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Computing eigenvalues of quasi-rational Said–Ball–Vandermonde matrices Adv. Comput. Math. (IF 1.7) Pub Date : 2024-08-22 Xiaoxiao Ma, Yingqing Xiao
This paper focuses on computing the eigenvalues of the generalized collocation matrix of the rational Said–Ball basis, also called as the quasi-rational Said–Ball–Vandermonde (q-RSBV) matrix, with high relative accuracy. To achieve this, we propose explicit expressions for the minors of the q-RSBV matrix and develop a high-precision algorithm to compute these parameters. Additionally, we present perturbation
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Morley type virtual element method for von Kármán equations Adv. Comput. Math. (IF 1.7) Pub Date : 2024-08-22 Devika Shylaja, Sarvesh Kumar
This paper analyses the nonconforming Morley type virtual element method to approximate a regular solution to the von Kármán equations that describes bending of very thin elastic plates. Local existence and uniqueness of a discrete solution to the non-linear problem is discussed. A priori error estimate in the energy norm is established under minimal regularity assumptions on the exact solution. Error
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Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems Adv. Comput. Math. (IF 1.7) Pub Date : 2024-08-19 Bernard Kapidani, Melina Merkel, Sebastian Schöps, Rafael Vázquez
Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a
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Convergence analysis of the Dirichlet-Neumann Waveform Relaxation algorithm for time fractional sub-diffusion and diffusion-wave equations in heterogeneous media Adv. Comput. Math. (IF 1.7) Pub Date : 2024-08-14 Soura Sana, Bankim C Mandal
This article presents a comprehensive study on the convergence behavior of the Dirichlet-Neumann Waveform Relaxation algorithm applied to solve the time fractional sub-diffusion and diffusion-wave equations in multiple subdomains, considering the presence of some heterogeneous media. Our analysis focuses on estimating the convergence rate of the algorithm and investigates how this estimate varies with
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Frame-normalizable sequences Adv. Comput. Math. (IF 1.7) Pub Date : 2024-08-09 Pu-Ting Yu
Let H be a separable Hilbert space and let \(\{x_{n}\}\) be a sequence in H that does not contain any zero elements. We say that \(\{x_{n}\}\) is a Bessel-normalizable or frame-normalizable sequence if the normalized sequence \({\bigl \{\frac{x_n}{\Vert x_n\Vert }\bigr \}}\) is a Bessel sequence or a frame for H, respectively. In this paper, several necessary and sufficient conditions for sequences
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SlabLU: a two-level sparse direct solver for elliptic PDEs Adv. Comput. Math. (IF 1.7) Pub Date : 2024-08-09 Anna Yesypenko, Per-Gunnar Martinsson
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Analysis of a WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data Adv. Comput. Math. (IF 1.7) Pub Date : 2024-08-08 Liyao Hao, Wenyi Tian
In this paper, we propose and analyze a second-order time-stepping numerical scheme for the inhomogeneous backward fractional Feynman-Kac equation with nonsmooth initial data. The complex parameters and time-space coupled Riemann-Liouville fractional substantial integral and derivative in the equation bring challenges on numerical analysis and computations. The nonlocal operators are approximated by
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Balanced truncation for quadratic-bilinear control systems Adv. Comput. Math. (IF 1.7) Pub Date : 2024-08-08 Peter Benner, Pawan Goyal
We discuss model order reduction (MOR) for large-scale quadratic-bilinear (QB) systems based on balanced truncation. The method for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. These Gramians are extended to a general nonlinear setting in Scherpen (Systems Control Lett. 21, 143-153 1993). These formulations of Gramians
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Weights for moments’ geometrical localization: a canonical isomorphism Adv. Comput. Math. (IF 1.7) Pub Date : 2024-08-06 Ana Alonso Rodríguez, Jessika Camaño, Eduardo De Los Santos, Francesca Rapetti
This paper deals with high order Whitney forms. We define a canonical isomorphism between two sets of degrees of freedom. This allows to geometrically localize the classical degrees of freedom, the moments, over the elements of a simplicial mesh. With such a localization, it is thus possible to associate, even with moments, a graph structure relating a field with its potential.
