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SIAM Journal on Numerical Analysis
Journal Prestige (SJR): 2.657  Citation Impact (citeScore): 3 Number of Followers: 7  Hybrid journal (It can contain Open Access articles)ISSN (Print) 0036-1429 - ISSN (Online) 1095-7170 Published by Society for Industrial and Applied Mathematics  [17 journals]
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- Coupling Parareal with Optimized Schwarz Waveform Relaxation for Parabolic
Problems-
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Authors: Duc Quang Bui, Caroline Japhet, Yvon Maday, Pascal Omnes
Pages: 913 - 939
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 913-939, June 2022.
We propose and analyze a new parallel paradigm that uses both the time and the space directions. The original approach couples the Parareal algorithm with incomplete optimized Schwarz waveform relaxation (OSWR) iterations. The analysis of this coupled method is presented for a one-dimensional advection-reaction-diffusion equation. We also prove a general convergence result for this method via energy estimates. Numerical results for two-dimensional advection-diffusion problems and for a diffusion equation with strong heterogeneities are presented to illustrate the performance of the coupled Parareal-OSWR algorithm.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-05-03T07:00:00Z
DOI: 10.1137/21M1419428
Issue No: Vol. 60, No. 3 (2022)
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- An Optimal Mass Transport Method for Random Genetic Drift
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Authors: José A. Carrillo, Lin Chen, Qi Wang
Pages: 940 - 969
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 940-969, June 2022.
We propose and analyze an optimal mass transport method for a random genetic drift problem driven by a Moran process under weak selection. The continuum limit, formulated as a reaction-advection-diffusion equation known as the Kimura equation, inherits degenerate diffusion from the discrete stochastic process that conveys to the blowup into Dirac-delta singularities and hence brings great challenges to both analytical and numerical studies. The proposed numerical method can quantitatively capture to the fullest possible extent the development of Dirac-delta singularities for genetic segregation on the one hand and preserve several sets of biologically relevant and computationally favored properties of the random genetic drift on the other. Moreover, the numerical scheme exponentially converges to the unique numerical stationary state in time at a rate independent of the mesh size up to a mesh error. Numerical evidence is given to illustrate and support these properties and to demonstrate the spatiotemporal dynamics of random genetic drift.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-05-05T07:00:00Z
DOI: 10.1137/20M1389431
Issue No: Vol. 60, No. 3 (2022)
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- A New Lagrange Multiplier Approach for Constructing Structure Preserving
Schemes, II. Bound Preserving-
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Authors: Qing Cheng, Jie Shen
Pages: 970 - 998
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 970-998, June 2022.
In the second part of this series, we use the Lagrange multiplier approach proposed in the first part [Comput. Methods Appl. Mech. Engr., 391 (2022), 114585] to construct efficient and accurate bound and/or mass preserving schemes for a class of semilinear and quasi-linear parabolic equations. We establish stability results under a general setting and carry out an error analysis for a second-order bound preserving scheme with a hybrid spectral discretization in space. We apply our approach to several typical PDEs which preserve bound and/or mass and also present ample numerical results to validate our approach.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-05-05T07:00:00Z
DOI: 10.1137/21M144877X
Issue No: Vol. 60, No. 3 (2022)
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- An Embedded Exponential-Type Low-Regularity Integrator for mKdV Equation
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Authors: Cui Ning, Yifei Wu, Xiaofei Zhao
Pages: 999 - 1025
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 999-1025, June 2022.
