Abstract: Abstract A time-fractional Allen-Cahn problem is considered, where the spatial domain Ω is a bounded subset of \(\mathbb {R}^{d}\) for some d ∈{1,2,3}. New bounds on certain derivatives of the solution are derived. These are used in the analysis of a numerical method (L1 discretization of the temporal fractional derivative on a graded mesh, with a standard finite element discretization of the spatial diffusion term, and Newton linearization of the nonlinear driving term), showing that the computed solution achieves the optimal rate of convergence in the Sobolev H1(Ω) norm. (Previous papers considered only convergence in L2(Ω).) Numerical results confirm our theoretical findings. PubDate: 2020-07-11

Abstract: Abstract In this paper, we present a novel approach to the approximate solution of elliptic partial differential equations on compact submanifolds of \(\mathbb {R}^{d}\) , particularly compact surfaces and the surface equation \({\Delta }_{\mathbb {M}} u - \lambda u=f\) . In the course of this, we reconsider differential operators on such submanifolds to deduce suitable penalty based functionals. These functionals are based on the residual of the equation in an integral representation, extended by a penalty on the first-order normal derivative. The general framework we develop is accompanied by error analysis and exemplified by numerical examples employing tensor product B-splines. PubDate: 2020-07-09

Abstract: Abstract Local superconvergence properties of the post-processed finite volume element method (FVEM) are studied. Some interpolation/extrapolation post-processing techniques are applied to a class of k th-order (k ≥ 2) FVE solutions for elliptic equations. A local analysis tool for the finite volume method is developed to analyze the proposed method, and some superconvergence results are established. The theoretical findings are supported by several numerical experiments. PubDate: 2020-07-06

Abstract: Abstract In this paper we consider fully discrete approximations with inf-sup stable mixed finite element methods in space to approximate the Navier-Stokes equations. A continuous downscaling data assimilation algorithm is analyzed in which measurements on a coarse scale are given represented by different types of interpolation operators. For the time discretization an implicit Euler scheme, an implicit and a semi-implicit second-order backward differentiation formula are considered. Uniform-in-time error estimates are obtained for all the methods for the error between the fully discrete approximation and the reference solution corresponding to the measurements. For the spatial discretization we consider both the Galerkin method and the Galerkin method with grad-div stabilization. For the last scheme error bounds in which the constants do not depend on inverse powers of the viscosity are obtained. PubDate: 2020-07-06

Abstract: Abstract This paper is concerned with the construction, analysis, and realization of a numerical method to approximate the solution of high-dimensional elliptic partial differential equations. We propose a new combination of an adaptive wavelet Galerkin method (AWGM) and the well-known hierarchical tensor (HT) format. The arising HT-AWGM is adaptive both in the wavelet representation of the low-dimensional factors and in the tensor rank of the HT representation. The point of departure is an adaptive wavelet method for the HT format using approximate Richardson iterations and an AWGM for elliptic problems. HT-AWGM performs a sequence of Galerkin solves based upon a truncated preconditioned conjugate gradient (PCG) algorithm in combination with a tensor-based preconditioner. Our analysis starts by showing convergence of the truncated conjugate gradient method. The next step is to add routines realizing the adaptive refinement. The resulting HT-AWGM is analyzed concerning convergence and complexity. We show that the performance of the scheme asymptotically depends only on the desired tolerance with convergence rates depending on the Besov regularity of low-dimensional quantities and the low-rank tensor structure of the solution. The complexity in the ranks is algebraic with powers of four stemming from the complexity of the tensor truncation. Numerical experiments show the quantitative performance. PubDate: 2020-07-06

Abstract: Abstract In this paper, we introduce some new accelerated hybrid algorithms for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition in a Hilbert space. The algorithms are constructed around the extragradient method, the inertial technique, the hybrid (or outer approximation) method, and the shrinking projection method. The algorithms are designed to work either with or without the prior knowledge of the Lipschitz-type constants of bifunction. Theorems of strong convergence are established under mild conditions. The results in this paper generalize, extend, and improve some known results in the field. Finally, several of numerical experiments are performed to support the obtained theoretical results. PubDate: 2020-06-23

