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Newton’s method for the parameterized generalized eigenvalue problem with nonsquare matrix pencils Adv. Comput. Math. (IF 1.748) Pub Date : 2021-04-10 Jiao-fen Li, Wen Li, Xue-feng Duan, Mingqing Xiao
The l parameterized generalized eigenvalue problems for the nonsquare matrix pencils, proposed by Chu et al.in 2006, can be formulated as an optimization problem on a corresponding complex product Stiefel manifold. In this paper, an effective and efficient algorithm based on the Riemannian Newton’s method is established to solve the underlying problem. Under our proposed framework, to solve the corresponding
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A stabilizer-free pressure-robust finite element method for the Stokes equations Adv. Comput. Math. (IF 1.748) Pub Date : 2021-04-08 Xiu Ye, Shangyou Zhang
In this paper, we introduce a new finite element method for solving the Stokes equations in the primary velocity-pressure formulation using H(div) finite elements to approximate velocity. Like other finite element methods with velocity discretized by H(div) conforming elements, our method has the advantages of an exact divergence-free velocity field and pressure-robustness. However, most of H(div)
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Multilevel correction adaptive finite element method for solving nonsymmetric eigenvalue problems Adv. Comput. Math. (IF 1.748) Pub Date : 2021-04-07 Fei Xu, Meiling Yue, Bin Zheng
Large-scale nonsymmetric eigenvalue problems are common in various fields of science and engineering computing. However, their efficient handling is challenging, and research on their solution algorithms is limited. In this study, a new multilevel correction adaptive finite element method is designed for solving nonsymmetric eigenvalue problems based on the adaptive refinement technique and multilevel
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Randomized linear algebra for model reduction—part II: minimal residual methods and dictionary-based approximation Adv. Comput. Math. (IF 1.748) Pub Date : 2021-03-26 Oleg Balabanov, Anthony Nouy
A methodology for using random sketching in the context of model order reduction for high-dimensional parameter-dependent systems of equations was introduced in Balabanov and Nouy (Part I; Advances in Computational Mathematics 45:2969–3019, 2019). Following this framework, we here construct a reduced model from a small, efficiently computable random object called a sketch of a reduced model, using
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p - and hp - virtual elements for the Stokes problem Adv. Comput. Math. (IF 1.748) Pub Date : 2021-03-23 A. Chernov, C. Marcati, L. Mascotto
We analyse the p- and hp-versions of the virtual element method (VEM) for the Stokes problem on polygonal domains. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside
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Joint spectral radius and ternary hermite subdivision Adv. Comput. Math. (IF 1.748) Pub Date : 2021-03-23 M. Charina, C. Conti, T. Mejstrik, J.-L. Merrien
In this paper we construct a family of ternary interpolatory Hermite subdivision schemes of order 1 with small support and \({\mathscr{H}}\mathcal {C}^{2}\)-smoothness. Indeed, leaving the binary domain, it is possible to derive interpolatory Hermite subdivision schemes with higher regularity than the existing binary examples. The family of schemes we construct is a two-parameter family whose \({\mathscr{H}}\mathcal
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The pointwise error estimates of two energy-preserving fourth-order compact schemes for viscous Burgers’ equation Adv. Comput. Math. (IF 1.748) Pub Date : 2021-03-15 Xuping Wang, Qifeng Zhang, Zhi-zhong Sun
A novel fourth-order three-point compact operator for the nonlinear convection term uux is provided in this paper. The operator makes the numerical analysis of higher-order difference schemes become possible for a wide class of nonlinear evolutionary equations under the unified framework. We take the classical viscous Burgers’ equation as an example and establish a new conservative fourth-order implicit
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Adaptive two- and three-dimensional multiresolution computations of resistive magnetohydrodynamics Adv. Comput. Math. (IF 1.748) Pub Date : 2021-02-23 Anna Karina Fontes Gomes, Margarete Oliveira Domingues, Odim Mendes, Kai Schneider