In this paper, we propose an embedded low-regularity integrator (ELRI) under a new framework for solving the modified Korteweg-de Vries (mKdV) equation under rough data. Different from the previous work [Wu and Zhao, BIT, Number. Math., (2021)], the present ELRI scheme is constructed based on an approximation of a scaled Schrödinger operator and a new strategy of iterative regularizing through the inverse Miura transform. Moreover, the ELRI scheme is explicitly defined in the physical space, and it is efficient under the Fourier pseudospectral discretization. By rigorous error analysis, we show that ELRI achieves first-order accuracy by requiring the boundedness of one additional spatial derivative of the solution. Numerical results are presented to show the accuracy and efficiency of ELRI.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-05-10T07:00:00Z
DOI: 10.1137/21M1408166
Issue No: Vol. 60, No. 3 (2022)
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- Stability and Error Analysis of IMEX SAV Schemes for the
Magneto-Hydrodynamic Equations-
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Authors: Xiaoli Li, Weilong Wang, Jie Shen
Pages: 1026 - 1054
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1026-1054, June 2022.
We construct and analyze first- and second-order implicit-explicit schemes based on the scalar auxiliary variable approach for the magneto-hydrodynamic equations. These schemes are linear, only require solving a sequence of linear differential equations with constant coefficients at each time step, and are unconditionally energy stable. We derive rigorous error estimates for the velocity, pressure, and magnetic field of the first-order scheme in the two-dimensional case without any condition on the time step. Numerical examples are presented to validate the proposed schemes.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-05-10T07:00:00Z
DOI: 10.1137/21M1430376
Issue No: Vol. 60, No. 3 (2022)
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- Local Convergence of the FEM for the Integral Fractional Laplacian
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Authors: Markus Faustmann, Michael Karkulik, Jens Markus Melenk
Pages: 1055 - 1082
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1055-1082, June 2022.
For first-order discretizations of the integral fractional Laplacian, we provide sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm. Our estimates have the form of a local best approximation error plus a global error measured in a weaker norm.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-05-12T07:00:00Z
DOI: 10.1137/20M1343853
Issue No: Vol. 60, No. 3 (2022)
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- Numerical Analysis for Maxwell Obstacle Problems in Electric Shielding
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Authors: Maurice Hensel, Irwin Yousept
Pages: 1083 - 1110
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1083-1110, June 2022.
This paper proposes and examines a finite element method (FEM) for a Maxwell obstacle problem in electric shielding. The model is given by a coupled system comprising the Faraday equation and an evolutionary variational inequality (VI) of Ampère--Maxwell-type. Based on the leapfrog (Yee) time-stepping and the Nédélec edge elements, we set up a fully discrete FEM where the obstacle is discretized in such a way that no additional nonlinear solver is required for the computation of the discrete VI. While the $L^2$-stability is achieved for the discrete solutions and the associated difference quotients, the scheme only guarantees the $L^1$-stability for the discrete magnetic curl field in the obstacle region. The lack of the global $L^2$-stability for the magnetic curl field is justified by the low regularity issue in Maxwell obstacle problems and turns to be the main challenge in the convergence analysis. Our convergence proof consists of two main stages. First, exploiting the $L^1$-stability in the obstacle region, we derive a convergence result towards a weaker system involving smooth feasible test functions. In the second step, we recover the original system by enlarging the feasible test function set through a specific constraint preserving mollification process in the spirit of Ern and Guermond [Comput. Methods Appl. Math., 16 (2016), pp. 51--75]. This paper is closed by three-dimensional numerical results of the proposed FEM confirming the theoretical convergence result and, in particular, the Faraday shielding effect.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-05-23T07:00:00Z
DOI: 10.1137/21M1427693
Issue No: Vol. 60, No. 3 (2022)
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- MINRES for Second-Order PDEs with Singular Data
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Authors: Thomas Führer, Norbert Heuer, Michael Karkulik
Pages: 1111 - 1135
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1111-1135, June 2022.
