knowledge on a parametrization of the geometrical variations. For this, we rely on morphometric techniques involving multidimensional scaling and couple them with reconstruction algorithms that make use of linear subspaces precomputed on a database of geometries. We prove the potential of the method on a synthetic test problem inspired by the reconstruction of blood flows and quantities of medical interest with Doppler ultrasound imaging.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-27T07:00:00Z
DOI: 10.1137/21M1430480
Issue No: Vol. 44, No. 3 (2022)

• A Well-Balanced Asymptotic Preserving Scheme for the Two-Dimensional
Rotating Shallow Water Equations with Nonflat Bottom Topography

Authors: Alexander Kurganov, Yongle Liu, Mária Lukáčová-Medviďová
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1655-A1680, June 2022.
We consider the two-dimensional rotating shallow water equations with nonflat bottom topography. We focus on the case of low Froude number, in which the system is stiff and conventional explicit numerical methods are extremely inefficient and often impractical. Our goal is to design a finite volume scheme, which is both asymptotic preserving (uniformly asymptotically consistent and stable for a broad range of low Froude numbers) and well-balanced (capable of exactly preserving geophysically relevant steady-state solutions). The goal is achieved in two steps. We first rewrite the studied equations in terms of perturbations of the steady state and then apply the flux splitting similar to the one used in [Liu, Chertok, and Kurganov J. Comput. Phys., 391 (2019), pp. 259-279]. We split the flux into the stiff and nonstiff parts and then use an implicit-explicit approach: apply an explicit second-order central-upwind scheme to the nonstiff part of the system while treating the stiff part implicitly. As the stiff part of the flux is linear, we reduce the implicit stage of the proposed method to solving a Poisson-type elliptic equation, which is discretized using a standard second-order central difference scheme. We prove the asymptotic preserving property of the developed scheme and conduct a series of numerical experiments, which demonstrate that our scheme outperforms the non-well-balanced asymptotic preserving scheme from [Liu, Chertok, and Kurganov J. Comput. Phys., 391 (2019), pp. 259-279].
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-24T07:00:00Z
DOI: 10.1137/21M141573X
Issue No: Vol. 44, No. 3 (2022)

• Optimizing Oblique Projections for Nonlinear Systems using Trajectories

Authors: Samuel E. Otto, Alberto Padovan, Clarence W. Rowley
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1681-A1702, June 2022.
Reduced-order modeling techniques, including balanced truncation and $\mathcal{H}_2$-optimal model reduction, exploit the structure of linear dynamical systems to produce models that accurately capture the dynamics. For nonlinear systems operating far away from equilibria, on the other hand, current approaches seek low-dimensional representations of the state that often neglect low-energy features that have high dynamical significance. For instance, low-energy features are known to play an important role in fluid dynamics, where they can be a driving mechanism for shear-layer instabilities. Neglecting these features leads to models with poor predictive accuracy despite being able to accurately encode and decode states. In order to improve predictive accuracy, we propose to optimize the reduced-order model to fit a collection of coarsely sampled trajectories from the original system. In particular, we optimize over the product of two Grassmann manifolds defining Petrov--Galerkin projections of the full-order governing equations. We compare our approach with existing methods including proper orthogonal decomposition, balanced truncation-based Petrov--Galerkin projection, quadratic-bilinear balanced truncation, and the quadratic-bilinear iterative rational Krylov algorithm. Our approach demonstrates significantly improved accuracy both on a nonlinear toy model and on an incompressible (nonlinear) axisymmetric jet flow with $10^5$ states.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-24T07:00:00Z
DOI: 10.1137/21M1425815
Issue No: Vol. 44, No. 3 (2022)

• A Massively Parallel Eulerian-Lagrangian Method for Advection-Dominated
Transport in Viscous Fluids

Authors: Nils Kohl, Marcus Mohr, Sebastian Eibl, Ulrich Rüde
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page C260-C285, June 2022.
Motivated by challenges in the Earth's mantle convection, we present a massively parallel implementation of an Eulerian-Lagrangian method for the advection-diffusion equation in the advection-dominated regime. The advection term is treated by a particle-based characteristics method coupled to a block-structured finite element framework. Its numerical and computational performance is evaluated in multiple two- and three-dimensional benchmarks, including curved geometries, discontinuous solutions, and pure advection, and it is applied to a coupled nonlinear system modeling buoyancy-driven convection in Stokes flow. We demonstrate the parallel performance in a strong and weak scaling experiment, with scalability to up to 147,456 parallel processes, solving for more than 5.2 x 10^{10} (52 billion) degrees of freedom per time-step.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-24T07:00:00Z
DOI: 10.1137/21M1402510
Issue No: Vol. 44, No. 3 (2022)

• Grouped Transformations and Regularization in High-Dimensional Explainable
ANOVA Approximation

Authors: Felix Bartel, Daniel Potts, Michael Schmischke
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1606-A1631, June 2022.
In this paper we propose a tool for high-dimensional approximation based on trigonometric polynomials where we allow only low-dimensional interactions of variables. In a general high-dimensional setting, it is already possible to deal with special sampling sets such as sparse grids or rank-1 lattices. This requires black-box access to the function, i.e., the ability to evaluate it at any point. Here, we focus on scattered data points and grouped frequency index sets along the dimensions. From there we propose a fast matrix-vector multiplication, the grouped Fourier transform, for high-dimensional grouped index sets. Those transformations can be used in the application of the previously introduced method of approximating functions with low superposition dimension based on the analysis of variance (ANOVA) decomposition where there is a one-to-one correspondence from the ANOVA terms to our proposed groups. The method is able to dynamically detect important sets of ANOVA terms in the approximation. In this paper, we consider the involved least-squares problem and add different forms of regularization: classical Tikhonov-regularization, namely, regularized least squares, and the technique of group lasso, which promotes sparsity in the groups. As for the latter, there are no explicit solution formulas, which is why we applied the fast iterative shrinkage-thresholding algorithm to obtain the minimizer. Moreover, we discuss the possibility of incorporating smoothness information into the least-squares problem. Numerical experiments in underdetermined, overdetermined, and noisy settings indicate the applicability of our algorithms. While we consider periodic functions, the idea can be directly generalized to nonperiodic functions as well.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-23T07:00:00Z
DOI: 10.1137/20M1374547
Issue No: Vol. 44, No. 3 (2022)

• Genetic Column Generation: Fast Computation of High-Dimensional
Multimarginal Optimal Transport Problems

Authors: Gero Friesecke, Andreas S. Schulz, Daniela Vögler
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1632-A1654, June 2022.
We introduce a simple, accurate, and extremely efficient method for numerically solving multimarginal optimal transport (MMOT) problems arising in density functional theory. The method relies on (i) the sparsity of optimal plans (for $N$ marginals discretized by $\ell$ gridpoints each, general Kantorovich plans require $\ell^N$ gridpoints, but the support of optimizers is of size $O(\ell\cdot N)$ [G. Friesecke and D. Vögler, SIAM J. Math. Anal., 50 (2018), pp. 3996--4019], (ii) the method of column generation (CG) from discrete optimization which is novel in the optimal transport context, and (iii) ideas from machine learning. The well-known bottleneck in CG consists in generating new candidate columns efficiently; we prove that in our context, finding the best new column is an NP-complete problem. To overcome this bottleneck we use a genetic learning method tailor-made for MMOT in which the dual state within CG plays the role of an “adversary” in loose similarity to Wasserstein generative adversarial networks (GANs). On a sequence of benchmark problems with up to 120 gridpoints and up to 30 marginals, our method always finds the exact optimizers. Moreover, empirically the number of computational steps needed to find them appears to scale only polynomially when both $N$ and $\ell$ are simultaneously increased (while keeping their ratio fixed to mimic a thermodynamic limit of the particle system).
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-23T07:00:00Z
DOI: 10.1137/21M140732X
Issue No: Vol. 44, No. 3 (2022)