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Online identification and control of PDEs via reinforcement learning methods Adv. Comput. Math. (IF 1.7) Pub Date : 2024-08-01 Alessandro Alla, Agnese Pacifico, Michele Palladino, Andrea Pesare
We focus on the control of unknown partial differential equations (PDEs). The system dynamics is unknown, but we assume we are able to observe its evolution for a given control input, as typical in a reinforcement learning framework. We propose an algorithm based on the idea to control and identify on the fly the unknown system configuration. In this work, the control is based on the state-dependent
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Two-grid stabilized finite element methods with backtracking for the stationary Navier-Stokes equations Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-30 Jing Han, Guangzhi Du
Based on local Gauss integral technique and backtracking technique, this paper presents and studies three kinds of two-grid stabilized finite element algorithms for the stationary Navier-Stokes equations. The proposed methods consist of deducing a coarse solution on the nonlinear system, updating the solution on a fine mesh via three different methods, and solving a linear correction problem on the
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Averaging property of wedge product and naturality in discrete exterior calculus Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-31 Mark D. Schubel, Daniel Berwick-Evans, Anil N. Hirani
In exterior calculus on smooth manifolds, the exterior derivative and wedge products are natural with respect to smooth maps between manifolds, that is, these operations commute with pullback. In discrete exterior calculus (DEC), simplicial cochains play the role of discrete forms, the coboundary operator serves as the discrete exterior derivative, and an antisymmetrized cup-like product provides a
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Analysis of the leapfrog-Verlet method applied to the Kuwabara-Kono force model in discrete element method simulations of granular materials Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-23 Gabriel Nóbrega Bufolo, Yuri Dumaresq Sobral
The discrete element method (DEM) is a numerical technique widely used to simulate granular materials. The temporal evolution of these simulations is often performed using a Verlet-type algorithm, because of its second order and its desirable property of better energy conservation. However, when dissipative forces are considered in the model, such as the nonlinear Kuwabara-Kono model, the Verlet method
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The $$L_q$$ -weighted dual programming of the linear Chebyshev approximation and an interior-point method Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-22 Yang Linyi, Zhang Lei-Hong, Zhang Ya-Nan
Given samples of a real or complex-valued function on a set of distinct nodes, the traditional linear Chebyshev approximation is to compute the minimax approximation on a prescribed linear functional space. Lawson’s iteration is a classical and well-known method for the task. However, Lawson’s iteration converges only linearly and in many cases, the convergence is very slow. In this paper, relying
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Randomized greedy magic point selection schemes for nonlinear model reduction Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-22 Ralf Zimmermann, Kai Cheng
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Adaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-19 Quirin Aumann, Steffen W. R. Werner
Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative
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On Krylov subspace methods for skew-symmetric and shifted skew-symmetric linear systems Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-19 Kui Du, Jia-Jun Fan, Xiao-Hui Sun, Fang Wang, Ya-Lan Zhang
Krylov subspace methods for solving linear systems of equations involving skew-symmetric matrices have gained recent attention. Numerical equivalences among Krylov subspace methods for nonsingular skew-symmetric linear systems have been given in Greif et al. [SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1071–1087]. In this work, we extend the results of Greif et al. to singular skew-symmetric linear
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A continuation method for fitting a bandlimited curve to points in the plane Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-16 Mohan Zhao, Kirill Serkh
In this paper, we describe an algorithm for fitting an analytic and bandlimited closed or open curve to interpolate an arbitrary collection of points in \(\mathbb {R}^{2}\). The main idea is to smooth the parametrization of the curve by iteratively filtering the Fourier or Chebyshev coefficients of both the derivative of the arc-length function and the tangential angle of the curve and applying smooth
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Augmented Lagrangian method for tensor low-rank and sparsity models in multi-dimensional image recovery Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-16 Hong Zhu, Xiaoxia Liu, Lin Huang, Zhaosong Lu, Jian Lu, Michael K. Ng
Multi-dimensional images can be viewed as tensors and have often embedded a low-rankness property that can be evaluated by tensor low-rank measures. In this paper, we first introduce a tensor low-rank and sparsity measure and then propose low-rank and sparsity models for tensor completion, tensor robust principal component analysis, and tensor denoising. The resulting tensor recovery models are further
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Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: a study using POD-Galerkin and dynamical low-rank approximation Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-16 Julian Koellermeier, Philipp Krah, Jonas Kusch
