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Limitations of neural network training due to numerical instability of backpropagation Adv. Comput. Math. (IF 1.7) Pub Date : 20240211
Clemens Karner, Vladimir Kazeev, Philipp Christian Petersen 
Analysis of a $$\varvec{P}_1\oplus \varvec{RT}_0$$ finite element method for linear elasticity with Dirichlet and mixed boundary conditions Adv. Comput. Math. (IF 1.7) Pub Date : 20240205
Hongpeng Li, Xu Li, Hongxing RuiIn this paper, we investigate a loworder robust numerical method for the linear elasticity problem. The method is based on a Bernardi–Raugellike \(\varvec{H}(\textrm{div})\)conforming method proposed first for the Stokes flows in [Li and Rui, IMA J. Numer. Anal. 42 (2022) 3711–3734]. Therein, the lowestorder \(\varvec{H}(\textrm{div})\)conforming Raviart–Thomas space (\(\varvec{RT}_0\)) was added

SemiLagrangian finite element exterior calculus for incompressible flows Adv. Comput. Math. (IF 1.7) Pub Date : 20240205
Wouter Tonnon, Ralf HiptmairWe develop a semiLagrangian discretization of the timedependent incompressible NavierStokes equations with free boundary conditions on arbitrary simplicial meshes. We recast the equations as a nonlinear transport problem for a momentum 1form and discretize in space using methods from finite element exterior calculus. Numerical experiments show that the linearly implicit fully discrete version of

Dictionarybased onlineadaptive structurepreserving model order reduction for parametric Hamiltonian systems Adv. Comput. Math. (IF 1.7) Pub Date : 20240205
Robin Herkert, Patrick Buchfink, Bernard HaasdonkClassical model order reduction (MOR) for parametric problems may become computationally inefficient due to large sizes of the required projection bases, especially for problems with slowly decaying Kolmogorov nwidths. Additionally, Hamiltonian structure of dynamical systems may be available and should be preserved during the reduction. In the current presentation, we address these two aspects by

On the balanced truncation error bound and sign parameters from arrowhead realizations Adv. Comput. Math. (IF 1.7) Pub Date : 20240131
Sean Reiter, Tobias Damm, Mark Embree, Serkan GugercinBalanced truncation and singular perturbation approximation for linear dynamical systems yield reduced order models that satisfy a wellknown error bound involving the Hankel singular values. We show that this bound holds with equality for singleinput, singleoutput systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a

Dual frames compensating for erasures—a noncanonical case Adv. Comput. Math. (IF 1.7) Pub Date : 20240125
Ljiljana Arambašić, Diana StoevaIn this paper, we study the problem of recovering a signal from frame coefficients with erasures. Suppose that erased coefficients are indexed by a finite set E. Starting from a frame \((x_n)_{n=1}^\infty \) and its arbitrary dual frame, we give sufficient conditions for constructing a dual frame of \((x_n)_{n\in E^c}\) so that the perfect reconstruction can be obtained from the preserved frame coefficients

An adaptive FEM for the elastic transmission eigenvalue problem with different elastic tensors and different mass densities Adv. Comput. Math. (IF 1.7) Pub Date : 20240117
Shixi Wang, Hai Bi, Yidu YangThe elastic transmission eigenvalue problem, arising from the inverse scattering theory, plays a critical role in the qualitative reconstruction methods for elastic media. This paper proposes and analyzes an a posteriori error estimator of the finite element method for solving the elastic transmission eigenvalue problem with different elastic tensors and different mass densities in \(\mathbb {R}^{d}~(d=2

Variational methods for solving numerically magnetostatic systems Adv. Comput. Math. (IF 1.7) Pub Date : 20240104
Patrick Ciarlet Jr., Erell JamelotIn this paper, we study some techniques for solving numerically magnetostatic systems. We consider fairly general assumptions on the magnetic permeability tensor. It is elliptic, but can be nonhermitian. In particular, we revisit existing classical variational methods and propose new numerical methods. The numerical approximation is either based on the classical edge finite elements or on continuous