• Differential Equation Based Path Integral for Open Quantum Systems

Authors: Geshuo Wang, Zhenning Cai
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B771-B804, June 2022.
We propose the differential equation based path integral method to simulate the real-time evolution of open quantum systems. In this method, a system of partial differential equations is derived based on the continuation of a classical numerical method called the iterative quasi-adiabatic propagator path integral (i-QuAPI). While the resulting system has infinite equations, we introduce a reasonable closure to obtain a series of finite systems. New numerical schemes can be derived by discretizing these differential equations. It is numerically verified that in certain cases, by selecting appropriate systems and applying suitable numerical schemes, the memory cost required in the i-QuAPI method can be significantly reduced.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-23T07:00:00Z
DOI: 10.1137/21M1439833
Issue No: Vol. 44, No. 3 (2022)

• A Nonlinear Elimination Preconditioned Inexact Newton Algorithm

Authors: Lulu Liu, Feng-Nan Hwang, Li Luo, Xiao-Chuan Cai, David E. Keyes
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1579-A1605, June 2022.
A nonlinear elimination preconditioned inexact Newton (NEPIN) algorithm is proposed for problems with localized strong nonlinearities. Due to unbalanced nonlinearities (nonlinear stiffness''), the traditional inexact Newton method often exhibits a long plateau in the norm of the nonlinear residual or even fails to converge. NEPIN implicitly removes the components causing trouble for the global convergence through a correction based on nonlinear elimination within a subspace that provides a modified direction for the global Newton iteration. Numerical experiments show that NEPIN can be more robust than global inexact Newton algorithms and maintain fast convergence even for challenging problems, such as full potential transonic flows. NEPIN complements several previously studied nonlinear preconditioners with which it compares favorably experimentally on a classic shocked duct flow problem considered herein. NEPIN is shown to be fairly insensitive to mesh resolution and “bad” subproblem identification based on the local Mach number or the local nonlinear residual for transonic flow over a wing.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-21T07:00:00Z
DOI: 10.1137/21M1416138
Issue No: Vol. 44, No. 3 (2022)

• Approximating Optimal feedback Controllers of Finite Horizon Control
Problems Using Hierarchical Tensor Formats

Authors: Mathias Oster, Leon Sallandt, Reinhold Schneider
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B746-B770, June 2022.
Controlling systems of ordinary differential equations is ubiquitous in science and engineering. For finding an optimal feedback controller, the value function and associated fundamental equations such as the Bellman equation and the Hamilton--Jacobi--Bellman equation are essential. The numerical treatment of these equations poses formidable challenges due to their non-linearity and their (possibly) high dimensionality. In this paper we consider a finite horizon control system with associated Bellman equation. After a time discretization, we obtain a sequence of short time horizon problems which we call local optimal control problems. For solving the local optimal control problems we apply two different methods; one is the well-known policy iteration, where a fixed-point iteration is required for every time step. The other algorithm borrows ideas from model predictive control by solving the local optimal control problem via open-loop control methods on a short time horizon, allowing us to replace the fixed-point iteration by an adjoint method. For high-dimensional systems we apply low-rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains and multipolynomials, together with high-dimensional quadrature, e.g., Monte Carlo. We prove a linear error propagation with respect to the time discretization and give numerical evidence by controlling a diffusion equation with an unstable reaction term and an Allen--Cahn equation.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-21T07:00:00Z
DOI: 10.1137/21M1412190
Issue No: Vol. 44, No. 3 (2022)

• Optimization-based Parametric Model Order Reduction via ${{\mathcal{H}_2} \otimes {\mathcal{L}_2}}$ First-order Necessary Conditions

Authors: Manuela Hund, Tim Mitchell, Petar Mlinarić, Jens Saak
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1554-A1578, June 2022.
In this paper, we generalize existing frameworks for ${{\mathcal{H}_2} \otimes {\mathcal{L}_2}}$-optimal model order reduction to a broad class of parametric linear time-invariant systems. To this end, we derive first-order necessary optimality conditions for a class of structured reduced-order models and then, building on those, propose a stability-preserving optimization-based method for computing locally ${{\mathcal{H}_2} \otimes {\mathcal{L}_2}}$-optimal reduced-order models. We also make a theoretical comparison to existing approaches in the literature and, in numerical experiments, show how our new method, with reasonable computational effort, produces stable optimized reduced-order models with significantly lower approximation errors.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-16T07:00:00Z
DOI: 10.1137/21M140290X
Issue No: Vol. 44, No. 3 (2022)

• Graph Refinement via Simultaneously Low-Rank and Sparse Approximation

Authors: Zhenyue Zhang, Zheng Zhai, Limin Li
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1525-A1553, June 2022.
Graphs play an important role in many fields of machine learning such as clustering. Many graph-based machine learning approaches assume that the graphs have hidden group structures. However, the group structures are unclear or noisy in applications generally. Graph refinement aims to clarify the underlying group structures. In this work, a novel approach, called Simultaneously Low-rank and Sparse Approximation (SLSA), is proposed for graph refinement, which imposes a strong cluster structure through strict sparse and low-rank assumptions simultaneously. This approach minimizes a nonconvex function. Fortunately, the optimization problem can be efficiently solved via an alternating iteration method, and the iterative method converges globally under a weak condition. A fast iterative algorithm is also given for large-scale sparse graphs, which costs $O(n)$ in each iteration. Compared with two other related methods for graph refinement, SLSA performs better on both synthetic and real-world data sets. Applications of the refinement method SLSA on several machine learning algorithms are discussed in detail. Numerical experiments show that the improvements of these algorithms are significant under the SLSA modifications and better than the improvements based on the refinements of other approaches.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-14T07:00:00Z
DOI: 10.1137/21M1446459
Issue No: Vol. 44, No. 3 (2022)

• Simulating Self-Avoiding Isometric Plate Bending

Authors: Sören Bartels, Frank Meyer, Christian Palus
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1475-A1496, June 2022.
Inspired by recent results on self-avoiding inextensible curves, we propose and experimentally investigate a numerical method for simulating isometric plate bending without self-intersections. We consider a nonlinear two-dimensional Kirchhoff plate model which is augmented via addition of a tangent-point energy. The resulting continuous model energy is finite if and only if the corresponding deformation is injective, i.e., neither includes self-intersections nor self-contact. We propose a finite element method method based on discrete Kirchhoff triangles for the spatial discretization and employ a semi-implicit gradient descent scheme for the minimization of the discretized energy functional. Practical properties of the proposed method are illustrated with numerous numerical simulations, exploring the model behavior in different settings and demonstrating that our method is capable of preventing noninjective deformations.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-13T07:00:00Z
DOI: 10.1137/21M1440001
Issue No: Vol. 44, No. 3 (2022)

• A Fast Petrov--Galerkin Spectral Method for the Multidimensional Boltzmann
Equation Using Mapped Chebyshev Functions