Geophysical flow simulations using hyperbolic shallow water moment equations require an efficient discretization of a potentially large system of PDEs, the so-called moment system. This calls for tailored model order reduction techniques that allow for efficient and accurate simulations while guaranteeing physical properties like mass conservation. In this paper, we develop the first model reduction
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Finding roots of complex analytic functions via generalized colleague matrices Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-15 H. Zhang, V. Rokhlin
We present a scheme for finding all roots of an analytic function in a square domain in the complex plane. The scheme can be viewed as a generalization of the classical approach to finding roots of a function on the real line, by first approximating it by a polynomial in the Chebyshev basis, followed by diagonalizing the so-called “colleague matrices.” Our extension of the classical approach is based
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Neural and spectral operator surrogates: unified construction and expression rate bounds Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-15 Lukas Herrmann, Christoph Schwab, Jakob Zech
Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising, e.g., as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for deep neural operator and generalized polynomial chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable
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Numerical analysis of a time discretized method for nonlinear filtering problem with Lévy process observations Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-15 Fengshan Zhang, Yongkui Zou, Shimin Chai, Yanzhao Cao
In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is an unnormalized probability density function of the filter solution. Then, we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up
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A sparse spectral method for fractional differential equations in one-spatial dimension Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-10 Ioannis P. A. Papadopoulos, Sheehan Olver
We develop a sparse spectral method for a class of fractional differential equations, posed on \(\mathbb {R}\), in one dimension. These equations may include sqrt-Laplacian, Hilbert, derivative, and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the second kind in conjunction with their Hilbert transforms. The former functions are supported on
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Pairwise ranking with Gaussian kernel Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-10 Guanhang Lei, Lei Shi
Regularized pairwise ranking with Gaussian kernels is one of the cutting-edge learning algorithms. Despite a wide range of applications, a rigorous theoretical demonstration still lacks to support the performance of such ranking estimators. This work aims to fill this gap by developing novel oracle inequalities for regularized pairwise ranking. With the help of these oracle inequalities, we derive
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Topological phase estimation method for reparameterized periodic functions Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-08 Thomas Bonis, Frédéric Chazal, Bertrand Michel, Wojciech Reise
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An adaptive finite element DtN method for the acoustic-elastic interaction problem Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-08 Lei Lin, Junliang Lv, Shuxin Li
Consider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable and isotropic elastic solid, which is immersed in a homogeneous compressible air/fluid. By the Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition is introduced and the model is formulated as a boundary value problem of acoustic-elastic interaction. Based on a duality argument technique
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Estimates for coefficients in Jacobi series for functions with limited regularity by fractional calculus Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-08 Guidong Liu, Wenjie Liu, Beiping Duan
In this paper, optimal estimates on the decaying rates of Jacobi expansion coefficients are obtained by fractional calculus for functions with algebraic and logarithmic singularities. This is inspired by the fact that integer-order derivatives fail to deal with singularity of fractional-type, while fractional calculus can. To this end, we first introduce new fractional Sobolev spaces defined as the
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On an accurate numerical integration for the triangular and tetrahedral spectral finite elements Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-03 Ziqing Xie, Shangyou Zhang
In the triangular/tetrahedral spectral finite elements, we apply a bilinear/trilinear transformation to map a reference square/cube to a triangle/tetrahedron, which consequently maps the \(\varvec{Q_k}\) polynomial space on the reference element to a finite element space of rational/algebraic functions on the triangle/tetrahedron. We prove that the resulting finite element space, even under this singular
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An adaptive time-stepping Fourier pseudo-spectral method for the Zakharov-Rubenchik equation Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-03 Bingquan Ji, Xuanxuan Zhou
An adaptive time-stepping scheme is developed for the Zakharov-Rubenchik system to resolve the multiple time scales accurately and to improve the computational efficiency during long-time simulations. The Crank-Nicolson formula and the Fourier pseudo-spectral method are respectively utilized for the temporal and spatial approximations. The proposed numerical method is proved to preserve the mass and