Fully adaptive computations of the resistive magnetohydrodynamic (MHD) equations are presented in two and three space dimensions using a finite volume discretization on locally refined dyadic grids. Divergence cleaning is used to control the incompressibility constraint of the magnetic field. For automatic grid adaptation a cell-averaged multiresolution analysis is applied which guarantees the precision
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A recovery-based linear C 0 finite element method for a fourth-order singularly perturbed Monge-Ampère equation Adv. Comput. Math. (IF 1.748) Pub Date : 2021-02-23 Hongtao Chen, Xiaobing Feng, Zhimin Zhang
This paper develops a new recovery-based linear C0 finite element method for approximating the weak solution of a fourth-order singularly perturbed Monge-Ampère equation, which is known as the vanishing moment approximation of the Monge-Ampère equation. The proposed method uses a gradient recovery technique to define a discrete Laplacian for a given linear C0 finite element function (offline), the
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Stable recovery of planar regions with algebraic boundaries in Bernstein form Adv. Comput. Math. (IF 1.748) Pub Date : 2021-02-16 Costanza Conti, Mariantonia Cotronei, Demetrio Labate, Wilfredo Molina
We present a new method for the stable reconstruction of a class of binary images from a small number of measurements. The images we consider are characteristic functions of algebraic domains, that is, domains defined as zero loci of bivariate polynomials, and we assume to know only a finite set of uniform samples for each image. The solution to such a problem can be set up in terms of linear equations
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A generalised complete flux scheme for anisotropic advection-diffusion equations Adv. Comput. Math. (IF 1.748) Pub Date : 2021-02-16 Hanz Martin Cheng, Jan ten Thije Boonkkamp
In this paper, we consider separating the discretisation of the diffusive and advective fluxes in the complete flux scheme. This allows the combination of several discretisation methods for the homogeneous flux with the complete flux (CF) method. In particular, we explore the combination of the hybrid mimetic mixed (HMM) method and the CF method, in order to utilise the advantages of each of these
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Hessian discretisation method for fourth-order semi-linear elliptic equations: applications to the von Kármán and Navier–Stokes models Adv. Comput. Math. (IF 1.748) Pub Date : 2021-02-16 Jérome Droniou, Neela Nataraj, Devika Shylaja
This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific
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Edge detection with trigonometric polynomial shearlets Adv. Comput. Math. (IF 1.748) Pub Date : 2021-02-08 Kevin Schober, Jürgen Prestin, Serhii A. Stasyuk
In this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products
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An adaptive multiscale hybrid-mixed method for the Oseen equations Adv. Comput. Math. (IF 1.748) Pub Date : 2021-02-02 Rodolfo Araya, Cristian Cárcamo, Abner H. Poza, Frédéric Valentin
A novel residual a posteriori error estimator for the Oseen equations achieves efficiency and reliability by including multilevel contributions in its construction. Originates from the Multiscale Hybrid Mixed (MHM) method, the estimator combines residuals from the skeleton of the first-level partition of the domain, along with the contributions from element-wise approximations. The second-level estimator
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Polynomial spline spaces of non-uniform bi-degree on T-meshes: combinatorial bounds on the dimension Adv. Comput. Math. (IF 1.748) Pub Date : 2021-02-02 Deepesh Toshniwal, Bernard Mourrain, Thomas J. R. Hughes
Polynomial splines are ubiquitous in the fields of computer-aided geometric design and computational analysis. Splines on T-meshes, especially, have the potential to be incredibly versatile since local mesh adaptivity enables efficient modeling and approximation of local features. Meaningful use of such splines for modeling and approximation requires the construction of a suitable spanning set of linearly
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An adaptive spectral graph wavelet method for PDEs on networks Adv. Comput. Math. (IF 1.748) Pub Date : 2021-02-02 Mani Mehra, Ankita Shukla, Günter Leugering
In this article, we propose an adaptive spectral graph wavelet method to solve partial differential equations on network-like structures using so-called spectral graph wavelets. The concept of spectral graph wavelets is based on the discrete graph Laplacian. The beauty of the method lies in the fact that the same operator is used for the approximation of differential operators and for the construction