Authors: Jingwei Hu, Xiaodong Huang, Jie Shen, Haizhao Yang
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1497-A1524, June 2022.
Numerical approximation of the Boltzmann equation presents a challenging problem due to its high-dimensional, nonlinear, and nonlocal collision operator. Among the deterministic methods, the Fourier--Galerkin spectral method stands out for its relative high accuracy and possibility of being accelerated by the fast Fourier transform. However, this method requires a domain truncation which is unphysical since the collision operator is defined in $\mathbb{R}^d$. In this paper, we introduce a Petrov--Galerkin spectral method for the Boltzmann equation in the unbounded domain. The basis functions (both test and trial functions) are carefully chosen mapped Chebyshev functions to obtain desired convergence and conservation properties. Furthermore, thanks to the close relationship of the Chebyshev functions and the Fourier cosine series, we are able to construct a fast algorithm with the help of the nonuniform fast Fourier transform. We demonstrate the superior accuracy of the proposed method in comparison to the Fourier spectral method through a series of two-dimensional and three-dimensional examples.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-13T07:00:00Z
DOI: 10.1137/21M1420721
Issue No: Vol. 44, No. 3 (2022)

• High-Dimensional Dynamic Stochastic Model Representation

Authors: Aryan Eftekhari, Simon Scheidegger
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page C210-C236, June 2022.
We propose a scalable method for computing global solutions of nonlinear, high-dimensional dynamic stochastic economic models. First, within a time iteration framework, we approximate economic policy functions using an adaptive, high-dimensional model representation scheme, combined with adaptive sparse grids to address the ubiquitous challenge of the curse of dimensionality. Moreover, the adaptivity within the individual component functions increases sparsity since grid points are added only where they are most needed, that is, in regions with steep gradients or at nondifferentiabilities. Second, we introduce a performant vectorization scheme for the interpolation compute kernel. Third, the algorithm is hybrid parallelized, leveraging both distributed- and shared-memory architectures. We observe significant speedups over the state-of-the-art techniques, and almost ideal strong scaling up to at least 1,000 compute nodes of a Cray XC50 system at the Swiss National Supercomputing Center. Finally, to demonstrate our method's broad applicability, we compute global solutions to two variates of a high-dimensional international real business cycle model up to 300 continuous state variables. In addition, we highlight a complementary advantage of the framework, which allows for a priori analysis of the model complexity.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-13T07:00:00Z
DOI: 10.1137/21M1392231
Issue No: Vol. 44, No. 3 (2022)

• Asynchronous Multiplicative Coarse-Space Correction

Authors: Guillaume Gbikpi-Benissan, Frédéric Magoulès
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page C237-C259, June 2022.
This paper introduces the multiplicative variant of the recently proposed asynchronous additive coarse-space correction method. The definition of an asynchronous extension of multiplicative correction is not straightforward; however, our analysis allows for usual asynchronous programming approaches. General asynchronous iterative models are explicitly devised both for shared or replicated coarse problems and for centralized or distributed ones. Convergence conditions are derived and shown to be satisfied for M-matrices, as also done for the additive case. Implementation aspects are discussed, which reveal the need for nonblocking synchronization for building the successive right-hand side vectors of the coarse problem. Optionally, a parameter allows for applying each coarse solution a maximum number of times, which has an impact on the algorithm efficiency. Numerical results on a high-speed homogeneous cluster confirm the practical efficiency of the asynchronous two-level method over its synchronous counterpart, even when it is not the case for the underlying one-level methods.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-13T07:00:00Z
DOI: 10.1137/21M1432107
Issue No: Vol. 44, No. 3 (2022)

• Parameter-Robust Preconditioning for Oseen Iteration Applied to Stationary
and Instationary Navier--Stokes Control

Authors: Santolo Leveque, John W. Pearson
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B694-B722, June 2022.
We derive novel, fast, and effective preconditioned iterative methods for steady and time-dependent Navier--Stokes control problems. Our approach may be applied to time-dependent problems which are discretized using backward Euler or Crank--Nicolson and is also a valuable candidate for Stokes control problems discretized using Crank--Nicolson. The key ingredients of the solver are a saddle-point type approximation for the linear systems, an inner iteration for the $(1,1)$-block accelerated by a preconditioner for convection-diffusion control, and an approximation to the Schur complement based on a potent commutator argument applied to an appropriate block matrix. Through a range of numerical experiments, we show the effectiveness of our approximations and observe their considerable parameter-robustness.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-07T07:00:00Z
DOI: 10.1137/21M1436531
Issue No: Vol. 44, No. 3 (2022)

• An Orthogonalization-Free Parallelizable Framework for All-Electron
Calculations in Density Functional Theory

Authors: Bin Gao, Guanghui Hu, Yang Kuang, Xin Liu
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B723-B745, June 2022.
All-electron calculations play an important role in density functional theory, in which improving computational efficiency is one of the most needed and challenging tasks. In the model formulations, both the nonlinear eigenvalue problem and the total energy minimization problem pursue orthogonal solutions. Most existing algorithms for solving these two models invoke orthogonalization process either explicitly or implicitly in each iteration. Their efficiency suffers from this process in view of its cubic complexity and low parallel scalability in terms of the number of electrons for large scale systems. To break through this bottleneck, we propose an orthogonalization-free algorithm framework based on the total energy minimization problem. It is shown that the desired orthogonality can be gradually achieved without invoking orthogonalization in each iteration. Moreover, this framework fully consists of BLAS operations and thus can be naturally parallelized. The global convergence of the proposed algorithm is established. We also present a preconditioning technique which can dramatically accelerate the convergence of the algorithm. The numerical experiments on all-electron calculations show the effectiveness and high scalability of the proposed algorithm.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-07T07:00:00Z
DOI: 10.1137/20M1355884
Issue No: Vol. 44, No. 3 (2022)

• Randomized Gram--Schmidt Process with Application to GMRES

Authors: Oleg Balabanov, Laura Grigori
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1450-A1474, June 2022.
A randomized Gram--Schmidt algorithm is developed for orthonormalization of high-dimensional vectors or QR factorization. The proposed process can be less computationally expensive than the classical Gram--Schmidt process while being at least as numerically stable as the modified Gram--Schmidt process. Our approach is based on random sketching, which is a dimension reduction technique consisting in estimation of inner products of high-dimensional vectors by inner products of their small efficiently computable random images, so-called sketches. In this way, an approximate orthogonality of the full vectors can be obtained by orthogonalization of their sketches. The proposed Gram--Schmidt algorithm can provide computational cost reduction in any architecture. The benefit of random sketching can be amplified by performing the nondominant operations in higher precision. In this case the numerical stability can be guaranteed with a working unit roundoff independent of the dimension of the problem. The proposed Gram--Schmidt process can be applied to Arnoldi iteration and results in new Krylov subspace methods for solving high-dimensional systems of equations or eigenvalue problems. Among them we chose the randomized GMRES method as a practical application of the methodology.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-06T07:00:00Z
DOI: 10.1137/20M138870X
Issue No: Vol. 44, No. 3 (2022)

• Gaussian Process Subspace Prediction for Model Reduction

Authors: Ruda Zhang, Simon Mak, David Dunson
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1428-A1449, June 2022.
Subspace-valued functions arise in a wide range of problems, including parametric reduced order modeling (PROM), parameter reduction, and subspace tracking. In PROM, each parameter point can be associated with a subspace, which is used for Petrov--Galerkin projections of large system matrices. Previous efforts to approximate such functions use interpolations on manifolds, which can be inaccurate and slow. To tackle this, we propose a novel Bayesian nonparametric model for subspace prediction: the Gaussian process subspace (GPS) model. This method is extrinsic and intrinsic at the same time: with multivariate Gaussian distributions on the Euclidean space, it induces a joint probability model on the Grassmann manifold, the set of fixed-dimensional subspaces. The GPS adopts a simple yet general correlation structure, and a principled approach for model selection. Its predictive distribution admits an analytical form, which allows for efficient subspace prediction over the parameter space. For PROM, the GPS provides a probabilistic prediction at a new parameter point that retains the accuracy of local reduced models, at a computational complexity that does not depend on system dimension, and thus is suitable for online computation. We give four numerical examples to compare our method to subspace interpolation, as well as two methods that interpolate local reduced models. Overall, GPS is the most data efficient, more computationally efficient than subspace interpolation, and gives smooth predictions with uncertainty quantification.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-02T07:00:00Z
DOI: 10.1137/21M1432739
Issue No: Vol. 44, No. 3 (2022)