A Grassmann manifold handbook: basic geometry and computational aspects Adv. Comput. Math. (IF 1.7) Pub Date : 20240105
Thomas Bendokat, Ralf Zimmermann, P.A. Absil 
A fractional osmosis model for image fusion Adv. Comput. Math. (IF 1.7) Pub Date : 20240108
Mohammed Hachama, Fatiha BoutaousThis paper introduces a novel model for image fusion that is based on a fractionalorder osmosis approach. The model incorporates a definition of osmosis energy that takes into account nonlocal pixel relationships using fractional derivatives and contrast change. The proposed model was subjected to theoretical and experimental investigation. The semigroup theory was used to demonstrate the existence

Numerical investigation of agentcontrolled pedestrian dynamics using a structurepreserving finite volume scheme Adv. Comput. Math. (IF 1.7) Pub Date : 20231228
JanFrederik Pietschmann, Ailyn Stötzner, Max WinklerWe provide a numerical realization of an optimal control problem for pedestrian motion with agents that was analyzed in Herzog et al. (Appl. Math. Optim. 88(3):87, 2023). The model consists of a regularized variant of Hughes’ model for pedestrian dynamics coupled to ordinary differential equations that describe the motion of agents which are able to influence the crowd via attractive forces. We devise

A threestep defectcorrection stabilized algorithm for incompressible flows with nonhomogeneous Dirichlet boundary conditions Adv. Comput. Math. (IF 1.7) Pub Date : 20231227
Bo Zheng, Yueqiang ShangAbstract Based on twogrid discretizations and quadratic equalorder finite elements for the velocity and pressure approximations, we develop a threestep defectcorrection stabilized algorithm for the incompressible NavierStokes equations, where nonhomogeneous Dirichlet boundary conditions are considered and high Reynolds numbers are allowed. In this developed algorithm, we first solve an artificial

Order two superconvergence of the CDG finite elements for nonself adjoint and indefinite elliptic equations Adv. Comput. Math. (IF 1.7) Pub Date : 20231222
Xiu Ye, Shangyou ZhangA conforming discontinuous Galerkin (CDG) finite element method is designed for solving second order nonself adjoint and indefinite elliptic equations. Unlike other discontinuous Galerkin (DG) methods, the numerical trace on the edge/triangle between two elements is not the average of two discontinuous \(P_k\) functions, but a lifted \(P_{k+2}\) function from four (eight in 3D) nearby \(P_k\) functions

Matching pursuit with unbounded parameter domains Adv. Comput. Math. (IF 1.7) Pub Date : 20231220
Wei Qu, Yanbo Wang, Xiaoyun SunIn various applications, the adoption of optimal energy matching pursuit with dictionary elements is common. When the dictionary elements are indexed by parameters within a bounded region, exhaustiontype algorithms can be employed. This article aims to investigate a process that converts the optimal parameter selection in unbounded regions to a bounded and closed (compact) subdomain. Such a process

Asymptotic convergence analysis and influence of initial guesses on composite Anderson acceleration Adv. Comput. Math. (IF 1.7) Pub Date : 20231213
Kewang Chen, Cornelis VuikAlthough Anderson acceleration AA(m) has been widely used to speed up nonlinear solvers, most authors are simply using and studying the stationary version of Anderson acceleration. The behavior and full potential of the nonstationary version of Anderson acceleration methods remain an open question. Motivated by the hybrid linear solver GMRESR (GMRES Recursive), we recently proposed a set of nonstationary

Interpolatory model reduction of quadraticbilinear dynamical systems with quadraticbilinear outputs Adv. Comput. Math. (IF 1.7) Pub Date : 20231214
Alejandro N. Diaz, Matthias Heinkenschloss, Ion Victor Gosea, Athanasios C. AntoulasThis paper extends interpolatory model reduction to systems with (up to) quadraticbilinear dynamics and quadraticbilinear outputs. These systems are referred to as QBQB systems and arise in a number of applications, including fluid dynamics, optimal control, and uncertainty quantification. In the interpolatory approach, the reduced order models (ROMs) are based on a PetrovGalerkin projection, and

Toward a certified greedy Loewner framework with minimal sampling Adv. Comput. Math. (IF 1.7) Pub Date : 20231205
Davide PradoveraWe propose a strategy for greedy sampling in the context of nonintrusive interpolationbased surrogate modeling for frequencydomain problems. We rely on a nonintrusive and cheap error indicator to drive the adaptive selection of the highfidelity samples on which the surrogate is based. We develop a theoretical framework to support our proposed indicator. We also present several practical approaches

Piecewise orthogonal collocation for computing periodic solutions of renewal equations Adv. Comput. Math. (IF 1.7) Pub Date : 20231207
Alessia Andò, Dimitri BredaWe extend the use of piecewise orthogonal collocation to computing periodic solutions of renewal equations, which are particularly important in modeling population dynamics. We prove convergence through a rigorous error analysis. Finally, we show some numerical experiments confirming the theoretical results and a couple of applications in view of bifurcation analysis.