Abstract: Abstract In this paper, we propose a novel overlapping domain decomposition method that can be applied to various problems in variational imaging such as total variation minimization. Most of recent domain decomposition methods for total variation minimization adopt the Fenchel–Rockafellar duality, whereas the proposed method is based on the primal formulation. Thus, the proposed method can be applied not only to total variation minimization but also to those with complex dual problems such as higher order models. In the proposed method, an equivalent formulation of the model problem with parallel structure is constructed using a custom overlapping domain decomposition scheme with the notion of essential domains. As a solver for the constructed formulation, we propose a decoupled augmented Lagrangian method for untying the coupling of adjacent subdomains. Convergence analysis of the decoupled augmented Lagrangian method is provided. We present implementation details and numerical examples for various model problems including total variation minimizations and higher order models. PubDate: 2020-06-22

Abstract: Abstract We present an hp-version of the C0-continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems. We derive a priori error bound in the H1-norm that is fully explicit with respect to the local time steps and the local approximation orders. Moreover, we prove that the hp-version of the C0-continuous Petrov-Galerkin time stepping method based on geometrically refined time steps and on linearly increasing approximation orders yields exponential rates of convergence for solutions with start-up singularities. Numerical examples confirm the theoretical results. PubDate: 2020-06-16

Abstract: Abstract In this work, we explore the advantages of end-to-end learning of multilayer maps offered by feedforward neural networks (FFNNs) for learning and predicting dynamics from transient flow data. While data-driven learning (and machine learning) in general depends on data quality and quantity relative to the underlying dynamics of the system, it is important for a given data-driven learning architecture to make the most of this available information. To this end, we focus on data-driven problems where there is a need to predict over reasonable time into the future with limited data availability. Such function time series prediction of full and reduced states is different from many applications of machine learning such as pattern recognition and parameter estimation that leverage large datasets. In this study, we interpret the suite of recently popular data-driven learning approaches that approximate the dynamics as Markov linear model in higher dimensional feature space as a multilayer architecture similar to neural networks. However, there exist a couple of key differences: (i) Markov linear models employ layer-wise learning in the sense of linear regression whereas neural networks represent end-to-end learning in the sense of nonlinear regression. We show through examples of data-driven modeling of canonical fluid flows that FFNN-like methods owe their success to leveraging the extended learning parameter space available in end-to-end learning without overfitting to the data. In this sense, the Markov linear models behave as shallow neural networks. (ii) The second major difference is that while the FFNN is by design a forward architecture, the class of Markov linear methods that approximate the Koopman operator is bi-directional, i.e., they incorporate both forward and backward maps in order to learn a linear map that can provide insight into spectral characteristics. In this study, we assess both reconstruction and predictive performance of temporally evolving dynamic using limited data for canonical nonlinear fluid flows including transient cylinder wake flow and the instability-driven dynamics of buoyant Boussinesq flow. PubDate: 2020-06-16

Abstract: Abstract We investigate an inverse and ill-posed problem for the two-dimensional inhomogeneous heat equation in the presence of a general source term. The goal here consists of recovering not only the temperature distribution but also the thermal flux from the measure data. With the appearance of the general source term, this model gets far worse than its homogeneous counterpart. Based on an analysis of the instability caused in the solution, we propose a kernel regularization scheme to stabilize the investigated problem. The Hölder-type convergence estimates are achieved under some appropriate a priori assumptions. We further numerically demonstrate the theoretical results by proposing a robust algorithm based on the 2-D fast Fourier transform. The numerical outputs exemplify the feasibility and efficiency of the proposed method. PubDate: 2020-06-16