• A Fast Time-Stepping Strategy for Dynamical Systems Equipped with a
Surrogate Model

Authors: Steven Roberts, Andrey A. Popov, Arash Sarshar, Adrian Sandu
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1405-A1427, June 2022.
Simulation of complex dynamical systems arising in many applications is computationally challenging due to their size and complexity. Model order reduction, machine learning, and other types of surrogate modeling techniques offer cheaper and simpler ways to describe the dynamics of these systems but are inexact and introduce additional approximation errors. In order to overcome the computational difficulties of the full complex models, on one hand, and the limitations of surrogate models, on the other, this work proposes a new accelerated time-stepping strategy that combines information from both. This approach is based on the multirate infinitesimal general-structure additive Runge--Kutta framework. The inexpensive surrogate model is integrated with a small time step to guide the solution trajectory, and the full model is treated with a large time step to occasionally correct for the surrogate model error and ensure convergence. We provide a theoretical error analysis, and several numerical experiments, to show that this approach can be significantly more efficient than using only the full or only the surrogate model for the integration.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-01T07:00:00Z
DOI: 10.1137/20M1386281
Issue No: Vol. 44, No. 3 (2022)

• PDE-Aware Deep Learning for Inverse Problems in Cardiac Electrophysiology

Authors: Riccardo Tenderini, Stefano Pagani, Alfio Quarteroni, Simone Deparis
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B605-B639, June 2022.
In this work, we present a PDE-aware deep learning model for the numerical solution to the inverse problem of electrocardiography. The model both leverages data availability and exploits the knowledge of a physically based mathematical model, expressed by means of partial differential equations (PDEs), to carry out the task at hand. The goal is to estimate the epicardial potential field from measurements of the electric potential at a discrete set of points on the body surface. The employment of deep learning techniques in this context is made difficult by the low amount of clinical data at disposal, as measuring cardiac potentials requires invasive procedures. Suitably exploiting the underlying physically based mathematical model allowed circumventing the data availability issue and led to the development of fast-training and low-complexity models. Physical awareness has been pursued by means of two elements: the projection of the epicardial potential onto a space-time reduced subspace, spanned by the numerical solutions of the governing PDEs, and the inclusion of a tensorial reduced basis solver of the forward problem in the network architecture. Numerical tests have been conducted only on synthetic data, obtained via a full order model approximation of the problem at hand, and two variants of the model have been addressed. Both proved to be accurate, up to an average $\ell^1$-norm relative error on epicardial activation maps of 3.5%, and both could be trained in $\approx$$15$ min. Nevertheless, some improvements, mostly concerning data generation, are necessary in order to bridge the gap with clinical applications.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-01T07:00:00Z
DOI: 10.1137/21M1438529
Issue No: Vol. 44, No. 3 (2022)

• A Well-Conditioned Weak Coupling of Boundary Element and High-Order Finite
Element Methods for Time-Harmonic Electromagnetic Scattering by
Inhomogeneous Objects

Authors: Ismaïl Badia, Boris Caudron, Xavier Antoine, Christophe Geuzaine
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B640-B667, June 2022.
The aim of this paper is to propose efficient weak coupling formulations between the boundary element method and the high-order finite element method for solving time-harmonic electromagnetic scattering problems. The approach is based on the use of a nonoverlapping domain decomposition method involving optimal transmission operators. The associated transmission conditions are constructed through a localization process based on complex rational Padé approximants of the nonlocal magnetic-to-electric operators. Numerical results are presented to validate and analyze the new approach for both homogeneous and inhomogeneous scatterers.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-01T07:00:00Z
DOI: 10.1137/21M1438293
Issue No: Vol. 44, No. 3 (2022)

• Bridging and Improving Theoretical and Computational Electrical Impedance
Tomography via Data Completion

Authors: Tan Bui-Thanh, Qin Li, Leonardo Zepeda-Nún͂ez
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B668-B693, June 2022.
In computational PDE-based inverse problems, a finite amount of data is collected to infer unknown parameters in the PDE. In order to obtain accurate inferences, the collected data must be informative about the unknown parameters. How to decide which data is most informative and how to efficiently sample it is the notoriously challenging task of optimal experimental design (OED). In this context, the best, and often infeasible, scenario is when the full input-to-output (ItO) map, i.e., an infinite amount of data, is available: This is the typical setting in many theoretical inverse problems, which is used to guarantee the unique parameter reconstruction. These two different settings have created a gap between computational and theoretical inverse problems, where finite and infinite amounts of data are used, respectively. In this article we aim to bridge this gap while circumventing the OED task. This is achieved by exploiting the structures of the ItO data from the underlying inverse problem, using the electrical impedance tomography (EIT) problem as an example. To accomplish our goal, we leverage the rank structure of the EIT model and formulate the ItO matrix\textemdash the discretized ItO map\textemdash as an $\mathcal{H}$-matrix whose off-diagonal blocks are low rank. This suggests that, when equipped with the matrix completion technique, one can recover the full ItO matrix, with high probability, from a subset of its entries sampled following the rank structure: The data in the diagonal blocks is informative and should be fully sampled, while data in the off-diagonal blocks can be subsampled. This recovered ItO matrix is then utilized to present the full ItO map up to a discretization error, paving the way to connect with the problem in the theoretical setting where the unique reconstruction of parameters is guaranteed. This strategy achieves two goals: (I) it bridges the gap between the finite- and infinite-dimensional settings for numerical and theoretical inverse problems and (II) it improves the quality of computational inverse solutions. We detail the theory for the EIT model and provide numerical verification to both EIT and optical tomography problems.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-01T07:00:00Z
DOI: 10.1137/21M141703X
Issue No: Vol. 44, No. 3 (2022)

• Mixed Methods for the Velocity-Pressure-Pseudostress Formulation of the
Stokes Eigenvalue Problem

Authors: Felipe Lepe, Gonzalo Rivera, Jesus Vellojin
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1358-A1380, June 2022.
In two and three dimensions, we analyze mixed finite element methods for a velocity-pressure-pseudostress formulation of the Stokes eigenvalue problem. The methods consist of two schemes: the velocity and pressure are approximated with piecewise polynomial, whereas for the pseudostress we consider two classic families of finite elements for ${H}(\div)$ spaces: the Raviart--Thomas and the Brezzi--Douglas--Marini elements. With the aid of the classic spectral theory for compact operators, we prove that our method does not introduce spurious modes. Also, we obtain convergence and error estimates for the proposed methods. We report numerical results to compare the accuracy and robustness between both numerical schemes.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-31T07:00:00Z
DOI: 10.1137/21M1402959
Issue No: Vol. 44, No. 3 (2022)

• High-Order Close Evaluation of Laplace Layer Potentials: A Differential
Geometric Approach

