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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
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Dictionary-based model reduction for state estimation Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-24 Anthony Nouy, Alexandre Pasco
We consider the problem of state estimation from a few linear measurements, where the state to recover is an element of the manifold \(\mathcal {M}\) of solutions of a parameter-dependent equation. The state is estimated using prior knowledge on \(\mathcal {M}\) coming from model order reduction. Variational approaches based on linear approximation of \(\mathcal {M}\), such as PBDW, yield a recovery
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Error estimates for POD-DL-ROMs: a deep learning framework for reduced order modeling of nonlinear parametrized PDEs enhanced by proper orthogonal decomposition Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-24 Simone Brivio, Stefania Fresca, Nicola Rares Franco, Andrea Manzoni
POD-DL-ROMs have been recently proposed as an extremely versatile strategy to build accurate and reliable reduced order models (ROMs) for nonlinear parametrized partial differential equations, combining (i) a preliminary dimensionality reduction obtained through proper orthogonal decomposition (POD) for the sake of efficiency, (ii) an autoencoder architecture that further reduces the dimensionality
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Artificial neural networks with uniform norm-based loss functions Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-23 Vinesha Peiris, Vera Roshchina, Nadezda Sukhorukova
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The improvement of the truncated Euler-Maruyama method for non-Lipschitz stochastic differential equations Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-22 Weijun Zhan, Yuyuan Li
This paper is concerned with the numerical approximations for stochastic differential equations with non-Lipschitz drift or diffusion coefficients. A modified truncated Euler-Maruyama discretization scheme is developed. Moreover, by establishing the criteria on stochastic C-stability and B-consistency of the truncated Euler-Maruyama method, we obtain the strong convergence and the convergence rate
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The adjoint double layer potential on smooth surfaces in $$\mathbb {R}^3$$ and the Neumann problem Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-19 J. Thomas Beale, Michael Storm, Svetlana Tlupova
We present a simple yet accurate method to compute the adjoint double layer potential, which is used to solve the Neumann boundary value problem for Laplace’s equation in three dimensions. An expansion in curvilinear coordinates leads us to modify the expression for the adjoint double layer so that the singularity is reduced when evaluating the integral on the surface. Then, to regularize the integral
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$$\mathcal {H}_2$$ optimal rational approximation on general domains Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-18 Alessandro Borghi, Tobias Breiten
Optimal model reduction for large-scale linear dynamical systems is studied. In contrast to most existing works, the systems under consideration are not required to be stable, neither in discrete nor in continuous time. As a consequence, the underlying rational transfer functions are allowed to have poles in general domains in the complex plane. In particular, this covers the case of specific conservative
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Computing equivariant matrices on homogeneous spaces for geometric deep learning and automorphic Lie algebras Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-11 Vincent Knibbeler
We develop an elementary method to compute spaces of equivariant maps from a homogeneous space G/H of a Lie group G to a module of this group. The Lie group is not required to be compact. More generally, we study spaces of invariant sections in homogeneous vector bundles, and take a special interest in the case where the fibres are algebras. These latter cases have a natural global algebra structure
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Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systems Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-11 Ion Victor Gosea, Serkan Gugercin, Steffen W. R. Werner
An essential tool in data-driven modeling of dynamical systems from frequency response measurements is the barycentric form of the underlying rational transfer function. In this work, we propose structured barycentric forms for modeling dynamical systems with second-order time derivatives using their frequency domain input-output data. By imposing a set of interpolation conditions, the systems’ transfer
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Numerical simulation of resistance furnaces by using distributed and lumped models Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-10 A. Bermúdez, D. Gómez, D. González
This work proposes a methodology that combines distributed and lumped models to simulate the current distribution within an indirect heat resistance furnace and, in particular, to calculate the current to be supplied for achieving a desired power output. The distributed model is a time-harmonic eddy current problem, which is solved numerically using the finite element method. The lumped model relies
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Robust space-time finite element methods for parabolic distributed optimal control problems with energy regularization Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-10 Ulrich Langer, Olaf Steinbach, Huidong Yang
As in our previous work (SINUM 59(2):660–674, 2021) we consider space-time tracking optimal control problems for linear parabolic initial boundary value problems that are given in the space-time cylinder \(Q = \Omega \times (0,T)\), and that are controlled by the right-hand side \(z_\varrho \) from the Bochner space \(L^2(0,T;H^{-1}(\Omega ))\). So it is natural to replace the usual \(L^2(Q)\) norm
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Robust low tubal rank tensor recovery using discrete empirical interpolation method with optimized slice/feature selection Adv. Comput. Math. (IF 1.7) Pub Date : 2024-04-06 Salman Ahmadi-Asl, Anh-Huy Phan, Cesar F. Caiafa, Andrzej Cichocki
In this paper, we extend the Discrete Empirical Interpolation Method (DEIM) to the third-order tensor case based on the t-product and use it to select important/significant lateral and horizontal slices/features. The proposed Tubal DEIM (TDEIM) is investigated both theoretically and numerically. In particular, the details of the error bounds of the proposed TDEIM method are derived. The experimental
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A block-randomized stochastic method with importance sampling for CP tensor decomposition Adv. Comput. Math. (IF 1.7) Pub Date : 2024-03-25 Yajie Yu, Hanyu Li
One popular way to compute the CANDECOMP/PARAFAC (CP) decomposition of a tensor is to transform the problem into a sequence of overdetermined least squares subproblems with Khatri-Rao product (KRP) structure involving factor matrices. In this work, based on choosing the factor matrix randomly, we propose a mini-batch stochastic gradient descent method with importance sampling for those special least