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Adaptive and optimal pointwise deconvolution density estimations by wavelets Adv. Comput. Math. (IF 1.748) Pub Date : 2021-02-02 Cong Wu, Xiaochen Zeng, Na Mi
This paper considers multivariate deconvolution density estimations under the local Hölder condition by wavelet methods. A pointwise lower bound of the deconvolution model is first investigated; then we provide a linear wavelet estimate to obtain the optimal convergence rate. The nonlinear wavelet estimator is introduced for adaptivity, which attains a nearly optimal rate (optimal up to a logarithmic
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A well-balanced high-order scheme on van Leer-type for the shallow water equations with temperature gradient and variable bottom topography Adv. Comput. Math. (IF 1.748) Pub Date : 2021-02-02 Nguyen Xuan Thanh, Mai Duc Thanh, Dao Huy Cuong
A well-balanced high-order scheme for shallow water equations with variable topography and temperature gradient is constructed. This scheme is of van Leer-type and is based on exact Riemann solvers. The scheme is shown to be able to capture almost exactly the stationary smooth solutions as well as stationary elementary discontinuities. Numerical tests show that the scheme gives a much better accuracy
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Efficient approximation of solutions of parametric linear transport equations by ReLU DNNs Adv. Comput. Math. (IF 1.748) Pub Date : 2021-01-28 Fabian Laakmann, Philipp Petersen
We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation of these functions suffers from a curse of dimension. We demonstrate that through their inherent compositionality
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Tensor completion via bilevel minimization with fixed-point constraint to estimate missing elements in noisy data Adv. Comput. Math. (IF 1.748) Pub Date : 2021-01-28 Souad Mohaoui, Abdelilah Hakim, Said Raghay
In this work, we consider the tensor completion problem of an incomplete and noisy observation. We introduce a novel completion model using bilevel minimization. Therefore, bilevel model-based denoising for the tensor completion problem is proposed. The denoising and completion tasks are fully separated. The upper-level directly addresses the completion problem with the truncated nuclear norm, while
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New convergence analysis of a primal-dual algorithm with large stepsizes Adv. Comput. Math. (IF 1.748) Pub Date : 2021-01-25 Zhi Li, Ming Yan
We consider a primal-dual algorithm for minimizing \(f({\mathbf {x}})+h\square l({\mathbf {A}}{\mathbf {x}})\) with Fréchet differentiable f and l∗. This primal-dual algorithm has two names in literature: Primal-Dual Fixed-Point algorithm based on the Proximity Operator (PDFP2O) and Proximal Alternating Predictor-Corrector (PAPC). In this paper, we prove its convergence under a weaker condition on
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Stability and error estimates for the variable step-size BDF2 method for linear and semilinear parabolic equations Adv. Comput. Math. (IF 1.748) Pub Date : 2021-01-21 Wansheng Wang, Mengli Mao, Zheng Wang
In this paper, stability and error estimates for time discretizations of linear and semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to the question: whether the upper bound of step-size ratios for the \(l^{\infty }(0,T;H)\)-stability of the BDF2 method for linear and semilinear parabolic
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Robin-Robin domain decomposition methods for the dual-porosity-conduit system Adv. Comput. Math. (IF 1.748) Pub Date : 2021-01-20 Jiangyong Hou, Wenjing Yan, Dan Hu, Zhengkang He
The recently developed dual-porosity-Stokes model describes a complicated dual-porosity-conduit system which uses a dual-porosity/permeability model to govern the flow in porous media coupled with free flow via four physical interface conditions. This system has important applications in unconventional reservoirs especially the multistage fractured horizontal wellbore problems. In this paper, we propose
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Counting the dimension of splines of mixed smoothness Adv. Comput. Math. (IF 1.748) Pub Date : 2021-01-17 Deepesh Toshniwal, Michael DiPasquale
In this paper, we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, “mixed smoothness” refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from homological algebra. These tools were first applied to the study of splines by Billera (Trans. Am. Math. Soc