Abstract: Abstract In this paper, we are concerned with multivariate Gegenbauer approximation of functions defined in the d-dimensional hypercube. Two new and sharper bounds for the coefficients of multivariate Gegenbauer expansion of analytic functions are presented based on two different extensions of the Bernstein ellipse. We then establish an explicit error bound for the multivariate Gegenbauer approximation associated with an ℓq ball index set in the uniform norm. We also consider the multivariate approximation of functions with finite regularity and derive the associated error bound on the full grid in the uniform norm. As an application, we extend our arguments to obtain some new tight bounds for the coefficients of tensorized Legendre expansions in the context of polynomial approximation of parametrized PDEs. PubDate: 2020-06-09

Abstract: Abstract In this paper, we consider the matrices approximated in \({\mathscr{H}}^{2}\) format. The direct solution, as well as the preconditioning of systems with such matrices, is a challenging problem. We propose a non-extensive sparse factorization of the \({\mathscr{H}}^{2}\) matrix that allows to substitute direct \({\mathscr{H}}^{2}\) solution with the solution of the system with an equivalent sparse matrix of the same size. The sparse factorization is constructed of parameters of the \({\mathscr{H}}^{2}\) matrix. In the numerical experiments, we show the consistency of this approach in comparison with the other approximate block low-rank hierarchical solvers, such as HODLR [3], H2Lib [5], and IFMM [11]. PubDate: 2020-06-08

Abstract: Abstract One of the most popular time-frequency representations is certainly the Wigner distribution. Its quadratic nature is, however, at the origin of unwanted interferences or artefacts. The desire to suppress these artefacts is the reason why engineers, mathematicians and physicists have been looking for related time-frequency distributions, many of them being members of the Cohen class. Among these, the Born-Jordan distribution has recently attracted the attention of many authors, since the so-called ghost frequencies are grandly damped, and the noise is, in general, reduced; it also seems to play a key role in quantum mechanics. The central insight relies on the kernel of such a distribution, which contains the sinus cardinalis sinc, the Fourier transform of the first B-spline B1. The idea is to replace the function B1 with the spline or order n, denoted by Bn, yielding the function (sinc)n when Fourier transformed, whose speed of decay at infinity increases with n. The related Cohen kernel is given by \({\Theta }^{n}(z_{1},z_{2})=\text {sinc}^{n}(z_{1}\cdot z_{2})\), \(n\in \mathbb {N}\), and the corresponding time-frequency distribution is called generalized Born-Jordan distribution of ordern. We show that this new representation has a great potential to damp unwanted interference effects and this damping effect increases with n. Our proofs of these properties require an interdisciplinary approach, using tools from both microlocal and time-frequency analysis. As a by-product, a new quantization rule and a related pseudo-differential calculus are investigated. PubDate: 2020-06-06

Abstract: Abstract This work is concerned with adaptive hybridizable discontinuous Galerkin methods of nonstationary convection diffusion problems. We address first the spatially semidiscrete case and then move to the fully discrete scheme by introducing a backward Euler discretization in time. More specifically, the computable a posteriori error estimator for the time-dependent problem is obtained by using the idea of elliptic reconstruction and conforming-nonconforming decomposition. In view of the method that has been employed in the time-dependent problem, we also obtain a computable a posteriori error estimator for the fully discrete scheme. Finally, two examples show the performance of the obtained a posteriori error estimators. PubDate: 2020-06-04

Abstract: Abstract We focus on a two-dimensional time-space diffusion equation with fractional derivatives in space. The use of Crank-Nicolson in time and finite differences in space leads to dense Toeplitz-like linear systems. Multigrid strategies that exploit such structure are particularly effective when the fractional orders are both close to 2. We seek to investigate how structure-based multigrid approaches can be efficiently extended to the case where only one of the two fractional orders is close to 2, i.e., when the fractional equation shows an intrinsic anisotropy. Precisely, we design a multigrid (block-banded–banded-block) preconditioner whose grid transfer operator is obtained with a semi-coarsening technique and that has relaxed Jacobi as smoother. The Jacobi relaxation parameter is estimated by using an automatic symbol-based procedure. A further improvement in the robustness of the proposed multigrid method is attained using the V-cycle with semi-coarsening as smoother inside an outer full-coarsening. Several numerical results confirm that the resulting multigrid preconditioner is computationally effective and outperforms current state of the art techniques. PubDate: 2020-06-03