Minimum residual methods such as the least-squares finite element method (FEM) or the discontinuous Petrov--Galerkin (DPG) method with optimal test functions usually exclude singular data, e.g., non-square-integrable loads. We consider a DPG method and a least-squares FEM for the Poisson problem. For both methods we analyze regularization approaches that allow the use of $H^{-1}$ loads and also study the case of point loads. For all cases we prove appropriate convergence orders. We present various numerical experiments that confirm our theoretical results. Our approach extends to general well-posed second-order problems.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-05-23T07:00:00Z
DOI: 10.1137/21M1457023
Issue No: Vol. 60, No. 3 (2022)
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- A Convergent Interacting Particle Method and Computation of KPP Front
Speeds in Chaotic Flows-
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Authors: Junlong Lyu, Zhongjian Wang, Jack Xin, Zhiwen Zhang
Pages: 1136 - 1167
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1136-1167, June 2022.
In this paper, we study the propagation speeds of reaction-diffusion-advection fronts in time-periodic cellular and chaotic flows with Kolmogorov--Petrovsky--Piskunov (KPP) nonlinearity. We first apply the variational principle to reduce the computation of KPP front speeds to a principal eigenvalue problem of a linear advection-diffusion operator with space-time periodic coefficient on a periodic domain. To this end, we develop efficient Lagrangian particle methods to compute the principal eigenvalue through the Feynman--Kac formula. By estimating the convergence rate of Feynman--Kac semigroups and the operator splitting method for approximating the linear advection-diffusion solution operators, we obtain convergence analysis for the proposed numerical method. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing KPP front speeds in time-periodic cellular and chaotic flows, especially the time-dependent Arnold--Beltrami--Childress flow and time-dependent Kolmogorov flow in three-dimensional space.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-05-26T07:00:00Z
DOI: 10.1137/21M1410786
Issue No: Vol. 60, No. 3 (2022)
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- On the Computation of Gaussian Quadrature Rules for Chebyshev Sets of
Linearly Independent Functions-
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Authors: Daan Huybrechs
Pages: 1168 - 1192
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1168-1192, June 2022.
We consider the computation of quadrature rules that are exact for a Chebyshev set of linearly independent functions on an interval $[a,b]$. A general theory of Chebyshev sets guarantees the existence of rules with a Gaussian property, in the sense that $2l$ basis functions can be integrated exactly with just $l$ points and weights. Moreover, all weights are positive, and the points lie inside the interval $[a,b]$. However, the points are not the roots of an orthogonal polynomial or any other known special function as in the case of regular Gaussian quadrature. The rules are characterized by a nonlinear system of equations, and earlier numerical methods have mostly focused on finding suitable starting values for a Newton iteration to solve this system. In this paper we describe an alternative scheme that is robust and generally applicable for so-called complete Chebyshev sets. These are ordered Chebyshev sets where the first $k$ elements also form a Chebyshev set for each $k$. The points of the quadrature rule are computed one by one, increasing the exactness of the rule in each step. Each step reduces to finding the unique root of a univariate and monotonic function. As such, the scheme of this paper is guaranteed to succeed. The quadrature rules are of interest for integrals with nonsmooth integrands that are not well approximated by polynomials.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-05-26T07:00:00Z
DOI: 10.1137/21M1456935
Issue No: Vol. 60, No. 3 (2022)
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- Flux-Mortar Mixed Finite Element Methods on NonMatching Grids
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Authors: Wietse M. Boon, Dennis Gläser, Rainer Helmig, Ivan Yotov
Pages: 1193 - 1225
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1193-1225, June 2022.
We investigate a mortar technique for mixed finite element approximations of a class of domain decomposition saddle point problems on nonmatching grids in which the variable associated with the essential boundary condition, referred to as flux, is chosen as the coupling variable. It plays the role of a Lagrange multiplier to impose weakly continuity of the variable associated with the natural boundary condition. The flux-mortar variable is incorporated with the use of a discrete extension operator. We present well-posedness and error analysis in an abstract setting under a set of suitable assumptions, followed by a nonoverlapping domain decomposition algorithm that reduces the global problem to a positive definite interface problem. The abstract theory is illustrated for Darcy flow, where the normal flux is the mortar variable used to impose continuity of pressure, and for Stokes flow, where the velocity vector is the mortar variable used to impose continuity of normal stress. In both examples, suitable discrete extension operators are developed and the assumptions from the abstract theory are verified. Numerical studies illustrating the theoretical results are presented for Darcy flow.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-05-31T07:00:00Z
DOI: 10.1137/20M1361407
Issue No: Vol. 60, No. 3 (2022)
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- Analysis of a Monte-Carlo Nystrom Method
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Authors: Florian Feppon, Habib Ammari
Pages: 1226 - 1250
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1226-1250, June 2022.