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Further analysis of multilevel Stein variational gradient descent with an application to the Bayesian inference of glacier ice models Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-03 Terrence Alsup, Tucker Hartland, Benjamin Peherstorfer, Noemi Petra
Multilevel Stein variational gradient descent is a method for particle-based variational inference that leverages hierarchies of surrogate target distributions with varying costs and fidelity to computationally speed up inference. The contribution of this work is twofold. First, an extension of a previous cost complexity analysis is presented that applies even when the exponential convergence rate
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Extrapolated regularization of nearly singular integrals on surfaces Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-01 J. Thomas Beale, Svetlana Tlupova
We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when one surface is close to another or to obtain values at grid points. We replace the singular kernel with a regularized version having a length parameter
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Convergence of projected subgradient method with sparse or low-rank constraints Adv. Comput. Math. (IF 1.7) Pub Date : 2024-07-02 Hang Xu, Song Li, Junhong Lin
Many problems in data science can be treated as recovering structural signals from a set of linear measurements, sometimes perturbed by dense noise or sparse corruptions. In this paper, we develop a unified framework of considering a nonsmooth formulation with sparse or low-rank constraint for meeting the challenges of mixed noises—bounded noise and sparse noise. We show that the nonsmooth formulations
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Stochastic modeling of stationary scalar Gaussian processes in continuous time from autocorrelation data Adv. Comput. Math. (IF 1.7) Pub Date : 2024-06-24 Martin Hanke
We consider the problem of constructing a vector-valued linear Markov process in continuous time, such that its first coordinate is in good agreement with given samples of the scalar autocorrelation function of an otherwise unknown stationary Gaussian process. This problem has intimate connections to the computation of a passive reduced model of a deterministic time-invariant linear system from given
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On relaxed inertial projection and contraction algorithms for solving monotone inclusion problems Adv. Comput. Math. (IF 1.7) Pub Date : 2024-06-18 Bing Tan, Xiaolong Qin
We present three novel algorithms based on the forward-backward splitting technique for the solution of monotone inclusion problems in real Hilbert spaces. The proposed algorithms work adaptively in the absence of the Lipschitz constant of the single-valued operator involved thanks to the fact that there is a non-monotonic step size criterion used. The weak and strong convergence and the R-linear convergence
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An efficient rotational pressure-correction schemes for 2D/3D closed-loop geothermal system Adv. Comput. Math. (IF 1.7) Pub Date : 2024-06-10 Jian Li, Jiawei Gao, Yi Qin
In this paper, the rotational pressure-correction schemes for the closed-loop geothermal system are developed and analyzed. The primary benefit of this projection method is to replace the incompressible condition. The system is considered consisting of two distinct regions, with the free flow region governed by the Navier–Stokes equations and the porous media region governed by Darcy’s law. At the
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Numerical methods for forward fractional Feynman–Kac equation Adv. Comput. Math. (IF 1.7) Pub Date : 2024-06-10 Daxin Nie, Jing Sun, Weihua Deng
Fractional Feynman–Kac equation governs the functional distribution of the trajectories of anomalous diffusion. The non-commutativity of the integral fractional Laplacian and time-space coupled fractional substantial derivative, i.e., \(\mathcal {A}^{s}{}_{0}\partial _{t}^{1-\alpha ,x}\ne {}_{0}\partial _{t}^{1-\alpha ,x}\mathcal {A}^{s}\), brings about huge challenges on the regularity and spatial
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Semi-active damping optimization of vibrational systems using the reduced basis method Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-31 Jennifer Przybilla, Igor Pontes Duff, Peter Benner
In this article, we consider vibrational systems with semi-active damping that are described by a second-order model. In order to minimize the influence of external inputs to the system response, we are optimizing some damping values. As minimization criterion, we evaluate the energy response, that is the \(\mathcal {H}_2\)-norm of the corresponding transfer function of the system. Computing the energy
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A rotational pressure-correction discontinuous Galerkin scheme for the Cahn-Hilliard-Darcy-Stokes system Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-30 Meiting Wang, Guang-an Zou, Jian Li
This paper is devoted to the numerical approximations of the Cahn-Hilliard-Darcy-Stokes system, which is a combination of the modified Cahn-Hilliard equation with the Darcy-Stokes equation. A novel discontinuous Galerkin pressure-correction scheme is proposed for solving the coupled system, which can achieve the desired level of linear, fully decoupled, and unconditionally energy stable. The developed
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Approximation in the extended functional tensor train format Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-28 Christoph Strössner, Bonan Sun, Daniel Kressner