Abstract: Abstract In this paper, we consider numerical approximations for solving the Cahn-Hilliard phase-field model with the Flory-Huggins-de Gennes free energy for homopolymer blends. We develop an efficient, second-order accurate, and unconditionally energy stable scheme that combines the SAV approach with the stabilization technique, in which the H1 norm is split from the total free energy and two extra linear stabilization terms are added to enhance the stability and keeping the required accuracy while using large time steps. The scheme is very easy to implement and non-iterative where one only needs to solve two decoupled fourth-order biharmonic equations with constant coefficients at each time step. We further prove the unconditional energy stability of the scheme rigorously. Through the comparisons with some other prevalent schemes like the non-stabilized-SAV and MSAV schemes for some benchmark numerical examples in 2D and 3D, we demonstrate the stability and the accuracy of the developed scheme numerically. PubDate: 2020-05-26

Abstract: Abstract We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally, energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order O(Δt2 + N−m), where Δt, N, and m are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes. PubDate: 2020-05-26

Abstract: Abstract The present work deals with rational model order reduction methods based on the single-point Least-Square (LS) Padé approximation techniques introduced in Bonizzoni et al. (ESAIM Math. Model. Numer. Anal., 52(4), 1261–1284 2018, Math. Comput. 89, 1229–1257 2020). Algorithmical aspects concerning the construction of rational LS-Padé approximants are described. In particular, we show that the computation of the Padé denominator can be carried out efficiently by solving an eigenvalue-eigenvector problem involving a Gramian matrix. The LS-Padé techniques are employed to approximate the frequency response map associated with two parametric time-harmonic acoustic wave problems, namely a transmission-reflection problem and a scattering problem. In both cases, we establish the meromorphy of the frequency response map. The Helmholtz equation with stochastic wavenumber is also considered. In particular, for Lipschitz functionals of the solution and their corresponding probability measures, we establish weak convergence of the measure derived from the LS-Padé approximant to the true one. 2D numerical tests are performed, which confirm the effectiveness of the approximation methods. PubDate: 2020-05-08

Abstract: Abstract In computational geometry, different ways of space partitioning have been developed, including the Voronoi diagram of points and the power diagram of balls. In this article, a generalized Voronoi partition of overlapping d-dimensional balls, called the boundary-partition-based diagram, is proposed. The definition, properties, and applications of this diagram are presented. Compared to the power diagram, this boundary-partition-based diagram is straightforward in the computation of the volume of overlapping balls, which avoids the possibly complicated construction of power cells. Furthermore, it can be applied to characterize singularities on molecular surfaces and to compute the medial axis that can potentially be used to classify molecular structures. PubDate: 2020-05-06

Abstract: Abstract A high-order cut finite element method is formulated for solving the elastic wave equation. Both a single domain problem and an interface problem are treated. The boundary or interface is allowed to cut through the background mesh. To avoid problems with small cuts, stabilizing terms are added to the bilinear forms corresponding to the mass and stiffness matrix. The stabilizing terms penalize jumps in normal derivatives over the faces of the elements cut by the boundary/interface. This ensures a stable discretization independently of how the boundary/interface cuts the mesh. Nitsche’s method is used to enforce boundary and interface conditions, resulting in symmetric bilinear forms. As a result of the symmetry, an energy estimate can be made and optimal order a priori error estimates are derived for the single domain problem. Finally, numerical experiments in two dimensions are presented that verify the order of accuracy and stability with respect to small cuts. PubDate: 2020-05-06