Authors: Hai Zhu, Shravan Veerapaneni
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1381-A1404, June 2022.
This paper presents a new approach for solving the close evaluation problem in three dimensions, commonly encountered while solving linear elliptic partial differential equations via potential theory. The goal is to evaluate layer potentials close to the boundary over which they are defined. The approach introduced here converts these nearly singular integrals on a patch of the boundary to a set of nonsingular line integrals on the patch boundary using the Stokes theorem on manifolds. A function approximation scheme based on harmonic polynomials is designed to express the integrand in a form that is suitable for applying the Stokes theorem. As long as the data---the boundary and the density function---is given in a high-order format, the double-layer potential and its derivatives can be evaluated with high-order accuracy using this scheme both on and off the boundary. In particular, we present numerical results demonstrating seventh-order convergence on a smooth, warped torus example achieving 10-digit accuracy in evaluating double-layer potential at targets that are arbitrarily close to the boundary.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-31T07:00:00Z
DOI: 10.1137/21M1423051
Issue No: Vol. 44, No. 3 (2022)

• Conditional A Posteriori Error Bounds for High Order Discontinuous
Galerkin Time Stepping Approximations of Semilinear Heat Models with
Blow-up

Authors: Stephen Metcalfe, Thomas P. Wihler
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1337-A1357, June 2022.
This work is concerned with the development of an adaptive numerical method for semilinear heat flow models featuring a general (possibly) nonlinear reaction term that may cause the solution to blow up in finite time. The fully discrete scheme consists of a high order discontinuous Galerkin time stepping method and a conforming finite element discretization in space. The proposed adaptive procedure is based on rigorously devised conditional a posteriori error bounds in the ${L}^\infty({L}^\infty)$ norm. Numerical experiments complement the theoretical results; in particular, we investigate whether exponential convergence to the blow-up time can be achieved via $hp$-adaptivity.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-24T07:00:00Z
DOI: 10.1137/21M1418964
Issue No: Vol. 44, No. 3 (2022)

• Sequential Active Learning of Low-Dimensional Model Representations for
Reliability Analysis

Authors: Max Ehre, Iason Papaioannou, Bruno Sudret, Daniel Straub
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B558-B584, June 2022.
To date, the analysis of high-dimensional, computationally expensive engineering models remains a difficult challenge in risk and reliability engineering. We use a combination of dimensionality reduction and surrogate modeling termed partial least squares--driven polynomial chaos expansion (PLS-PCE) to render such problems feasible. Standalone surrogate models typically perform poorly for reliability analysis. Therefore, in a previous work, we have used PLS-PCEs to reconstruct the intermediate densities of a sequential importance sampling approach to reliability analysis. Here, we extend this approach with an active learning procedure that allows for improved error control at each importance sampling level. To this end, we formulate an estimate of the combined estimation error for both the subspace identified in the dimension reduction step and the surrogate model constructed therein. With this, it is possible to adapt the training set so as to optimally learn the subspace representation and the surrogate model constructed therein. The approach is gradient-free and thus can be directly applied to black box--type models. We demonstrate the performance of this approach with a series of low- (2 dimensions) to high- (869 dimensions) dimensional example problems featuring a number of well-known caveats for reliability methods besides high dimensions and expensive computational models: strongly nonlinear limit-state functions, multiple relevant failure regions, and small probabilities of failure.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-24T07:00:00Z
DOI: 10.1137/21M1416758
Issue No: Vol. 44, No. 3 (2022)

• Compact Exponential Conservative Approaches for the Schrödinger Equation
in the Semiclassical Regimes

Authors: Jiaxiang Cai, Hua Liang
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B585-B604, June 2022.
In this paper we propose several efficient conservative prediction-correction methods for studying the Schrödinger equation in the semiclassical regimes with small parameter $\varepsilon$. In the prediction step, the linear part of the equation is integrated exactly, which allows the $\varepsilon$-oscillatory solution to be captured effectively, while only a local nonlinear system needs to be solved at each spatial grid point. Besides, the compact representation saves the storage requirement and computational cost. In the correction step, the prediction solution is modified to admit the mass/energy conservation law or both of them by adding supplementary variables to the original equation. This procedure is inexpensive since only algebraic nonlinear equations need to be solved. Extensive numerical tests suggest the meshing stategies for obtaining correct physical quantities: $\tau=\mathcal{O}(\varepsilon)$ and $h=\mathcal{O}(\varepsilon)$ for the equation with defocusing nonlinearities or weak $\mathcal{O}(\varepsilon)$--focusing/defocusing nonlinearities, and $\tau$ independent of $\varepsilon$ and $h=\mathcal{O}(\varepsilon)$ for the linear equation. Some numerical experiments for the equation in two and three dimensions are also carried out to demonstrate the power of the present methods in simulation of Bose--Einstein condensation.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-24T07:00:00Z
DOI: 10.1137/21M1439122
Issue No: Vol. 44, No. 3 (2022)

• Data-Driven Algorithms for Signal Processing with Trigonometric Rational
Functions

Authors: Heather Wilber, Anil Damle, Alex Townsend
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page C185-C209, June 2022.
Rational approximation schemes for reconstructing periodic signals from samples with poorly separated spectral content are described. These methods are automatic and adaptive, requiring no tuning or manual parameter selection. Collectively, they form a framework for fitting trigonometric rational models to data that is robust to various forms of corruption, including additive Gaussian noise, perturbed sampling grids, and missing data. Our approach combines a variant of Prony's method with a modified version of the adaptive Antoulas--Anderson algorithm. Using representations in both frequency and time space, a collection of algorithms is described for adaptively computing with trigonometric rationals. This includes procedures for differentiation, filtering, convolution, and more. A new MATLAB software system based on these algorithms is introduced. Its effectiveness is illustrated with synthetic and practical examples drawn from applications including biomedical monitoring, acoustic denoising, and feature detection.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-24T07:00:00Z
DOI: 10.1137/21M1420277
Issue No: Vol. 44, No. 3 (2022)

• Linearization of the Travel Time Functional in Porous Media Flows

Authors: Paul Houston, Connor J. Rourke, Kristoffer G. van der Zee
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B531-B557, June 2022.
The travel time functional measures the time taken for a particle trajectory to travel from a given initial position to the boundary of the domain. Such evaluation is paramount in the postclosure safety assessment of deep geological storage facilities for radioactive waste where leaked, nonsorbing solutes can be transported to the surface of the site by the surrounding groundwater. The accurate simulation of this transport can be attained using standard dual-weighted-residual techniques to derive goal-oriented a posteriori error bounds. This work provides a key aspect in obtaining a suitable error estimate for the travel time functional: the evaluation of its Gâteaux derivative. A mixed finite element method is implemented to approximate Darcy's equations, and numerical experiments are presented to test the performance of the proposed error estimator. In particular, we consider a test case inspired by the Sellafield site located in Cumbria, UK.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-23T07:00:00Z
DOI: 10.1137/21M1451105
Issue No: Vol. 44, No. 3 (2022)

• A Framework for Error-Bounded Approximate Computing, with an Application
to Dot Products

Authors: James Diffenderfer, Daniel Osei-Kuffuor, Harshitha Menon
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1290-A1314, June 2022.
Approximate computing techniques, which trade off the computation accuracy of an algorithm for better performance and energy efficiency, have been successful in reducing computation and power costs in several domains. However, error sensitive applications in high-performance computing are unable to benefit from existing approximate computing strategies that are not developed with guaranteed error bounds. While approximate computing techniques can be developed for individual high-performance computing applications by domain specialists, this often requires additional theoretical analysis and potentially extensive software modification. Hence, the development of low-level error-bounded approximate computing strategies that can be introduced into any high-performance computing application without requiring additional analysis or significant software alterations is desirable. In this paper, we provide a contribution in this direction by proposing a general framework for designing error-bounded approximate computing strategies and apply it to the dot product kernel to develop \bf qdot---an error-bounded approximate dot product kernel. Following the introduction of qdot, we perform a theoretical analysis that yields a deterministic bound on the relative approximation error introduced by qdot. Empirical tests are performed to illustrate the tightness of the derived error bound and to demonstrate the effectiveness of qdot on a synthetic dataset, as well as two scientific benchmarks---the conjugate gradient (CG) and power methods. In some instances, using qdot for the dot products in CG can result in many components being quantized to half precision without increasing the iteration count required for convergence to the same solution as CG using a double precision dot product.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-19T07:00:00Z
DOI: 10.1137/21M1406994
Issue No: Vol. 44, No. 3 (2022)