This paper considers a Monte-Carlo Nystrom method for solving integral equations of the second kind, whereby the values $(z(y_i))_{1\leq i\leq N}$ of the solution $z$ at a set of $N$ random and independent points $(y_i)_{1\leq i\leq N}$ are approximated by the solution $(z_{N,i})_{1\leq i\leq N}$ of a discrete $N$-dimensional linear system obtained by replacing the integral with the empirical average over the samples $(y_i)_{1\leq i\leq N}$. Under the unique assumption that the integral equation admits a unique solution $z(y)$, we prove the invertibility of the linear system for sufficiently large $N$ with probability one, and the convergence of the solution $(z_{N,i})_{1\leq i\leq N}$ towards the point values $(z(y_i))_{1\leq i\leq N}$ in a mean-square sense at a rate $O(N^{-\frac{1}{2}})$. For a particular choices of kernels, the discrete linear system arises as the Foldy--Lax approximation for the scattered field generated by a system of $N$ sources emitting waves at the points $(y_i)_{1\leq i\leq N}$. In this context, our result can be equivalently considered as a proof of the well-posedness of the Foldy--Lax approximation for systems of $N$ point scatterers, and of its convergence as $N\rightarrow +\infty$ in a mean-square sense to the solution of a Lippmann--Schwinger equation characterizing the effective medium. The convergence of Monte-Carlo solutions at the rate $O(N^{-1/2})$ is numerically illustrated on one-dimensional examples and for solving a two-dimensional Lippmann--Schwinger equation.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-01T07:00:00Z
DOI: 10.1137/21M1432338
Issue No: Vol. 60, No. 3 (2022)
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- Optimal Maximum Norm Estimates for Virtual Element Methods
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Authors: Wen-Ming He, Hailong Guo
Pages: 1251 - 1280
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1251-1280, June 2022.
The maximum norm error estimations for virtual element methods are studied. To establish the error estimations, we prove higher local regularity based on delicate analysis of Green's functions and high-order local error estimations for the partition of the virtual element solutions. The maximum norm of the exact gradient and the gradient of the projection of the virtual element solutions are proved to achieve optimal convergence results. For high-order virtual element methods, we establish the optimal convergence results in $L^{\infty}$ norm. Our theoretical discoveries are validated by a numerical example on general polygonal meshes.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-02T07:00:00Z
DOI: 10.1137/21M1420186
Issue No: Vol. 60, No. 3 (2022)
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- Convergence of Anisotropic Mesh Adaptation via Metric Optimization
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Authors: Hugh A. Carson, Steven Allmaras, Marshall Galbraith, David Darmofal
Pages: 1281 - 1306
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1281-1306, June 2022.
Adaptive finite element methods (AFEMs) are an increasingly common means of automatically controlling error in numerical simulations. Proofs of convergence and rate of convergence exist for AFEMs; however, these proofs typically rely upon a nested structure for the sequence of meshes. A metric adaptive finite element method (MAFEM) utilizes the continuous mesh model and instead seeks to optimize a Riemannian metric field for a given cost, from which a mesh is generated. This meshing process results in a sequence of nonnested meshes. In this paper we introduce a proof of convergence for a class of MAFEM, utilizing an optimization statement to relate the error on the sequence of meshes. In addition, we prove that such a sequence of meshes will demonstrate the optimal asymptotic rate of convergence for a given polynomial order. Finally some numerical results demonstrate the performance of the algorithm for a singularly perturbed linear advection diffusion problem.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-02T07:00:00Z
DOI: 10.1137/20M1338721
Issue No: Vol. 60, No. 3 (2022)
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- Conforming Finite Element DIVDIV Complexes and the Application for the
Linearized Einstein--Bianchi System-
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Authors: Jun Hu, Yizhou Liang, Rui Ma
Pages: 1307 - 1330
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1307-1330, June 2022.