This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, the adaptivity of
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An unfitted finite element method with direct extension stabilization for time-harmonic Maxwell problems on smooth domains Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-22 Fanyi Yang, Xiaoping Xie
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An optimal control framework for adaptive neural ODEs Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-23 Joubine Aghili, Olga Mula
In recent years, the notion of neural ODEs has connected deep learning with the field of ODEs and optimal control. In this setting, neural networks are defined as the mapping induced by the corresponding time-discretization scheme of a given ODE. The learning task consists in finding the ODE parameters as the optimal values of a sampled loss minimization problem. In the limit of infinite time steps
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Variable transformations in combination with wavelets and ANOVA for high-dimensional approximation Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-23 Daniel Potts, Laura Weidensager
We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low-dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis on the torus. With a variable transformation, we are able to transform the approximation rates and fast algorithms from the torus to other domains. We perform
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Error analysis of a collocation method on graded meshes for a fractional Laplacian problem Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-20 Minghua Chen, Weihua Deng, Chao Min, Jiankang Shi, Martin Stynes
The numerical solution of a 1D fractional Laplacian boundary value problem is studied. Although the fractional Laplacian is one of the most important and prominent nonlocal operators, its numerical analysis is challenging, partly because the problem’s solution has in general a weak singularity at the boundary of the domain. To solve the problem numerically, we use piecewise linear collocation on a
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A sparse approximation for fractional Fourier transform Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-20 Fang Yang, Jiecheng Chen, Tao Qian, Jiman Zhao
The paper promotes a new sparse approximation for fractional Fourier transform, which is based on adaptive Fourier decomposition in Hardy-Hilbert space on the upper half-plane. Under this methodology, the local polynomial Fourier transform characterization of Hardy space is established, which is an analog of the Paley-Wiener theorem. Meanwhile, a sparse fractional Fourier series for chirp \( L^2 \)
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An adaptive certified space-time reduced basis method for nonsmooth parabolic partial differential equations Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-15 Marco Bernreuther, Stefan Volkwein
In this paper, a nonsmooth semilinear parabolic partial differential equation (PDE) is considered. For a reduced basis (RB) approach, a space-time formulation is used to develop a certified a-posteriori error estimator. This error estimator is adopted to the presence of the discrete empirical interpolation method (DEIM) as approximation technique for the nonsmoothness. The separability of the estimated
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A Lagrangian approach for solving an axisymmetric thermo-electromagnetic problem. Application to time-varying geometry processes Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-08 Marta Benítez, Alfredo Bermúdez, Pedro Fontán, Iván Martínez, Pilar Salgado
The aim of this work is to introduce a thermo-electromagnetic model for calculating the temperature and the power dissipated in cylindrical pieces whose geometry varies with time and undergoes large deformations; the motion will be a known data. The work will be a first step towards building a complete thermo-electromagnetic-mechanical model suitable for simulating electrically assisted forming processes
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Optimally convergent mixed finite element methods for the time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-08 Xinyue Gao, Yi Qin, Jian Li
In this paper, a new time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise is developed and studied. This model considers heat transfer between the free flow in the pipe region and the porous media flow in the porous media region. Darcy’s law and stochastic Navier-Stokes equations are used to control the flows in the pipe and porous media regions, respectively. The
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Local behaviors of Fourier expansions for functions of limited regularities Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-09 Shunfeng Yang, Shuhuang Xiang
Based on the explicit formula of the pointwise error of Fourier projection approximation and by applying van der Corput-type Lemma, optimal convergence rates for periodic functions with different degrees of smoothness are established. It shows that the convergence rate enjoys a decay rate one order higher in the smooth parts than that at the singularities. In addition, it also depends on the distance
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On the maximum principle and high-order, delay-free integrators for the viscous Cahn–Hilliard equation Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-03 Hong Zhang, Gengen Zhang, Ziyuan Liu, Xu Qian, Songhe Song