• Recovering Wavelet Coefficients from Binary Samples Using Fast Transforms

Authors: Vegard Antun
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1315-A1336, June 2022.
Recovering a signal (function) from finitely many binary or Fourier samples is one of the core problems in modern imaging, and by now there exist a plethora of methods for recovering a signal from such samples. Examples of methods which can utilize wavelet reconstruction include generalized sampling, infinite-dimensional compressive sensing, the parameterized-background data-weak (PBDW) method, etc. However, for any of these methods to be applied in practice, accurate and fast modeling of an $N \times M$ section of the infinite-dimensional change-of-basis matrix between the sampling basis (Fourier or Walsh--Hadamard samples) and the wavelet reconstruction basis is paramount. Building on the work of Gataric and Poon [SIAM J. Sci. Comput., 38 (2016), pp. A1075--A1099], we derive an algorithm which bypasses the $NM$ storage requirement and the $\mathcal{O}(NM)$ computational cost of matrix-vector multiplication with this matrix and its adjoint when using Walsh--Hadamard samples and wavelet reconstruction. The proposed algorithm computes the matrix-vector multiplication in $\mathcal{O}(N\log N)$ operations and has a storage requirement of $\mathcal{O}(2^q)$, where $N=2^{dq} M$ (usually $q \in \{1,2\}$) and $d=1,2$ is the dimension. As matrix-vector multiplications are the computational bottleneck for iterative algorithms used by the mentioned reconstruction methods, the proposed algorithm speeds up the reconstruction of wavelet coefficients from Walsh--Hadamard samples considerably.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-19T07:00:00Z
DOI: 10.1137/21M1427188
Issue No: Vol. 44, No. 3 (2022)

• Edge-Promoting Adaptive Bayesian Experimental Design for X-ray Imaging

Authors: Tapio Helin, Nuutti Hyvönen, Juha-Pekka Puska
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B506-B530, June 2022.
This work considers sequential edge-promoting Bayesian experimental design for (discretized) linear inverse problems, exemplified by X-ray tomography. The process of computing a total variation--type reconstruction of the absorption inside the imaged body via lagged diffusivity iteration is interpreted in the Bayesian framework. Assuming a Gaussian additive noise model, this leads to an approximate Gaussian posterior with a covariance structure that contains information on the location of edges in the posterior mean. The next projection geometry is then chosen through A- or D-optimal Bayesian design, which corresponds to minimizing the trace or the determinant of the updated posterior covariance matrix that accounts for the new projection. Two- and three-dimensional numerical examples based on simulated data demonstrate the functionality of the introduced approach.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-19T07:00:00Z
DOI: 10.1137/21M1409330
Issue No: Vol. 44, No. 3 (2022)

• Explicit Time Stepping for the Wave Equation using CutFEM with Discrete
Extension

Authors: Erik Burman, Peter Hansbo, Mats G. Larson
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1254-A1289, June 2022.
In this paper we develop a fully explicit cut finite element method for the wave equation. The method is based on using a standard leap frog scheme combined with an extension operator that defines the nodal values outside of the domain in terms of the nodal values inside the domain. We show that the mass matrix associated with the extended finite element space can be lumped leading to a fully explicit scheme. We derive stability estimates for the method and provide optimal order a priori error estimates. Finally, we present some illustrating numerical examples.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-12T07:00:00Z
DOI: 10.1137/20M137937X
Issue No: Vol. 44, No. 3 (2022)

• A Posteriori Estimator for the Adaptive Solution of a Quasi-Static
Fracture Phase-Field Model with Irreversibility Constraints

Authors: Mirjam Walloth, Winnifried Wollner
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B479-B505, June 2022.
Within this article, we develop a residual-type a posteriori error estimator for a time discrete quasi-static phase-field fracture model. Particular emphasis is given to the robustness of the error estimator for the variational inequality governing the phase-field evolution with respect to the phase-field regularization parameter $\epsilon$. The article concludes with numerical examples highlighting the performance of the proposed a posteriori error estimators on three standard test cases: the single edge notched tension and shear test as well as the L-shaped panel test.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-12T07:00:00Z
DOI: 10.1137/21M1427437
Issue No: Vol. 44, No. 3 (2022)

• Doubly Monotonic Constraint on Interpolators: Bridging Second-Order to
Singularity Preservation to Cancel “Numerical Wetting” in Transport
Schemes

Authors: Christina Paulin, Antoine Llor, Thibaud Vazquez-Gonzalez, Jean-Philippe Perlat, Éric Heulhard de Montigny
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1227-A1253, June 2022.
Monotonic interpolation and its avatars are major ingredients of many numerical schemes for solving partial differential equations (PDEs) under total variation diminishing (TVD) or similar constraints. However, despite over forty years of extensive study of principles and applications, a key aspect of monotonic interpolant design can still appear somewhat empirical: how does a monotonic interpolator connect the limiting cases of smooth (differentiable) and singular (limited) functions in a consistent and possibly canonical way' The present study aims at providing understanding in the basic but important case of per-cell monotonic one-dimensional scalar reconstruction and at applying it to second-order accurate transport. First, a general mapping of bounded monotonic functions in elliptic coordinates is built. Then, the usual “single-slope” second-order monotonic interpolants are continued into “slope-and-bound” monotonic interpolants. Finally, a critical constraint is introduced, the “double monotonicity,” in order to build various slope-and-bound monotonic interpolators from this set of interpolants. With these slope-and-bound interpolators, standard numerical tests show a complete cancellation of the “numerical wetting” that usual TVD transport schemes produce. When transporting scalar fields of compact support, this effect---not to be confused with usual numerical diffusion---is the low-level contamination that spreads linearly in time over all the regions of the computational domain where nonvanishing transport is present. Removal of numerical wetting is of particular importance in many industrial and academic applications, notably at “phase disappearance” episodes in multiphase flows or “wet-dry” transitions in shallow water flows. Improvement of the “numerical erosion” of extrema is also observed. The general principles exposed here can be extended to multidimensional settings, high-order schemes, and other PDEs.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-10T07:00:00Z
DOI: 10.1137/21M140314X
Issue No: Vol. 44, No. 3 (2022)

• Adaptive GDSW Coarse Spaces of Reduced Dimension for Overlapping Schwarz
Methods

Authors: Alexander Heinlein, Axel Klawonn, Jascha Knepper, Oliver Rheinbach, Olof B. Widlund
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1176-A1204, June 2022.
A new reduced-dimension adaptive generalized Dryja--Smith--Widlund (GDSW) overlapping Schwarz method for linear second-order elliptic problems in three dimensions is introduced. It is robust with respect to large contrasts of the coefficients of the partial differential equations. The condition number bound of the new method is shown to be independent of the coefficient contrast and only dependent on a user-prescribed tolerance. The interface of the nonoverlapping domain decomposition is partitioned into nonoverlapping patches. The new coarse space is obtained by selecting a few eigenvectors of certain local eigenproblems which are defined on these patches. These eigenmodes are energy-minimally extended to the interior of the nonoverlapping subdomains and added to the coarse space. By using a new interface decomposition, the reduced-dimension adaptive GDSW overlapping Schwarz method usually has a smaller coarse space than existing GDSW and adaptive GDSW domain decomposition methods. A robust condition number estimate is proven for the new reduced-dimension adaptive GDSW method which is also valid for existing adaptive GDSW methods. Numerical results for the equations of isotropic linear elasticity in three dimensions confirming the theoretical findings are presented.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-09T07:00:00Z
DOI: 10.1137/20M1364540
Issue No: Vol. 44, No. 3 (2022)