This paper presents the first family of conforming finite element $\ddiv\ddiv$ complexes on tetrahedral grids in three dimensions. In these complexes, finite element spaces of $H(\ddiv\ddiv,\Omega;\Sbb)$ are from a recent article [L. Chen and X. Huang, Math. Comp., 91 (2022), pp. 1107--1142] while finite element spaces of both $H(\sym\ccurl,\Omega;\Tbb)$ and $H^1(\Omega;\R^3)$ are newly constructed here. It is proved that these finite element complexes are exact. As a result, they can be used to discretize the linearized Einstein--Bianchi system within the dual formulation.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-02T07:00:00Z
DOI: 10.1137/21M1404235
Issue No: Vol. 60, No. 3 (2022)
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- Time Stability of Strong Boundary Conditions in Finite-Difference Schemes
for Hyperbolic Systems-
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Authors: Nek Sharan, Peter T. Brady, Daniel Livescu
Pages: 1331 - 1362
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1331-1362, June 2022.
A framework to construct time-stable finite-difference schemes that apply boundary conditions strongly (or exactly) is presented for hyperbolic systems. A strong time-stability definition that applies to problems with homogeneous as well as nonhomogeneous boundary data is introduced. Sufficient conditions for strong time stability and conservation are derived for the linear advection equation and coupled system of hyperbolic equations using the energy method. Explicit boundary stencils and norms that satisfy those sufficient conditions are derived for various orders of accuracy. The discretization uses nonsquare derivative operators to allow stability and conservation conditions in terms of boundary data at grid points where physical boundary condition is directly injected and solution values at the rest of the grid points. Various linear and nonlinear numerical tests that verify the accuracy and stability of the derived stencils are presented.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-06T07:00:00Z
DOI: 10.1137/21M1419957
Issue No: Vol. 60, No. 3 (2022)
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- A Note on Sampling Recovery of Multivariate Functions in the Uniform Norm
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Authors: Kateryna Pozharska, Tino Ullrich
Pages: 1363 - 1384
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1363-1384, June 2022.
We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Surprisingly, a certain weighted least squares recovery operator which uses random samples from a tailored distribution leads to near-optimal results in several relevant situations. The results are stated in terms of the decay of related singular numbers of the compact embedding into $L_2(D)$ multiplied with the supremum of the Christoffel function of the subspace spanned by the first $m$ singular functions. As an application we obtain new recovery guarantees for Sobolev type spaces related to Jacobi type differential operators, on the one hand, and classical multivariate periodic Sobolev type spaces with general smoothness weight on the other hand. By applying a recently introduced subsampling technique related to Weaver's conjecture we mostly lose a $\sqrt{\log n}$ factor, compared to the optimal worst-case error, and sometimes even less.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-09T07:00:00Z
DOI: 10.1137/21M1410580
Issue No: Vol. 60, No. 3 (2022)
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- Neural Parametric Fokker--Planck Equation
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Authors: Shu Liu, Wuchen Li, Hongyuan Zha, Haomin Zhou
Pages: 1385 - 1449
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1385-1449, June 2022.