The stabilization approach has been known to permit large time-step sizes while maintaining stability. However, it may “slow down the convergence rate” or cause “delayed convergence” if the time-step rescaling is not well resolved. By considering a fourth-order-in-space viscous Cahn–Hilliard (VCH) equation, we propose a class of up to the fourth-order single-step methods that are able to capture the
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Dominant subspaces of high-fidelity polynomial structured parametric dynamical systems and model reduction Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-03 Pawan Goyal, Igor Pontes Duff, Peter Benner
In this work, we investigate a model order reduction scheme for high-fidelity nonlinear structured parametric dynamical systems. More specifically, we consider a class of nonlinear dynamical systems whose nonlinear terms are polynomial functions, and the linear part corresponds to a linear structured model, such as second-order, time-delay, or fractional-order systems. Our approach relies on the Volterra
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Unconditional superconvergence analysis of a structure-preserving finite element method for the Poisson-Nernst-Planck equations Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-06 Huaijun Yang, Meng Li
In this paper, a linearized structure-preserving Galerkin finite element method is investigated for Poisson-Nernst-Planck (PNP) equations. By making full use of the high accuracy estimation of the bilinear element, the mean value technique and rigorously dealing with the coupled nonlinear term, not only the unconditionally optimal error estimate in \(L^2\)-norm but also the unconditionally superclose
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Stray field computation by inverted finite elements: a new method in micromagnetic simulations Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-07 Tahar Z. Boulmezaoud, Keltoum Kaliche
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Inverse problem for determining free parameters of a reduced turbulent transport model for tokamak plasma Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-02 Louis Lamérand, Didier Auroux, Philippe Ghendrih, Francesca Rapetti, Eric Serre
Two-dimensional transport codes for the simulation of tokamak plasma are reduced version of full 3D fluid models where plasma turbulence has been smoothed out by averaging. One of the main issues nowadays in such reduced models is the accurate modelling of transverse transport fluxes resulting from the averaging of stresses due to fluctuations. Transverse fluxes are assumed driven by local gradients
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Computation of Laplacian eigenvalues of two-dimensional shapes with dihedral symmetry Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-02 David Berghaus, Robert Stephen Jones, Hartmut Monien, Danylo Radchenko
We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues \(\lambda (n)\) of shapes with n edges that are of the form \(\lambda (n) \sim x\sum _{k=0}^{\infty
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Fast numerical integration of highly oscillatory Bessel transforms with a Cauchy type singular point and exotic oscillators Adv. Comput. Math. (IF 1.7) Pub Date : 2024-05-02 Hongchao Kang, Qi Xu, Guidong Liu
In this article, we propose an efficient hybrid method to calculate the highly oscillatory Bessel integral \(\int _{0}^{1} \frac{f(x)}{x-\tau } J_{m} (\omega x^{\gamma } )\textrm{d}x\) with the Cauchy type singular point, where \( 0< \tau < 1, m \ge 0, 2\gamma \in N^{+}. \) The hybrid method is established by combining the complex integration method with the Clenshaw– Curtis– Filon– type method. Based
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Spatial best linear unbiased prediction: a computational mathematics approach for high dimensional massive datasets Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-30 Julio Enrique Castrillón-Candás
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Hermite kernel surrogates for the value function of high-dimensional nonlinear optimal control problems Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-29 Tobias Ehring, Bernard Haasdonk
Numerical methods for the optimal feedback control of high-dimensional dynamical systems typically suffer from the curse of dimensionality. In the current presentation, we devise a mesh-free data-based approximation method for the value function of optimal control problems, which partially mitigates the dimensionality problem. The method is based on a greedy Hermite kernel interpolation scheme and
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Stability analysis for electromagnetic waveguides. Part 2: non-homogeneous waveguides Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-29 Leszek Demkowicz, Jens M. Melenk, Jacob Badger, Stefan Henneking
This paper is a continuation of Melenk et al., “Stability analysis for electromagnetic waveguides. Part 1: acoustic and homogeneous electromagnetic waveguides” (2023) [5], extending the stability results for homogeneous electromagnetic (EM) waveguides to the non-homogeneous case. The analysis is done using perturbation techniques for self-adjoint operators eigenproblems. We show that the non-homogeneous
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Energy stable and maximum bound principle preserving schemes for the Allen-Cahn equation based on the Saul’yev methods Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-29 Xuelong Gu, Yushun Wang, Wenjun Cai
The energy dissipation law and maximum bound principle are significant characteristics of the Allen-Chan equation. To preserve discrete counterpart of these properties, the linear part of the target system is usually discretized implicitly, resulting in a large linear or nonlinear system of equations. The fast Fourier transform is commonly used to solve the resulting linear or nonlinear systems with