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A probabilistic reduced basis method for parameter-dependent problems Adv. Comput. Math. (IF 1.7) Pub Date : 2024-03-13 Marie Billaud-Friess, Arthur Macherey, Anthony Nouy, Clémentine Prieur
Probabilistic variants of model order reduction (MOR) methods have recently emerged for improving stability and computational performance of classical approaches. In this paper, we propose a probabilistic reduced basis method (RBM) for the approximation of a family of parameter-dependent functions. It relies on a probabilistic greedy algorithm with an error indicator that can be written as an expectation
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Structured interpolation for multivariate transfer functions of quadratic-bilinear systems Adv. Comput. Math. (IF 1.7) Pub Date : 2024-03-12 Peter Benner, Serkan Gugercin, Steffen W. R. Werner
High-dimensional/high-fidelity nonlinear dynamical systems appear naturally when the goal is to accurately model real-world phenomena. Many physical properties are thereby encoded in the internal differential structure of these resulting large-scale nonlinear systems. The high dimensionality of the dynamics causes computational bottlenecks, especially when these large-scale systems need to be simulated
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New low-order mixed finite element methods for linear elasticity Adv. Comput. Math. (IF 1.7) Pub Date : 2024-03-06 Xuehai Huang, Chao Zhang, Yaqian Zhou, Yangxing Zhu
New low-order \({H}({{\text {div}}})\)-conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the \({(d+1)}\)-order normal-normal face bubble space. The reduced counterpart has only \({d(d+1)}^{{2}}\) degrees of freedom. Basis functions are explicitly given in terms of
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Conditioning and spectral properties of isogeometric collocation matrices for acoustic wave problems Adv. Comput. Math. (IF 1.7) Pub Date : 2024-03-04 Elena Zampieri, Luca F. Pavarino
The conditioning and spectral properties of the mass and stiffness matrices for acoustic wave problems are here investigated when isogeometric analysis (IGA) collocation methods in space and Newmark methods in time are employed. Theoretical estimates and extensive numerical results are reported for the eigenvalues and condition numbers of the acoustic mass and stiffness matrices in the reference square
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A space–time DG method for the Schrödinger equation with variable potential Adv. Comput. Math. (IF 1.7) Pub Date : 2024-03-01 Sergio Gómez, Andrea Moiola
We present a space–time ultra-weak discontinuous Galerkin discretization of the linear Schrödinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal h-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials or as a novel quasi-Trefftz
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Limitations of neural network training due to numerical instability of backpropagation Adv. Comput. Math. (IF 1.7) Pub Date : 2024-02-11 Clemens Karner, Vladimir Kazeev, Philipp Christian Petersen
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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 : 2024-02-05 Hongpeng Li, Xu Li, Hongxing Rui
In this paper, we investigate a low-order robust numerical method for the linear elasticity problem. The method is based on a Bernardi–Raugel-like \(\varvec{H}(\textrm{div})\)-conforming method proposed first for the Stokes flows in [Li and Rui, IMA J. Numer. Anal. 42 (2022) 3711–3734]. Therein, the lowest-order \(\varvec{H}(\textrm{div})\)-conforming Raviart–Thomas space (\(\varvec{RT}_0\)) was added
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Semi-Lagrangian finite element exterior calculus for incompressible flows Adv. Comput. Math. (IF 1.7) Pub Date : 2024-02-05 Wouter Tonnon, Ralf Hiptmair
We develop a semi-Lagrangian discretization of the time-dependent incompressible Navier-Stokes equations with free boundary conditions on arbitrary simplicial meshes. We recast the equations as a nonlinear transport problem for a momentum 1-form and discretize in space using methods from finite element exterior calculus. Numerical experiments show that the linearly implicit fully discrete version of
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Dictionary-based online-adaptive structure-preserving model order reduction for parametric Hamiltonian systems Adv. Comput. Math. (IF 1.7) Pub Date : 2024-02-05 Robin Herkert, Patrick Buchfink, Bernard Haasdonk
Classical 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 n-widths. 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
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On the balanced truncation error bound and sign parameters from arrowhead realizations Adv. Comput. Math. (IF 1.7) Pub Date : 2024-01-31 Sean Reiter, Tobias Damm, Mark Embree, Serkan Gugercin
Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a
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Dual frames compensating for erasures—a non-canonical case Adv. Comput. Math. (IF 1.7) Pub Date : 2024-01-25 Ljiljana Arambašić, Diana Stoeva
In 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
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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 : 2024-01-17 Shixi Wang, Hai Bi, Yidu Yang
The 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
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Variational methods for solving numerically magnetostatic systems Adv. Comput. Math. (IF 1.7) Pub Date : 2024-01-04 Patrick Ciarlet Jr., Erell Jamelot
In 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
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A Grassmann manifold handbook: basic geometry and computational aspects Adv. Comput. Math. (IF 1.7) Pub Date : 2024-01-05 Thomas Bendokat, Ralf Zimmermann, P.-A. Absil
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A fractional osmosis model for image fusion Adv. Comput. Math. (IF 1.7) Pub Date : 2024-01-08 Mohammed Hachama, Fatiha Boutaous
This paper introduces a novel model for image fusion that is based on a fractional-order 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
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Numerical investigation of agent-controlled pedestrian dynamics using a structure-preserving finite volume scheme Adv. Comput. Math. (IF 1.7) Pub Date : 2023-12-28 Jan-Frederik Pietschmann, Ailyn Stötzner, Max Winkler
We 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
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A three-step defect-correction stabilized algorithm for incompressible flows with non-homogeneous Dirichlet boundary conditions Adv. Comput. Math. (IF 1.7) Pub Date : 2023-12-27 Bo Zheng, Yueqiang Shang
Abstract Based on two-grid discretizations and quadratic equal-order finite elements for the velocity and pressure approximations, we develop a three-step defect-correction stabilized algorithm for the incompressible Navier-Stokes equations, where non-homogeneous Dirichlet boundary conditions are considered and high Reynolds numbers are allowed. In this developed algorithm, we first solve an artificial
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Order two superconvergence of the CDG finite elements for non-self adjoint and indefinite elliptic equations Adv. Comput. Math. (IF 1.7) Pub Date : 2023-12-22 Xiu Ye, Shangyou Zhang
A conforming discontinuous Galerkin (CDG) finite element method is designed for solving second order non-self 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
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Matching pursuit with unbounded parameter domains Adv. Comput. Math. (IF 1.7) Pub Date : 2023-12-20 Wei Qu, Yanbo Wang, Xiaoyun Sun
In 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, exhaustion-type 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) sub-domain. Such a process
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Asymptotic convergence analysis and influence of initial guesses on composite Anderson acceleration Adv. Comput. Math. (IF 1.7) Pub Date : 2023-12-13 Kewang Chen, Cornelis Vuik
Although 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 non-stationary version of Anderson acceleration methods remain an open question. Motivated by the hybrid linear solver GMRESR (GMRES Recursive), we recently proposed a set of non-stationary
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Interpolatory model reduction of quadratic-bilinear dynamical systems with quadratic-bilinear outputs Adv. Comput. Math. (IF 1.7) Pub Date : 2023-12-14 Alejandro N. Diaz, Matthias Heinkenschloss, Ion Victor Gosea, Athanasios C. Antoulas
This paper extends interpolatory model reduction to systems with (up to) quadratic-bilinear dynamics and quadratic-bilinear outputs. These systems are referred to as QB-QB 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 Petrov-Galerkin projection, and
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Toward a certified greedy Loewner framework with minimal sampling Adv. Comput. Math. (IF 1.7) Pub Date : 2023-12-05 Davide Pradovera
We propose a strategy for greedy sampling in the context of non-intrusive interpolation-based surrogate modeling for frequency-domain problems. We rely on a non-intrusive and cheap error indicator to drive the adaptive selection of the high-fidelity samples on which the surrogate is based. We develop a theoretical framework to support our proposed indicator. We also present several practical approaches
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Piecewise orthogonal collocation for computing periodic solutions of renewal equations Adv. Comput. Math. (IF 1.7) Pub Date : 2023-12-07 Alessia Andò, Dimitri Breda
We 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.