In this paper, we develop and analyze numerical methods for high-dimensional Fokker--Planck equations by leveraging generative models from deep learning. Our starting point is a formulation of the Fokker--Planck equation as a system of ordinary differential equations (ODEs) on finite-dimensional parameter space with the parameters inherited from generative models such as normalizing flows. We call such ODEs neural parametric Fokker--Planck equations. The fact that the Fokker--Planck equation can be viewed as the $L^2$-Wasserstein gradient flow of Kullback--Leibler (KL) divergence allows us to derive the ODEs as the constrained $L^2$-Wasserstein gradient flow of KL divergence on the set of probability densities generated by neural networks. For numerical computation, we design a variational semi-implicit scheme for the time discretization of the proposed ODE. Such an algorithm is sampling-based, which can readily handle the Fokker--Planck equations in higher dimensional spaces. Moreover, we also establish bounds for the asymptotic convergence analysis of the neural parametric Fokker--Planck equation as well as the error analysis for both the continuous and discrete versions. Several numerical examples are provided to illustrate the performance of the proposed algorithms and analysis.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-14T07:00:00Z
DOI: 10.1137/20M1344986
Issue No: Vol. 60, No. 3 (2022)
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- Adaptive FEM for Parameter-Errors in Elliptic Linear-Quadratic Parameter
Estimation Problems-
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Authors: Roland Becker, Michael Innerberger, Dirk Praetorius
Pages: 1450 - 1471
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1450-1471, June 2022.
We consider an elliptic linear-quadratic parameter estimation problem with a finite number of parameters. A novel a priori bound for the parameter error is proved and, based on this bound, an adaptive finite element method driven by an a posteriori error estimator is presented. Unlike prior results in the literature, our estimator, which is composed of standard energy error residual estimators for the state equation and suitable co-state problems, reflects the faster convergence of the parameter error compared to the (co-)state variables. We show optimal convergence rates of our method; in particular and unlike prior works, we prove that the estimator decreases with a rate that is the sum of the best approximation rates of the state and co-state variables. Experiments confirm that our method matches the convergence rate of the parameter error.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-16T07:00:00Z
DOI: 10.1137/21M1458077
Issue No: Vol. 60, No. 3 (2022)
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- Exponential Integrators for Quasilinear Wave-Type Equations
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Authors: Benjamin Dörich, Marlis Hochbruck
Pages: 1472 - 1493
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1472-1493, June 2022.
In this paper we propose two exponential integrators of first and second order applied to a class of quasilinear wave-type equations. The analytical framework is an extension of the classical Kato framework and covers quasilinear Maxwell's equations in full space and on a smooth domain as well as a class of quasilinear wave equations. In contrast to earlier works, we do not assume regularity of the solution but only on the data. From this we deduce a well-posedness result upon which we base our error analysis. Compared to existing results, our error bounds require less regularity in space and in time. We include numerical examples to confirm our theoretical findings.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-21T07:00:00Z
DOI: 10.1137/21M1410579
Issue No: Vol. 60, No. 3 (2022)
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- A Stabilization of a Continuous Limit of the Ensemble Kalman Inversion
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Authors: Dieter Armbruster, Michael Herty, Giuseppe Visconti
Pages: 1494 - 1515
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1494-1515, June 2022.
The ensemble Kalman filter (EnKF) belongs to the class of iterative particle filtering methods and can be used for solving control-to-observable inverse problems. In this context, the EnKF is known as ensemble Kalman inversion (EKI). In recent years several continuous limits in the number of iterations and particles have been performed in order to study properties of the method. In particular, a one-dimensional linear stability analysis reveals possible drawbacks in the phase space of moments provided by the continuous limits of the EKI but is observed also in the multidimensional setting. In this work we address this issue by introducing a stabilization of the dynamics which leads to a method with globally asymptotically stable solutions. We illustrate the performance of the stabilized version by using test inverse problems from the literature and comparing it with the classical continuous limit formulation of the method.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-21T07:00:00Z
DOI: 10.1137/21M1414000
Issue No: Vol. 60, No. 3 (2022)
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- A High Order Finite Difference Method for the Elastic Wave Equation in
Bounded Domains with Nonconforming Interfaces-
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Authors: Lu Zhang, Siyang Wang
Pages: 1516 - 1547
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1516-1547, June 2022.