Approximation of curvebased sleeve functions in high dimensions Adv. Comput. Math. (IF 1.7) Pub Date : 20231130
Robert Beinert 
A posteriori error analysis and adaptivity for a VEM discretization of the Navier–Stokes equations Adv. Comput. Math. (IF 1.7) Pub Date : 20231130
Claudio Canuto, Davide RossoWe consider the virtual element method (VEM) introduced by Beirão da Veiga et al. in 2016 for the numerical solution of the steady, incompressible Navier–Stokes equations; the method has arbitrary order \({k} \ge {2}\) and guarantees divergencefree velocities. For such discretization, we develop a residualbased a posteriori error estimator, which is a combination of standard terms in VEM analysis

Continuousstage adapted exponential methods for chargedparticle dynamics with arbitrary magnetic fields Adv. Comput. Math. (IF 1.7) Pub Date : 20231130
Ting Li, Bin WangThis paper is devoted to the numerical symplectic approximation of the chargedparticle dynamics (CPD) with a homogeneous magnetic field and its extension to a nonhomogeneous magnetic field. By utilizing continuousstage methods and exponential integrators, a general class of symplectic methods is formulated for CPD under a homogeneous magnetic field. Based on the derived symplectic conditions, two

Live load matrix recovery from scattering data in linear elasticity Adv. Comput. Math. (IF 1.7) Pub Date : 20231128
Juan Antonio Barceló, Carlos Castro, Mari Cruz VilelaWe study the numerical approximation of the inverse scattering problem in the twodimensional homogeneous isotropic linear elasticity with an unknown linear load given by a square matrix. For both backscattering data and fixedangle scattering data, we show how to obtain numerical approximations of the socalled Born approximations and propose new iterative algorithms that provide sequences of approximations

$$\mathcal {H}$$ inverses for RBF interpolation Adv. Comput. Math. (IF 1.7) Pub Date : 20231123
Niklas Angleitner, Markus Faustmann, Jens Markus MelenkWe consider the interpolation problem for a class of radial basis functions (RBFs) that includes the classical polyharmonic splines (PHS). We show that the inverse of the system matrix for this interpolation problem can be approximated at an exponential rate in the block rank in the \(\mathcal {H}\)matrix format, if the block structure of the \(\mathcal {H}\)matrix arises from a standard clustering

Iterative twogrid methods for discontinuous Galerkin finite element approximations of semilinear elliptic problem Adv. Comput. Math. (IF 1.7) Pub Date : 20231122
Jiajun Zhan, Liuqiang Zhong, Jie PengIn this paper, we design and analyze the iterative twogrid methods for the discontinuous Galerkin (DG) discretization of semilinear elliptic partial differential equations (PDEs). We first present an iterative twogrid method that is just like the classical iterative twogrid methods for nonsymmetric or indefinite linear elliptic PDEs, namely, to solve a semilinear problem on the coarse space and

An unconditionally stable and $$L^2$$ optimal quadratic finite volume scheme over triangular meshes for anisotropic elliptic equations Adv. Comput. Math. (IF 1.7) Pub Date : 20231122
Xiaoxin Wu, Weifeng Qiu, Kejia PanIn this paper, we propose an unconditionally stable and \(L^2\) optimal quadratic finite volume (FV) scheme for solving the twodimensional anisotropic elliptic equation on triangular meshes. In quadratic FV schemes, the construction of the dual partition is closely related to the \(L^2\) error estimate. While many dual partitions over triangular meshes have been investigated in the literature, only