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Approximation of curve-based sleeve functions in high dimensions Adv. Comput. Math. (IF 1.7) Pub Date : 2023-11-30 Robert Beinert
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A posteriori error analysis and adaptivity for a VEM discretization of the Navier–Stokes equations Adv. Comput. Math. (IF 1.7) Pub Date : 2023-11-30 Claudio Canuto, Davide Rosso
We 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 divergence-free velocities. For such discretization, we develop a residual-based a posteriori error estimator, which is a combination of standard terms in VEM analysis
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Continuous-stage adapted exponential methods for charged-particle dynamics with arbitrary magnetic fields Adv. Comput. Math. (IF 1.7) Pub Date : 2023-11-30 Ting Li, Bin Wang
This paper is devoted to the numerical symplectic approximation of the charged-particle dynamics (CPD) with a homogeneous magnetic field and its extension to a non-homogeneous magnetic field. By utilizing continuous-stage 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
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Live load matrix recovery from scattering data in linear elasticity Adv. Comput. Math. (IF 1.7) Pub Date : 2023-11-28 Juan Antonio Barceló, Carlos Castro, Mari Cruz Vilela
We study the numerical approximation of the inverse scattering problem in the two-dimensional homogeneous isotropic linear elasticity with an unknown linear load given by a square matrix. For both backscattering data and fixed-angle scattering data, we show how to obtain numerical approximations of the so-called Born approximations and propose new iterative algorithms that provide sequences of approximations
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$$\mathcal {H}$$ -inverses for RBF interpolation Adv. Comput. Math. (IF 1.7) Pub Date : 2023-11-23 Niklas Angleitner, Markus Faustmann, Jens Markus Melenk
We 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
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Iterative two-grid methods for discontinuous Galerkin finite element approximations of semilinear elliptic problem Adv. Comput. Math. (IF 1.7) Pub Date : 2023-11-22 Jiajun Zhan, Liuqiang Zhong, Jie Peng
In this paper, we design and analyze the iterative two-grid methods for the discontinuous Galerkin (DG) discretization of semilinear elliptic partial differential equations (PDEs). We first present an iterative two-grid method that is just like the classical iterative two-grid methods for nonsymmetric or indefinite linear elliptic PDEs, namely, to solve a semilinear problem on the coarse space and
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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 : 2023-11-22 Xiaoxin Wu, Weifeng Qiu, Kejia Pan
In this paper, we propose an unconditionally stable and \(L^2\) optimal quadratic finite volume (FV) scheme for solving the two-dimensional 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
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Estimation of quadrature errors for layer potentials evaluated near surfaces with spherical topology Adv. Comput. Math. (IF 1.7) Pub Date : 2023-11-23 Chiara Sorgentone, Anna-Karin Tornberg
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The Butzer-Kozakiewicz article on Riemann derivatives of 1954 and its influence Adv. Comput. Math. (IF 1.7) Pub Date : 2023-11-23 P. L. Butzer, R. L. Stens
The article on Riemann derivatives by P. L. Butzer and W. Kozakiewicz of 1954 was the basis to generalizations of the classical scalar-valued 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
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Max filtering with reflection groups Adv. Comput. Math. (IF 1.7) Pub Date : 2023-11-21 Dustin G. Mixon, Daniel Packer
Given a finite-dimensional 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.