• Lightning Stokes Solver

Authors: Pablo D. Brubeck, Lloyd N. Trefethen
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1205-A1226, June 2022.
Gopal and Trefethen recently introduced “lightning solvers” for the 2D Laplace and Helmholtz equations, based on rational functions with poles exponentially clustered near singular corners. Making use of the Goursat representation in terms of analytic functions, we apply these methods to the biharmonic equation, specifically to 2D Stokes flow. Solutions to model problems are computed to 10-digit accuracy in less than a second of laptop time. As an illustration of the high accuracy, we resolve two or more counterrotating Moffatt eddies near a singular corner.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-09T07:00:00Z
DOI: 10.1137/21M1408579
Issue No: Vol. 44, No. 3 (2022)

• Improving “Fast Iterative Shrinkage-Thresholding Algorithm”: Faster,
Smarter, and Greedier

Authors: Jingwei Liang, Tao Luo, Carola-Bibiane Schönlieb
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1069-A1091, June 2022.
The “fast iterative shrinkage-thresholding algorithm,” a.k.a. FISTA, is one of the most well known first-order optimization scheme in the literature, as it achieves the worst-case $O(1/k^2)$ optimal convergence rate for objective function value. However, despite such an optimal theoretical convergence rate, in practice the (local) oscillatory behavior of FISTA often damps its efficiency. Over the past years, various efforts have been made in the literature to improve the practical performance of FISTA, such as monotone FISTA, restarting FISTA, and backtracking strategies. In this paper, we propose a simple yet effective modification to the original FISTA scheme which has two advantages: It allows us to (1) prove the convergence of generated sequence and (2) design a so-called lazy-start strategy which can be up to an order faster than the original scheme. Moreover, we propose novel adaptive and greedy strategies which further improve the practical performance. The advantages of the proposed schemes are tested through problems arising from inverse problems, machine learning, and signal/image processing.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-05T07:00:00Z
DOI: 10.1137/21M1395685
Issue No: Vol. 44, No. 3 (2022)

• Energy-Preserving Continuous-Stage Exponential Runge--Kutta Integrators
for Efficiently Solving Hamiltonian Systems

Authors: Lijie Mei, Li Huang, Xinyuan Wu
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1092-A1115, June 2022.
As one of the most important properties, energy preservation is a natural requirement for numerical integrators of Hamiltonian systems. Considering the limited second-order accuracy of the existing exponential average vector filed (or discrete gradient) method, this paper is devoted to developing high-order energy-preserving exponential integrators for general semilinear Hamiltonian systems. To this end, we first formulate and analyze the continuous-stage exponential Runge--Kutta (ERK) method. After deriving the order conditions, energy-preserving conditions, and symmetric conditions for the continuous-stage ERK method, we construct a novel class of fourth-order symmetric energy-preserving continuous-stage ERK methods with a free parameter based on the established conditions and present a convergence theorem for the fixed-point iteration associated with the continuous-stage ERK method. Furthermore, we extend the proposed continuous-stage ERK method to noncanonical Hamiltonian systems and discuss implementation issues in detail. Finally, numerical results from the FPU problem, the charged-particle dynamics, and the Klein--Gordon equation demonstrate the high efficiency, good energy-preserving behavior, and applicability of large time stepsizes of the proposed fourth-order energy-preserving continuous-stage ERK method.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-05T07:00:00Z
DOI: 10.1137/21M1412475
Issue No: Vol. 44, No. 3 (2022)

• Alternating Linear Scheme in a Bayesian Framework for Low-Rank Tensor
Approximation

Authors: Clara Menzen, Manon Kok, Kim Batselier
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1116-A1144, June 2022.
Multiway data often naturally occurs in a tensorial format which can be approximately represented by a low-rank tensor decomposition. This is useful because complexity can be significantly reduced and the treatment of large-scale data sets can be facilitated. In this paper, we find a low-rank representation for a given tensor by solving a Bayesian inference problem. This is achieved by dividing the overall inference problem into subproblems where we sequentially infer the posterior distribution of one tensor decomposition component at a time. This leads to a probabilistic interpretation of the well-known iterative algorithm alternating linear scheme (ALS). In this way, the consideration of measurement noise is enabled, as well as the incorporation of application-specific prior knowledge and the uncertainty quantification of the low-rank tensor estimate. To compute the low-rank tensor estimate from the posterior distributions of the tensor decomposition components, we present an algorithm that performs the unscented transform in tensor train format.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-05T07:00:00Z
DOI: 10.1137/20M1386414
Issue No: Vol. 44, No. 3 (2022)

• Robust Identification of Differential Equations by Numerical Techniques
from a Single Set of Noisy Observation

Authors: Yuchen He, Sung-Ha Kang, Wenjing Liao, Hao Liu, Yingjie Liu
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1145-A1175, June 2022.
We propose robust methods to identify the underlying Partial Differential Equation (PDE) from a given single set of noisy time-dependent data. We assume that the governing equation of the PDE is a linear combination of a few linear and nonlinear differential terms in a prescribed dictionary. Noisy data make such identification particularly challenging. Our objective is to develop robust methods against a high level of noise and approximate the underlying noise-free dynamics well. We first introduce a Successively Denoised Differentiation (SDD) scheme to stabilize the amplified noise in numerical differentiation. SDD effectively denoises the given data and the corresponding derivatives. Second, we present two algorithms for PDE identification: Subspace pursuit Time evolution (ST) error and Subspace pursuit Cross-validation (SC). Our general strategy is to first find a candidate set using the Subspace Pursuit (SP) greedy algorithm, then choose the best one via time evolution or cross-validation. ST uses a multishooting numerical time evolution and selects the PDE which yields the least evolution error. SC evaluates the cross-validation error in the least-squares fitting and picks the PDE that gives the smallest validation error. We present various numerical experiments to validate our methods. Both methods are efficient and robust to noise.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-05T07:00:00Z
DOI: 10.1137/20M134513X
Issue No: Vol. 44, No. 3 (2022)

• A Robust Algebraic Domain Decomposition Preconditioner for Sparse Normal
Equations

Authors: Hussam Al Daas, Pierre Jolivet, Jennifer A. Scott
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1047-A1068, June 2022.
Solving the normal equations corresponding to large sparse linear least-squares problems is an important and challenging problem. For very large problems, an iterative solver is needed and, in general, a preconditioner is required to achieve good convergence. In recent years, a number of preconditioners have been proposed. These are largely serial, and reported results demonstrate that none of the commonly used preconditioners for the normal equations matrix is capable of solving all sparse least-squares problems. Our interest is thus in designing new preconditioners for the normal equations that are efficient and robust and can be implemented in parallel. Our proposed preconditioners can be constructed efficiently and algebraically without any knowledge of the problem and without any assumption on the least-squares matrix except that it is sparse. We exploit the structure of the symmetric positive definite normal equations matrix and use the concept of algebraic local symmetric positive semidefinite splittings to introduce two-level Schwarz preconditioners for least-squares problems. The condition number of the preconditioned normal equations matrix is shown to be theoretically bounded independently of the number of subdomains in the splitting. This upper bound can be adjusted using a single parameter $\tau$ that the user can specify. We discuss how the new preconditioners can be implemented on top of the PETSc library using only 150 lines of Fortran, C, or Python code. Problems arising from practical applications are used to compare the performance of the proposed new preconditioner with that of other preconditioners.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-03T07:00:00Z
DOI: 10.1137/21M1434891
Issue No: Vol. 44, No. 3 (2022)