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A stable second-order BDF scheme for the three-dimensional Cahn–Hilliard–Hele–Shaw system Adv. Comput. Math. (IF 1.748) Pub Date : 2021-01-12 Yibao Li, Qian Yu, Weiwei Fang, Binhu Xia, Junseok Kim
We propose a stable scheme to solve numerically the Cahn–Hilliard–Hele–Shaw system in three-dimensional space. In the proposed scheme, we discretize the space and time derivative terms by combining with backward differentiation formula, which turns out to be both second-order accurate in space and time. Using this method, a set of linear elliptic equations are solved instead of the complicated and
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Symplectic geometry and connectivity of spaces of frames Adv. Comput. Math. (IF 1.748) Pub Date : 2021-01-12 Tom Needham, Clayton Shonkwiler
Frames provide redundant, stable representations of data which have important applications in signal processing. We introduce a connection between symplectic geometry and frame theory and show that many important classes of frames have natural symplectic descriptions. Symplectic tools seem well-adapted to addressing a number of important questions about frames; in this paper, we focus on the frame
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Frame approximation with bounded coefficients Adv. Comput. Math. (IF 1.748) Pub Date : 2021-01-12 Ben Adcock, Mohsen Seifi
Due to their flexibility, frames of Hilbert spaces are attractive alternatives to bases in approximation schemes for problems where identifying a basis is not straightforward or even feasible. Computing a best approximation using frames, however, can be challenging since it requires solving an ill-conditioned linear system. One consequence of this ill-conditioning is that the coefficients of such a
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Virtual element methods for the three-field formulation of time-dependent linear poroelasticity Adv. Comput. Math. (IF 1.748) Pub Date : 2021-01-04 Raimund Bürger, Sarvesh Kumar, David Mora, Ricardo Ruiz-Baier, Nitesh Verma
A virtual element discretisation for the numerical approximation of the three-field formulation of linear poroelasticity introduced in R. Oyarzúa and R. Ruiz-Baier, (SIAM J. Numer. Anal. 54 2951–2973, 2016) is proposed. The treatment is extended to include also the transient case. Appropriate poroelasticity projector operators are introduced and they assist in deriving energy bounds for the time-dependent
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Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method Adv. Comput. Math. (IF 1.748) Pub Date : 2020-12-17 Moreno Pintore, Federico Pichi, Martin Hess, Gianluigi Rozza, Claudio Canuto
The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even
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Sampling-free model reduction of systems with low-rank parameterization Adv. Comput. Math. (IF 1.748) Pub Date : 2020-11-19 Christopher Beattie, Serkan Gugercin, Zoran Tomljanović
We consider the reduction of parametric families of linear dynamical systems having an affine parameter dependence that allow for low-rank variation in the state matrix. Usual approaches for parametric model reduction typically involve exploring the parameter space to identify representative parameter values and the associated models become the principal focus of model reduction methodology. These
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Balanced truncation of linear time-invariant systems over finite-frequency ranges Adv. Comput. Math. (IF 1.748) Pub Date : 2020-11-16 Peter Benner, Xin Du, Guanghong Yang, Dan Ye
This paper discusses model order reduction of linear time-invariant (LTI) systems over limited frequency intervals within the framework of balanced truncation. Two new frequency-dependent balanced truncation methods are developed, one is single-frequency (SF)-type frequency-dependent balanced truncation to cope with the cases that only a single dominating point of the operating frequency interval is
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Non-conforming Crouzeix-Raviart element approximation for Stekloff eigenvalues in inverse scattering Adv. Comput. Math. (IF 1.748) Pub Date : 2020-11-11 Yidu Yang, Yu Zhang, Hai Bi
In this paper, we use the non-conforming Crouzeix-Raviart element method to solve a Stekloff eigenvalue problem arising in inverse scattering. The weak formulation corresponding to this problem is non-self-adjoint and indefinite, and its Crouzeix-Raviart element discretization does not meet the condition of the Strang lemma. We use the standard duality technique to prove an extension of the Strang