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Multilevel Monte Carlo simulation for the Heston stochastic volatility model Adv. Comput. Math. (IF 1.7) Pub Date : 2023-11-20 Chao Zheng
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Solving the backward problem for time-fractional wave equations by the quasi-reversibility regularization method Adv. Comput. Math. (IF 1.7) Pub Date : 2023-10-25 Jin Wen, Zhi-Yuan Li, Yong-Ping Wang
This paper is devoted to the backward problem of determining the initial value and initial velocity simultaneously in a time-fractional 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 Mittag-Leffler functions provided that the two fixed measurement times are sufficiently close. Since this problem
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A novel robust fractional-time anisotropic diffusion for multi-frame image super-resolution Adv. Comput. Math. (IF 1.7) Pub Date : 2023-10-25 Anouar Ben-loghfyry, Abdelilah Hakim
In this paper, we propose an image multi-frame Super Resolution (SR) method based on fractional-time Caputo derivative combined with Weickert-type 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
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On flexible block Chebyshev-Davidson method for solving symmetric generalized eigenvalue problems Adv. Comput. Math. (IF 1.7) Pub Date : 2023-10-24 Cun-Qiang Miao, Lan Cheng
In a recent work (J. Sci. Comput. 85 (2020), no. 3), the author generalized the Chebyshev-Davidson method appeared in standard eigenvalue problems to symmetric generalized eigenvalue problems. The theoretical derivation indicates that the Chebyshev-Davidson method for symmetric generalized eigenvalue problems only admits local convergence; thus, in this paper, we adopt a flexible strategy to improve
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Mass- and energy-conserving Gauss collocation methods for the nonlinear Schrödinger equation with a wave operator Adv. Comput. Math. (IF 1.7) Pub Date : 2023-10-23 Shu Ma, Jilu Wang, Mingyan Zhang, Zhimin Zhang
A 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 d-dimensional 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
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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 : 2023-10-17 Dhiraj Patel, S. Sivananthan
The 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
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Arbitrary high-order structure-preserving methods for the quantum Zakharov system Adv. Comput. Math. (IF 1.7) Pub Date : 2023-10-17 Gengen Zhang, Chaolong Jiang
In this paper, we present a new methodology to develop arbitrary high-order structure-preserving 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
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A second-order scheme based on blended BDF for the incompressible MHD system Adv. Comput. Math. (IF 1.7) Pub Date : 2023-10-03 Shuaijun Liu, Pengzhan Huang, Yinnian He
For the incompressible MHD equations, we present a fully discrete second-order-in-time scheme based on a blended BDF and extrapolation treatments for nonlinear terms. The proposed scheme is more accurate than the two-step BDF with additional nominal computational time and is still A-stable. Then, unconditional stability, long-time stability, and optimal convergence rate of the scheme are presented
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An efficient approximation to the stochastic Allen-Cahn equation with random diffusion coefficient field and multiplicative noise Adv. Comput. Math. (IF 1.7) Pub Date : 2023-09-27 Xiao Qi, Yanrong Zhang, Chuanju Xu
This paper studies the stochastic Allen-Cahn equation involving random diffusion coefficient field and multiplicative force noise. A new time-stepping 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
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The stability and convergence analysis of finite difference methods for the fractional neutron diffusion equation Adv. Comput. Math. (IF 1.7) Pub Date : 2023-09-14 Daopeng Yin, Yingying Xie, Liquan Mei
For the time-fractional 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 L1-type 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
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Interaction with an obstacle in the 2D focusing nonlinear Schrödinger equation Adv. Comput. Math. (IF 1.7) Pub Date : 2023-09-13 Oussama Landoulsi, Svetlana Roudenko, Kai Yang
We 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
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Calibration of P-values for calibration and for deviation of a subpopulation from the full population Adv. Comput. Math. (IF 1.7) Pub Date : 2023-09-04 Mark Tygert
The 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 open-access Journal of Big Data during 2021), propose graphical methods and summary statistics, without extensively calibrating formal significance tests. The summary metrics and
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Convergence of non-stationary semi-discrete RBF schemes for the heat and wave equation Adv. Comput. Math. (IF 1.7) Pub Date : 2023-08-31 Raymond Brummelhuis
We 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 non-stationary 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 \(\nu > n + 2\).