We develop a stable finite difference method for the elastic wave equation in bounded media, where the material properties can be discontinuous at curved interfaces. The governing equation is discretized in second order form by a fourth or sixth order accurate summation-by-parts operator. The mesh size is determined by the velocity structure of the material, resulting in nonconforming grid interfaces with hanging nodes. We use order-preserving interpolation and the ghost point technique to couple adjacent mesh blocks in an energy-conserving manner, which is supported by a fully discrete stability analysis. In our previous work for the wave equation, two pairs of order-preserving interpolation operators were needed when imposing the interface conditions weakly by a penalty technique. Here, we only use one pair in the ghost point method. In numerical experiments, we demonstrate that the convergence rate is optimal and is the same as when a globally uniform mesh is used in a single domain. In addition, with a predictor-corrector time integration method, we obtain time stepping stability with stepsize almost the same as that given by the usual Courant--Friedrichs--Lewy condition.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-27T07:00:00Z
DOI: 10.1137/21M1422586
Issue No: Vol. 60, No. 3 (2022)
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- A Unified Convergence Analysis for the Fractional Diffusion Equation
Driven by Fractional Gaussian Noise with Hurst Index $H\in(0,1)$-
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Authors: Daxin Nie, Weihua Deng
Pages: 1548 - 1573
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1548-1573, June 2022.
Here, we provide a unified framework for numerical analysis of the stochastic nonlinear fractional diffusion equation driven by fractional Gaussian noise with Hurst index $H\in(0,1)$. A novel estimate of the second moment of the stochastic integral with respect to fractional Brownian motion is constructed, which greatly contributes to the regularity analyses of the solution in time and space for $H\in(0,1)$. Then we use the spectral Galerkin method and backward Euler convolution quadrature to discretize the fractional Laplacian and Riemann--Liouville fractional derivative, respectively. The sharp error estimates of the built numerical scheme are also obtained. Finally, the extensive numerical experiments verify the theoretical results.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-28T07:00:00Z
DOI: 10.1137/21M1422616
Issue No: Vol. 60, No. 3 (2022)
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- A Mass Conserving Mixed $hp$-FEM Scheme for Stokes Flow. Part III:
Implementation and Preconditioning-
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Authors: Mark Ainsworth, Charles Parker
Pages: 1574 - 1606
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1574-1606, June 2022.
This is the third part in a series on a mass conserving, high order, mixed finite element method for 2D Stokes flow. In this part, we study a block-diagonal preconditioner for the indefinite Schur complement system arising from the discretization of the Stokes equations using these elements. The underlying finite element method is uniformly stable in both the mesh size $h$ and the polynomial order $p$, and we prove bounds on the eigenvalues of the preconditioned system which are independent of $h$ and grow modestly in $p$. The analysis relates the Schur complement system to an appropriate variational setting with subspaces for which exact sequence properties and inf-sup stability hold. Several numerical examples demonstrate agreement with the theoretical results.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-30T07:00:00Z
DOI: 10.1137/21M1433927
Issue No: Vol. 60, No. 3 (2022)
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- A Well-Posed First Order System Least Squares Formulation of the
Instationary Stokes Equations-
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Authors: Gregor Gantner, Rob Stevenson
Pages: 1607 - 1629
Abstract: SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1607-1629, June 2022.
In this paper, a well-posed simultaneous space-time first order system least squares formulation is constructed of the instationary incompressible Stokes equations with slip boundary conditions. As a consequence of this well-posedness, the minimization over any conforming triple of finite element spaces for velocities, pressure, and stress tensor gives a quasi-best approximation from that triple. The formulation is practical in the sense that all norms in the least squares functional can be efficiently evaluated. Being of least squares type, the formulation comes with an efficient and reliable a posteriori error estimator. In addition, a priori error estimates are derived, and numerical results are presented.
Citation: SIAM Journal on Numerical Analysis
PubDate: 2022-06-30T07:00:00Z
DOI: 10.1137/21M1432600
Issue No: Vol. 60, No. 3 (2022)
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