• Polynomial Chaos Expansions for Stiff Random ODEs

Authors: Wenjie Shi, Daniel M. Tartakovsky
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page A1021-A1046, June 2022.
Generalized polynomial chaos (gPC), often combined with mono-implicit Runge--Kutta (MIRK) methods, is widely used to solve random ODEs. We investigate the impact of stiffness of random ODEs on the gPC performance. We start by extending pragmatic definitions of stiffness used in deterministic ODEs to their random counterparts. Then we introduce gPC with parallel MIRK schemes to solve random stiff ODEs, in which a suitable parallelism partially alleviates the curse of dimensionality. Our stiffness analysis comprises two parts: (i) the relationship between Jacobians of random ODEs and the corresponding gPC equations and (ii) stiffness of the gPC equations. It provides a direct way to determine whether a random ODE and/or the corresponding gPC equations are stiff. This theoretical analysis plays a key role in designing numerical implementations not only of gPC but also of other methods of stochastic computation, e.g., Monte Carlo simulations and stochastic collocation. A series of computational experiments is used to demonstrate the agreement between our theoretical analysis and numerical results and to establish gPC with parallel MIRK as a feasible and effective tool for solving stiff random ODEs.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-02T07:00:00Z
DOI: 10.1137/21M1432545
Issue No: Vol. 44, No. 3 (2022)

• The Sampling Method for Inverse Exterior Stokes Problems

Authors: Meng Liu, Jiaqing Yang
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B429-B456, June 2022.
This paper is concerned with an inverse problem to recover an impenetrable solid in an unbounded stationary flow which is modeled by the Stokes equation with a Dirichlet or Robin boundary condition. We propose three simple imaging techniques for detecting the shape and location of the solid by extending the well-known linear sampling method, factorization method, and generalized linear sampling method for wave equations, where only the velocity field measurements are taken on a closed surface around the solid. Several numerical examples are carried out to illustrate the effectiveness of the reduced inversion algorithm.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-02T07:00:00Z
DOI: 10.1137/21M1390591
Issue No: Vol. 44, No. 3 (2022)

• A Decoupling and Linearizing Discretization for Weakly Coupled
Poroelasticity with Nonlinear Permeability

Authors: Robert Altmann, Roland Maier
Abstract: SIAM Journal on Scientific Computing, Volume 44, Issue 3, Page B457-B478, June 2022.
We analyze a semiexplicit time discretization scheme of first order for poroelasticity with nonlinear permeability provided that the elasticity model and the flow equation are only weakly coupled. The approach leads to a decoupling of the equations and, at the same time, linearizes the nonlinearity without the need of further inner iteration steps. Hence, the computational speedup is twofold without a loss in the convergence rate. We prove optimal first-order error estimates by considering a related delay system and investigate the method numerically for different examples with various types of nonlinear displacement-permeability relations.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-05-02T07:00:00Z
DOI: 10.1137/21M1413985
Issue No: Vol. 44, No. 3 (2022)

• Monolithic Multigrid for a Reduced-Quadrature Discretization of
Poroelasticity

Authors: James H. Adler, Yunhui He, Xiaozhe Hu, Scott MacLachlan, Peter Ohm
Abstract: SIAM Journal on Scientific Computing, Ahead of Print.
Advanced finite-element discretizations and preconditioners for models of poroelasticity have attracted significant attention in recent years. The equations of poroelasticity offer significant challenges in both areas, due to the potentially strong coupling between unknowns in the system, saddle-point structure, and the need to account for wide ranges of parameter values, including limiting behavior such as incompressible elasticity. This paper was motivated by an attempt to develop monolithic multigrid preconditioners for the discretization developed in [C. Rodrigo et al., Comput. Methods App. Mech. Engrg, 341 (2018), pp. 467--484]; we show here why this is a difficult task and, as a result, we modify the discretization in [Rodrigo et al.] through the use of a reduced-quadrature approximation, yielding a more “solver-friendly” discretization. Local Fourier analysis is used to optimize parameters in the resulting monolithic multigrid method, allowing a fair comparison between the performance and costs of methods based on Vanka and Braess--Sarazin relaxation. Numerical results are presented to validate the local Fourier analysis predictions and demonstrate efficiency of the algorithms. Finally, a comparison to existing block-factorization preconditioners is also given.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-06-24T07:00:00Z
DOI: 10.1137/21M1429072

• Robust Multigrid Techniques for Augmented Lagrangian Preconditioning of
Incompressible Stokes Equations with Extreme Viscosity Variations

Authors: Yu-hsuan Shih, Georg Stadler, Florian Wechsung
Abstract: SIAM Journal on Scientific Computing, Ahead of Print.
We present augmented Lagrangian Schur complement preconditioners and robust multigrid methods for incompressible Stokes problems with extreme viscosity variations. Such Stokes systems arise, for instance, upon linearization of nonlinear viscous flow problems, and they can have severely inhomogeneous and anisotropic coefficients. Using an augmented Lagrangian formulation for the incompressibility constraint makes the Schur complement easier to approximate but results in a nearly singular (1,1)-block in the Stokes system. We present eigenvalue estimates for the quality of the Schur complement approximation. To cope with the near-singularity of the (1,1)-block, we extend a multigrid scheme with a discretization-dependent smoother and transfer operators from triangular/tetrahedral to the quadrilateral/hexahedral finite element discretizations $[\mathbb{Q}_k]^d\times \mathbb{P}_{k-1}^{\text{disc}}$, $k\geq 2$, $d=2,3$. Using numerical examples with scalar and with anisotropic fourth-order tensor viscosity arising from linearization of a viscoplastic constitutive relation, we confirm the robustness of the multigrid scheme and the overall efficiency of the solver. We present scalability results using up to 28,672 parallel tasks for problems with up to 1.6 billion unknowns and a viscosity contrast up to ten orders of magnitude.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-04-13T07:00:00Z
DOI: 10.1137/21M1430698

• Multilevel Spectral Domain Decomposition

Authors: Peter Bastian, Robert Scheichl, Linus Seelinger, Arne Strehlow
Abstract: SIAM Journal on Scientific Computing, Ahead of Print.
Highly heterogeneous, anisotropic coefficients, e.g., in the simulation of carbon-Fiber composite components, can lead to extremely challenging finite element systems. Direct solvers for the resulting large and sparse linear systems suffer from severe memory requirements and limited parallel scalability, while iterative solvers in general lack robustness. Two-level spectral domain decomposition methods can provide such robustness for symmetric positive definite linear systems by using coarse spaces based on independent generalized eigenproblems in the subdomains. Rigorous condition number bounds are independent of mesh size, number of subdomains, and coefficient contrast. However, their parallel scalability is still limited by the fact that (in order to guarantee robustness) the coarse problem is solved via a direct method. In this paper, we introduce a multilevel variant in the context of subspace correction methods and provide a general convergence theory for its robust convergence for abstract, elliptic variational problems. Assumptions of the theory are verified for conforming as well as for discontinuous Galerkin methods applied to a scalar diffusion problem. Numerical results illustrate the performance of the method for two- and three-dimensional problems and for various discretization schemes, in the context of scalar diffusion and linear elasticity.
Citation: SIAM Journal on Scientific Computing
PubDate: 2022-01-31T08:00:00Z
DOI: 10.1137/21M1427231

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