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A posteriori error estimates for the time-dependent Navier-Stokes system coupled with the convection-diffusion-reaction equation Adv. Comput. Math. (IF 1.7) Pub Date : 2023-08-14 Jad Dakroub, Joanna Faddoul, Pascal Omnes, Toni Sayah
In this paper we study the a posteriori error estimates for the time dependent Navier-Stokes system coupled with the convection-diffusion-reaction 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
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Simpler is better: a comparative study of randomized pivoting algorithms for CUR and interpolative decompositions Adv. Comput. Math. (IF 1.7) Pub Date : 2023-08-07 Yijun Dong, Per-Gunnar Martinsson
Matrix skeletonizations like the interpolative and CUR decompositions provide a framework for low-rank 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 low-rank decompositions with orthonormal bases, including
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Splitting scheme for backward doubly stochastic differential equations Adv. Comput. Math. (IF 1.7) Pub Date : 2023-08-03 Feng Bao, Yanzhao Cao, He Zhang
A 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 first-order finite difference
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Numerical analysis of nonlinear degenerate parabolic problems with application to eddy current models Adv. Comput. Math. (IF 1.7) Pub Date : 2023-08-02 Ramiro Acevedo, Christian Gómez, Paulo Navia
This 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 quasi-optimal error estimates for the approximation. Finally,
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A fast time domain solver for the equilibrium Dyson equation Adv. Comput. Math. (IF 1.7) Pub Date : 2023-08-02 Jason Kaye, Hugo U. R. Strand
We consider the numerical solution of the real-time equilibrium Dyson equation, which is used in calculations of the dynamical properties of quantum many-body systems. We show that this equation can be written as a system of coupled, nonlinear, convolutional Volterra integro-differential equations, for which the kernel depends self-consistently on the solution. As is typical in the numerical solution
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Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations Adv. Comput. Math. (IF 1.7) Pub Date : 2023-07-31 Ben Moseley, Andrew Markham, Tarje Nissen-Meyer
Recently, physics-informed 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 mesh-free solutions of differential equations and their ability to carry out forward and inverse modelling within the same optimisation problem. Whilst
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Interior penalty discontinuous Galerkin methods for the velocity-pressure formulation of the Stokes spectral problem Adv. Comput. Math. (IF 1.7) Pub Date : 2023-07-27 Felipe Lepe
In 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
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High-order corrected trapezoidal rules for a class of singular integrals Adv. Comput. Math. (IF 1.7) Pub Date : 2023-07-26 Federico Izzo, Olof Runborg, Richard Tsai
We present a family of high-order trapezoidal rule-based 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
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Adaptive finite element approximation of optimal control problems with the integral fractional Laplacian Adv. Comput. Math. (IF 1.7) Pub Date : 2023-07-24 Zhou Zhaojie, Wang Qiming
In 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
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Adaptive and local regularization for data fitting by tensor-product spline surfaces Adv. Comput. Math. (IF 1.7) Pub Date : 2023-07-24 Sandra Merchel, Bert Jüttler, Dominik Mokriš
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Full recovery from point values: an optimal algorithm for Chebyshev approximability prior Adv. Comput. Math. (IF 1.7) Pub Date : 2023-07-24 Simon Foucart
Given 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 worst-case 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
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An adaptive discontinuous finite volume element method for the Allen-Cahn equation Adv. Comput. Math. (IF 1.7) Pub Date : 2023-07-21 Jian Li, Jiyao Zeng, Rui Li
In this paper, the discontinuous finite volume element method (DFVEM) is considered to solve the Allen-Cahn 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 semi-discrete and fully discrete scheme are
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Phase retrieval and system identification in dynamical sampling via Prony’s method Adv. Comput. Math. (IF 1.7) Pub Date : 2023-07-21 Robert Beinert, Marzieh Hasannasab
Phase 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 well-posed dynamical sampling into a severe ill-posed inverse problem. In the existing literature
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A posteriori error estimates for wave maps into spheres Adv. Comput. Math. (IF 1.7) Pub Date : 2023-07-17 Jan Giesselmann, Elena Mäder-Baumdicker, David Jakob Stonner
We provide a posteriori error estimates in the energy norm for temporal semi-discretisations 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 time-adaptive numerical simulations based on the a posteriori error estimators for