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Exact posterior computation for the binomial–Kumaraswamy model Adv. Comput. Math. (IF 1.748) Pub Date : 2020-11-09 J. A. A. Andrade
In Bayesian analysis, the well-known beta–binomial model is largely used as a conjugate structure, and the beta prior distribution is a natural choice to model parameters defined in the (0,1) range. The Kumaraswamy distribution has been used as a natural alternative to the beta distribution and has received great attention in statistics in the past few years, mainly due to the simplicity and the great
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Second-order energy stable schemes for the new model of the Cahn-Hilliard-MHD equations Adv. Comput. Math. (IF 1.748) Pub Date : 2020-11-06 Rui Chen, Hui Zhang
To simulate the two-phase flow of conducting fluids, we propose a coupled model of the Cahn-Hilliard equations and the inductionless and incompressible magnetohydrodynamic (MHD) equations. The model describes the dynamic behavior of conducting fluid under the influence of magnetic field. Based on the “invariant energy quadratization” method, we propose two fully discrete time-marching schemes which
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Superconvergent gradient recovery for nonlinear Poisson-Nernst-Planck equations with applications to the ion channel problem Adv. Comput. Math. (IF 1.748) Pub Date : 2020-10-29 Ying Yang, Ming Tang, Chun Liu, Benzhuo Lu, Liuqiang Zhong
Poisson-Nernst-Planck equations are widely used to describe the electrodiffusion of ions in a solvated biomolecular system. An error estimate in H1 norm is obtained for a piecewise finite element approximation to the solution of the nonlinear steady-state Poisson-Nernst-Planck equations. Some superconvergence results are also derived by using the gradient recovery technique for the equations. Numerical
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Numerical analysis of a finite element method for the electromagnetic concentrator model Adv. Comput. Math. (IF 1.748) Pub Date : 2020-10-27 Yunqing Huang, Jichun Li
In this paper, we consider the electromagnetic concentrator model obtained through transformation optics. This model is formed by a system of coupled time-dependent Maxwell’s equations with three unknowns, which makes the analysis and simulation much more challenging compared to the standard Maxwell equations. In our previous work (W. Yang, J. Li, Y. Huang, and B. He, Commun. Comput. Phys., 25(1),
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Fast randomized matrix and tensor interpolative decomposition using CountSketch Adv. Comput. Math. (IF 1.748) Pub Date : 2020-10-26 Osman Asif Malik, Stephen Becker
We propose a new fast randomized algorithm for interpolative decomposition of matrices which utilizes CountSketch. We then extend this approach to the tensor interpolative decomposition problem introduced by Biagioni et al. (J. Comput. Phys. 281(C), 116–134 (2015)). Theoretical performance guarantees are provided for both the matrix and tensor settings. Numerical experiments on both synthetic and real
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Unconditionally optimal convergence analysis of second-order BDF Galerkin finite element scheme for a hybrid MHD system Adv. Comput. Math. (IF 1.748) Pub Date : 2020-09-18 Yuan Li, Chunfang Zhai
In this paper, a second-order backward differentiation formula (BDF) scheme for a hybrid MHD system is considered. Being different with the steady and nonstationary MHD equations, the hybrid MHD system is coupled by the time-dependent Navier-Stokes equations and the steady Maxwell equations. By using the standard extrapolation technique for the nonlinear terms, the proposed BDF scheme is a semi-implicit
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Virtual element methods for nonlocal parabolic problems on general type of meshes Adv. Comput. Math. (IF 1.748) Pub Date : 2020-09-09 D. Adak, S. Natarajan
In this paper, we consider the discretization of a parabolic nonlocal problem within the framework of the virtual element method. Using the fixed point argument, we prove that the fully discrete scheme has a unique solution. The presence of the nonlocal term makes the problem nonlinear, and the resulting nonlinear equations are solved using the Newton method. The computational cost of the Jacobian