Estimation of quadrature errors for layer potentials evaluated near surfaces with spherical topology Adv. Comput. Math. (IF 1.7) Pub Date : 20231123
Chiara Sorgentone, AnnaKarin Tornberg 
The ButzerKozakiewicz article on Riemann derivatives of 1954 and its influence Adv. Comput. Math. (IF 1.7) Pub Date : 20231123
P. L. Butzer, R. L. StensThe article on Riemann derivatives by P. L. Butzer and W. Kozakiewicz of 1954 was the basis to generalizations of the classical scalarvalued derivatives to Taylor, Peano, and Riemann derivatives in the setting of semigroup theory. The present paper gives an overview of the 1954 article, describes its influence, and integrates it into the literature on related problems. It also describes the state

Max filtering with reflection groups Adv. Comput. Math. (IF 1.7) Pub Date : 20231121
Dustin G. Mixon, Daniel PackerGiven a finitedimensional real inner product space V and a finite subgroup G of linear isometries, max filtering affords a bilipschitz Euclidean embedding of the orbit space V/G. We identify the max filtering maps of minimum distortion in the setting where G is a reflection group. Our analysis involves an interplay between Coxeter’s classification and semidefinite programming.

Multilevel Monte Carlo simulation for the Heston stochastic volatility model Adv. Comput. Math. (IF 1.7) Pub Date : 20231120
Chao Zheng 
Solving the backward problem for timefractional wave equations by the quasireversibility regularization method Adv. Comput. Math. (IF 1.7) Pub Date : 20231025
Jin Wen, ZhiYuan Li, YongPing WangThis paper is devoted to the backward problem of determining the initial value and initial velocity simultaneously in a timefractional wave equation, with the aid of extra measurement data at two fixed times. Uniqueness results are obtained by using the analyticity and the asymptotics of the MittagLeffler functions provided that the two fixed measurement times are sufficiently close. Since this problem

A novel robust fractionaltime anisotropic diffusion for multiframe image superresolution Adv. Comput. Math. (IF 1.7) Pub Date : 20231025
Anouar Benloghfyry, Abdelilah HakimIn this paper, we propose an image multiframe Super Resolution (SR) method based on fractionaltime Caputo derivative combined with Weickerttype diffusion process idea. We provide the existence and uniqueness results with a detailed discretization using the finite difference scheme. Our approach is based on anisotropic diffusion behavior with coherence enhancing diffusion tensor together with the

On flexible block ChebyshevDavidson method for solving symmetric generalized eigenvalue problems Adv. Comput. Math. (IF 1.7) Pub Date : 20231024
CunQiang Miao, Lan ChengIn a recent work (J. Sci. Comput. 85 (2020), no. 3), the author generalized the ChebyshevDavidson method appeared in standard eigenvalue problems to symmetric generalized eigenvalue problems. The theoretical derivation indicates that the ChebyshevDavidson method for symmetric generalized eigenvalue problems only admits local convergence; thus, in this paper, we adopt a flexible strategy to improve

Mass and energyconserving Gauss collocation methods for the nonlinear Schrödinger equation with a wave operator Adv. Comput. Math. (IF 1.7) Pub Date : 20231023
Shu Ma, Jilu Wang, Mingyan Zhang, Zhimin ZhangA fully discrete finite element method with a Gauss collocation in time is proposed for solving the nonlinear Schrödinger equation with a wave operator in the ddimensional torus, \(d\in \{1,2,3\}\). Based on Gauss collocation method in time and the scalar auxiliary variable technique, the proposed method preserves both mass and energy conservations at the discrete level. Existence and uniqueness of

Random sampling of signals concentrated on compact set in localized reproducing kernel subspace of $$L^p(\mathbb R^n)$$ Adv. Comput. Math. (IF 1.7) Pub Date : 20231017
Dhiraj Patel, S. SivananthanThe paper is devoted to studying the stability of random sampling in a localized reproducing kernel space. We show that if the sampling set on \(\Omega \) (compact) discretizes the integral norm of simple functions up to a given error, then the sampling set is stable for the set of functions concentrated on \(\Omega \). Moreover, we prove with an overwhelming probability that \(\mathcal {O}(\mu (\Omega

Arbitrary highorder structurepreserving methods for the quantum Zakharov system Adv. Comput. Math. (IF 1.7) Pub Date : 20231017
Gengen Zhang, Chaolong JiangIn this paper, we present a new methodology to develop arbitrary highorder structurepreserving methods for solving the quantum Zakharov system. The key ingredients of our method are as follows: (i) the original Hamiltonian energy is reformulated into a quadratic form by introducing a new quadratic auxiliary variable; (ii) based on the energy variational principle, the original system is then rewritten

A secondorder scheme based on blended BDF for the incompressible MHD system Adv. Comput. Math. (IF 1.7) Pub Date : 20231003
Shuaijun Liu, Pengzhan Huang, Yinnian HeFor the incompressible MHD equations, we present a fully discrete secondorderintime scheme based on a blended BDF and extrapolation treatments for nonlinear terms. The proposed scheme is more accurate than the twostep BDF with additional nominal computational time and is still Astable. Then, unconditional stability, longtime stability, and optimal convergence rate of the scheme are presented

An efficient approximation to the stochastic AllenCahn equation with random diffusion coefficient field and multiplicative noise Adv. Comput. Math. (IF 1.7) Pub Date : 20230927
Xiao Qi, Yanrong Zhang, Chuanju XuThis paper studies the stochastic AllenCahn equation involving random diffusion coefficient field and multiplicative force noise. A new timestepping method based on auxiliary variable approach is proposed and analyzed. The proposed method is efficient thanks to its low computational complexity. Furthermore, it is unconditionally stable in the sense that a discrete energy is dissipative when the multiplicative

The stability and convergence analysis of finite difference methods for the fractional neutron diffusion equation Adv. Comput. Math. (IF 1.7) Pub Date : 20230914
Daopeng Yin, Yingying Xie, Liquan MeiFor the timefractional neutron diffusion equation with a Caputo derivative of order \( \varvec{\alpha } \in ~(\varvec{0},\frac{\varvec{1}}{\varvec{2}}) \), we give the optimal error bounds of L1type schemes under the spatial \( L^{\infty } \)norm with lower regularity solution than typical \( \varvec{\partial }_{\varvec{t}}^{\varvec{l}} \varvec{u}(\varvec{x},\varvec{t})  \le \varvec{C} (\varv

Interaction with an obstacle in the 2D focusing nonlinear Schrödinger equation Adv. Comput. Math. (IF 1.7) Pub Date : 20230913
Oussama Landoulsi, Svetlana Roudenko, Kai YangWe present a numerical study of solutions to the 2d cubic and quintic focusing nonlinear Schrödinger equation in the exterior of a smooth, compact and strictly convex obstacle (a disk) with Dirichlet boundary condition. We first investigate the effect of the obstacle on the behavior of solutions traveling towards the obstacle at different angles and with different velocities directions. We introduce

Calibration of Pvalues for calibration and for deviation of a subpopulation from the full population Adv. Comput. Math. (IF 1.7) Pub Date : 20230904
Mark TygertThe author’s recent research papers, “Cumulative deviation of a subpopulation from the full population” and “A graphical method of cumulative differences between two subpopulations” (both published in volume 8 of Springer’s openaccess Journal of Big Data during 2021), propose graphical methods and summary statistics, without extensively calibrating formal significance tests. The summary metrics and

Convergence of nonstationary semidiscrete RBF schemes for the heat and wave equation Adv. Comput. Math. (IF 1.7) Pub Date : 20230831
Raymond BrummelhuisWe give a detailed analysis of the convergence in Sobolev norm of the method of lines for the classical heat and wave equations on \(\mathbb {R }^n \) using nonstationary radial basis function interpolation on regular grids \(h \mathbb {Z }^n \) (scaled cardinal interpolation), for basis functions whose native space is a Sobolev space of order \(\nu / 2 \) with \(\nu > n + 2\).