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On the computation of Nash and Pareto equilibria for some bi-objective control problems for the wave equation Adv. Comput. Math. (IF 1.748) Pub Date : 2020-09-02 Pitágoras Pinheiro de Carvalho, Enrique Fernández-Cara, Juan Bautista Límaco Ferrel
This paper deals with the numerical implementation of a systematic method for solving bi-objective optimal control problems for wave equations. More precisely, we look for Nash and Pareto equilibria which respectively correspond to appropriate noncooperative and cooperative strategies in multi-objective optimal control. The numerical methods described here consist of a combination of the following:
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Analysis of adaptive synchrosqueezing transform with a time-varying parameter Adv. Comput. Math. (IF 1.748) Pub Date : 2020-08-29 Jian Lu, Qingtang Jiang, Lin Li
The synchrosqueezing transform (SST) was developed recently to separate the components of non-stationary multicomponent signals. The continuous wavelet transform-based SST (WSST) reassigns the scale variable of the continuous wavelet transform of a signal to the frequency variable and sharpens the time-frequency representation. The WSST with a time-varying parameter, called the adaptive WSST, was introduced
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A unified analysis of a class of quadratic finite volume element schemes on triangular meshes Adv. Comput. Math. (IF 1.748) Pub Date : 2020-08-27 Yanhui Zhou, Jiming Wu
This paper presents a general framework for the coercivity analysis of a class of quadratic finite volume element (FVE) schemes on triangular meshes for solving elliptic boundary value problems. This class of schemes covers all the existing quadratic schemes of Lagrange type. With the help of a new mapping from the trial function space to the test function space, we find that each element matrix can
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A characterization of supersmoothness of multivariate splines Adv. Comput. Math. (IF 1.748) Pub Date : 2020-08-26 Michael S. Floater, Kaibo Hu
We consider spline functions over simplicial meshes in \(\mathbb {R}^{n}\). We assume that the spline pieces join together with some finite order of smoothness but the pieces themselves are infinitely smooth. Such splines can have extra orders of smoothness at a vertex, a property known as supersmoothness, which plays a role in the construction of multivariate splines and in the finite element method
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An integral equation–based numerical method for the forced heat equation on complex domains Adv. Comput. Math. (IF 1.748) Pub Date : 2020-08-20 Fredrik Fryklund, Mary Catherine A. Kropinski, Anna-Karin Tornberg
Integral equation–based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations
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A mixed virtual element method for the vibration problem of clamped Kirchhoff plate Adv. Comput. Math. (IF 1.748) Pub Date : 2020-08-10 Jian Meng, Liquan Mei
In this paper, we give a presentation of virtual element method for the approximation of the vibration problem of clamped Kirchhoff plate, which involves the biharmonic eigenvalue problem. Following the theory of Babǔska and Osborn, the error estimates of the discrete scheme for the degree k ≥ 2 of polynomials are standard results. However, when considering the case k = 1, we can not apply the technical
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Difference potentials method for models with dynamic boundary conditions and bulk-surface problems Adv. Comput. Math. (IF 1.748) Pub Date : 2020-07-31 Yekaterina Epshteyn, Qing Xia
In this work, we consider parabolic models with dynamic boundary conditions and parabolic bulk-surface problems in 3D. Such partial differential equations–based models describe phenomena that happen both on the surface and in the bulk/domain. These problems may appear in many applications, ranging from cell dynamics in biology, to grain growth models in polycrystalline materials. Using difference potentials
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Spectral collocation method for nonlinear Caputo fractional differential system Adv. Comput. Math. (IF 1.748) Pub Date : 2020-07-29 Zhendong Gu
A spectral collocation method is developed to solve a nonlinear Caputo fractional differential system. The main idea is to solve the corresponding system of weakly singular nonlinear Volterra integral equations (VIEs). The convergence analysis in matrix form shows that the presented method has spectral convergence. Numerical experiments are carried out to confirm theoretical results.