A posteriori error estimates for the timedependent NavierStokes system coupled with the convectiondiffusionreaction equation Adv. Comput. Math. (IF 1.7) Pub Date : 20230814
Jad Dakroub, Joanna Faddoul, Pascal Omnes, Toni SayahIn this paper we study the a posteriori error estimates for the time dependent NavierStokes system coupled with the convectiondiffusionreaction equation. The problem is discretized in time using the implicit Euler method and in space using the finite element method. We establish a posteriori error estimates with two types of computable error indicators, the first one linked to the space discretization

Simpler is better: a comparative study of randomized pivoting algorithms for CUR and interpolative decompositions Adv. Comput. Math. (IF 1.7) Pub Date : 20230807
Yijun Dong, PerGunnar MartinssonMatrix skeletonizations like the interpolative and CUR decompositions provide a framework for lowrank approximation in which subsets of a given matrix’s columns and/or rows are selected to form approximate spanning sets for its column and/or row space. Such decompositions that rely on “natural” bases have several advantages over traditional lowrank decompositions with orthonormal bases, including

Splitting scheme for backward doubly stochastic differential equations Adv. Comput. Math. (IF 1.7) Pub Date : 20230803
Feng Bao, Yanzhao Cao, He ZhangA splitting scheme is proposed for a class of backward doubly stochastic differential equations (BDSDEs). The main idea is to decompose the backward doubly stochastic differential equation into a backward stochastic differential equation and a stochastic differential equation, which are much easier to solve than the BDSDE itself. The two equations are then approximated by firstorder finite difference

Numerical analysis of nonlinear degenerate parabolic problems with application to eddy current models Adv. Comput. Math. (IF 1.7) Pub Date : 20230802
Ramiro Acevedo, Christian Gómez, Paulo NaviaThis paper deals with the numerical analysis for a family of nonlinear degenerate parabolic problems. The model is spatially discretized using a finite element method; an implicit Euler scheme is employed for time discretization. We deduce sufficient conditions to ensure that the fully discrete problem has a unique solution and to prove quasioptimal error estimates for the approximation. Finally,

A fast time domain solver for the equilibrium Dyson equation Adv. Comput. Math. (IF 1.7) Pub Date : 20230802
Jason Kaye, Hugo U. R. StrandWe consider the numerical solution of the realtime equilibrium Dyson equation, which is used in calculations of the dynamical properties of quantum manybody systems. We show that this equation can be written as a system of coupled, nonlinear, convolutional Volterra integrodifferential equations, for which the kernel depends selfconsistently on the solution. As is typical in the numerical solution

Finite basis physicsinformed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations Adv. Comput. Math. (IF 1.7) Pub Date : 20230731
Ben Moseley, Andrew Markham, Tarje NissenMeyerRecently, physicsinformed neural networks (PINNs) have offered a powerful new paradigm for solving problems relating to differential equations. Compared to classical numerical methods, PINNs have several advantages, for example their ability to provide meshfree solutions of differential equations and their ability to carry out forward and inverse modelling within the same optimisation problem. Whilst

Interior penalty discontinuous Galerkin methods for the velocitypressure formulation of the Stokes spectral problem Adv. Comput. Math. (IF 1.7) Pub Date : 20230727
Felipe LepeIn this paper, we analyze discontinuous Galerkin methods based in the interior penalty method in order to approximate the eigenvalues and eigenfunctions of the Stokes eigenvalue problem. The considered methods in this work are based in discontinuous polynomials approximations for the velocity field and the pressure fluctuation in two and three dimensions. The methods under consideration are symmetric

Highorder corrected trapezoidal rules for a class of singular integrals Adv. Comput. Math. (IF 1.7) Pub Date : 20230726
Federico Izzo, Olof Runborg, Richard TsaiWe present a family of highorder trapezoidal rulebased quadratures for a class of singular integrals, where the integrand has a point singularity. The singular part of the integrand is expanded in a Taylor series involving terms of increasing smoothness. The quadratures are based on the trapezoidal rule, with the quadrature weights for Cartesian nodes close to the singularity judiciously corrected

Adaptive finite element approximation of optimal control problems with the integral fractional Laplacian Adv. Comput. Math. (IF 1.7) Pub Date : 20230724
Zhou Zhaojie, Wang QimingIn this paper, we study an adaptive finite element approximation of optimal control problems with integral fractional Laplacian and pointwise control constraints. The state variable is approximated by piecewise linear polynomials, and the control variable is implicitly discretized. Upper and lower bounds of a posteriori error estimates for finite element approximation of the optimal control problem

Adaptive and local regularization for data fitting by tensorproduct spline surfaces Adv. Comput. Math. (IF 1.7) Pub Date : 20230724
Sandra Merchel, Bert Jüttler, Dominik Mokriš 
Full recovery from point values: an optimal algorithm for Chebyshev approximability prior Adv. Comput. Math. (IF 1.7) Pub Date : 20230724
Simon FoucartGiven pointwise samples of an unknown function belonging to a certain model set, one seeks in optimal recovery to recover this function in a way that minimizes the worstcase error of the recovery procedure. While it is often known that such an optimal recovery procedure can be chosen to be linear, e.g., when the model set is based on approximability by a subspace of continuous functions, a construction

An adaptive discontinuous finite volume element method for the AllenCahn equation Adv. Comput. Math. (IF 1.7) Pub Date : 20230721
Jian Li, Jiyao Zeng, Rui LiIn this paper, the discontinuous finite volume element method (DFVEM) is considered to solve the AllenCahn equation which contains strong nonlinearity. The method is based on the DFVEM in space and the backward Euler method in time. The energy stability and unique solvability of the proposed fully discrete scheme are derived. The error estimates for the semidiscrete and fully discrete scheme are

Phase retrieval and system identification in dynamical sampling via Prony’s method Adv. Comput. Math. (IF 1.7) Pub Date : 20230721
Robert Beinert, Marzieh HasannasabPhase retrieval in dynamical sampling is a novel research direction, where an unknown signal has to be recovered from the phaseless measurements with respect to a dynamical frame, i.e., a sequence of sampling vectors constructed by the repeated action of an operator. The loss of the phase here turns the wellposed dynamical sampling into a severe illposed inverse problem. In the existing literature

A posteriori error estimates for wave maps into spheres Adv. Comput. Math. (IF 1.7) Pub Date : 20230717
Jan Giesselmann, Elena MäderBaumdicker, David Jakob StonnerWe provide a posteriori error estimates in the energy norm for temporal semidiscretisations of wave maps into spheres that are based on the angular momentum formulation. Our analysis is based on novel weak–strong stability estimates which we combine with suitable reconstructions of the numerical solution. We present timeadaptive numerical simulations based on the a posteriori error estimators for

A DCANewton method for quartic minimization over the sphere Adv. Comput. Math. (IF 1.7) Pub Date : 20230710
Shenglong Hu, Yong Wang, Jinling ZhouIn this paper, a method for quartic minimization over the sphere is studied. It is based on an equivalent difference of convex (DC) reformulation of this problem in the matrix variable. This derivation also induces a global optimality certification for the quartic minimization over the sphere. An algorithm with the subproblem being solved by a semismooth Newton method is then proposed for solving the

On the stability of the representation of finite rank operators Adv. Comput. Math. (IF 1.7) Pub Date : 20230710
J. M. Carnicer, E. Mainar, J. M. PeñaThe stability of the representation of finite rank operators in terms of a basis is analyzed. A conditioning is introduced as a measure of the stability properties. This conditioning improves some other conditionings because it is closer to the Lebesgue function. Improved bounds for the conditioning of the Fourier sums with respect to an orthogonal basis are obtained, in particular, for Legendre, Chebyshev

Deep learning theory of distribution regression with CNNs Adv. Comput. Math. (IF 1.7) Pub Date : 20230707
Zhan Yu, DingXuan ZhouWe establish a deep learning theory for distribution regression with deep convolutional neural networks (DCNNs). Deep learning based on structured deep neural networks has been powerful in practical applications. Generalization analysis for regression with DCNNs has been carried out very recently. However, for the distribution regression problem in which the input variables are probability measures

Ellipsoidal conformal and area/volumepreserving parameterizations and associated optimal mass transportations Adv. Comput. Math. (IF 1.7) Pub Date : 20230703
JiaWei Lin, Tiexiang Li, WenWei Lin, TsungMing Huang 
Exponentially fitted methods with a local energy conservation law Adv. Comput. Math. (IF 1.7) Pub Date : 20230703
Dajana Conte, Gianluca FrascaCacciaA new exponentially fitted version of the discrete variational derivative method for the efficient solution of oscillatory complex Hamiltonian partial differential equations is proposed. When applied to the nonlinear Schrödinger equation, this scheme has discrete conservation laws of charge and energy. The new method is compared with other conservative schemes from the literature on a benchmark problem