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A sparse FFT approach for ODE with random coefficients Adv. Comput. Math. (IF 1.748) Pub Date : 2020-07-19 Maximilian Bochmann, Lutz Kämmerer, Daniel Potts
The paper presents a general strategy to solve ordinary differential equations (ODE), where some coefficient depend on the spatial variable and on additional random variables. The approach is based on the application of a recently developed dimension-incremental sparse fast Fourier transform. Since such algorithms require periodic signals, we discuss periodization strategies and associated necessary
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Solving partial differential equations on (evolving) surfaces with radial basis functions Adv. Comput. Math. (IF 1.748) Pub Date : 2020-07-16 Holger Wendland, Jens Künemund
Meshfree, kernel-based spatial discretisations are recent tools to discretise partial differential equations on surfaces. The goals of this paper are to analyse and compare three different meshfree kernel-based methods for the spatial discretisation of semi-linear parabolic partial differential equations (PDEs) on surfaces, i.e. on smooth, compact, connected, orientable, and closed (d − 1)-dimensional
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Optimal H 1 spatial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation Adv. Comput. Math. (IF 1.748) Pub Date : 2020-07-11 Chaobao Huang, Martin Stynes
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
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Ambient residual penalty approximation of partial differential equations on embedded submanifolds Adv. Comput. Math. (IF 1.748) Pub Date : 2020-07-09 L.-B. Maier
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
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Local superconvergence of post-processed high-order finite volume element solutions Adv. Comput. Math. (IF 1.748) Pub Date : 2020-07-06 Wenming He, Zhimin Zhang, Qingsong Zou
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
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Error analysis of fully discrete mixed finite element data assimilation schemes for the Navier-Stokes equations Adv. Comput. Math. (IF 1.748) Pub Date : 2020-07-06 Bosco García-Archilla, Julia Novo
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
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HT-AWGM: a hierarchical Tucker–adaptive wavelet Galerkin method for high-dimensional elliptic problems Adv. Comput. Math. (IF 1.748) Pub Date : 2020-07-06 Mazen Ali, Karsten Urban
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
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Accelerated hybrid methods for solving pseudomonotone equilibrium problems Adv. Comput. Math. (IF 1.748) Pub Date : 2020-06-23 Dang Van Hieu, Pham Kim Quy, La Thi Hong, Le Van Vy
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
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An overlapping domain decomposition framework without dual formulation for variational imaging problems Adv. Comput. Math. (IF 1.748) Pub Date : 2020-06-22 Jongho Park
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
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An hp -version of the C 0 -continuous Petrov-Galerkin time stepping method for nonlinear second-order initial value problems Adv. Comput. Math. (IF 1.748) Pub Date : 2020-06-16 Yichen Wei, Lijun Yi
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
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Assessment of end-to-end and sequential data-driven learning for non-intrusive modeling of fluid flows Adv. Comput. Math. (IF 1.748) Pub Date : 2020-06-16 Shivakanth Chary Puligilla, Balaji Jayaraman
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
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Determination of temperature distribution and thermal flux for two-dimensional inhomogeneous sideways heat equations Adv. Comput. Math. (IF 1.748) Pub Date : 2020-06-16 Tran Nhat Luan, Tra Quoc Khanh
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
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Analysis of multivariate Gegenbauer approximation in the hypercube Adv. Comput. Math. (IF 1.748) Pub Date : 2020-06-09 Haiyong Wang, Lun Zhang
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
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