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An adaptive heavy ball method for ill-posed inverse problems arXiv.cs.NA Pub Date : 2024-04-04 Qinian Jin, Qin Huang
In this paper we consider ill-posed inverse problems, both linear and nonlinear, by a heavy ball method in which a strongly convex regularization function is incorporated to detect the feature of the sought solution. We develop ideas on how to adaptively choose the step-sizes and the momentum coefficients to achieve acceleration over the Landweber-type method. We then analyze the method and establish
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Conditional Pseudo-Reversible Normalizing Flow for Surrogate Modeling in Quantifying Uncertainty Propagation arXiv.cs.NA Pub Date : 2024-03-31 Minglei Yang, Pengjun Wang, Ming Fan, Dan Lu, Yanzhao Cao, Guannan Zhang
We introduce a conditional pseudo-reversible normalizing flow for constructing surrogate models of a physical model polluted by additive noise to efficiently quantify forward and inverse uncertainty propagation. Existing surrogate modeling approaches usually focus on approximating the deterministic component of physical model. However, this strategy necessitates knowledge of noise and resorts to auxiliary
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Robust Numerical Algebraic Geometry arXiv.cs.NA Pub Date : 2024-03-27 Emma R. Cobian, Jonathan D. Hauenstein, Charles W. Wampler
The field of numerical algebraic geometry consists of algorithms for numerically solving systems of polynomial equations. When the system is exact, such as having rational coefficients, the solution set is well-defined. However, for a member of a parameterized family of polynomial systems where the parameter values may be measured with imprecision or arise from prior numerical computations, uncertainty
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An exactly curl-free finite-volume scheme for a hyperbolic compressible barotropic two-phase model arXiv.cs.NA Pub Date : 2024-03-27 Laura Río-Martín, Firas Dhaouadi, Michael Dumbser
We present a new second order accurate structure-preserving finite volume scheme for the solution of the compressible barotropic two-phase model of Romenski et. al in multiple space dimensions. The governing equations fall into the wider class of symmetric hyperbolic and thermodynamically compatible (SHTC) systems and consist of a set of first-order hyperbolic partial differential equations (PDE).
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Convergence rates under a range invariance condition with application to electrical impedance tomography arXiv.cs.NA Pub Date : 2024-03-27 Barbara Kaltenbacher
This paper is devoted to proving convergence rates of variational and iterative regularization methods under variational source conditions VSCs for inverse problems whose linearization satisfies a range invariance condition. In order to achieve this, often an appropriate relaxation of the problem needs to be found that is usually based on an augmentation of the set of unknowns and leads to a particularly
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Improving Efficiency of Parallel Across the Method Spectral Deferred Corrections arXiv.cs.NA Pub Date : 2024-03-27 Gayatri Čaklović, Thibaut Lunet, Sebastian Götschel, Daniel Ruprecht
Parallel-across-the method time integration can provide small scale parallelism when solving initial value problems. Spectral deferred corrections (SDC) with a diagonal sweeper, which is closely related to iterated Runge-Kutta methods proposed by Van der Houwen and Sommeijer, can use a number of threads equal to the number of quadrature nodes in the underlying collocation method. However, convergence
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Generalized convergence of the deep BSDE method: a step towards fully-coupled FBSDEs and applications in stochastic control arXiv.cs.NA Pub Date : 2024-03-27 Balint Negyesi, Zhipeng Huang, Cornelis W. Oosterlee
We are concerned with high-dimensional coupled FBSDE systems approximated by the deep BSDE method of Han et al. (2018). It was shown by Han and Long (2020) that the errors induced by the deep BSDE method admit a posteriori estimate depending on the loss function, whenever the backward equation only couples into the forward diffusion through the Y process. We generalize this result to fully-coupled
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Wirtinger gradient descent methods for low-dose Poisson phase retrieval arXiv.cs.NA Pub Date : 2024-03-27 Benedikt Diederichs, Frank Filbir, Patricia Römer
The problem of phase retrieval has many applications in the field of optical imaging. Motivated by imaging experiments with biological specimens, we primarily consider the setting of low-dose illumination where Poisson noise plays the dominant role. In this paper, we discuss gradient descent algorithms based on different loss functions adapted to data affected by Poisson noise, in particular in the
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Global convergence of iterative solvers for problems of nonlinear magnetostatics arXiv.cs.NA Pub Date : 2024-03-27 Herbert Egger, Felix Engertsberger, Bogdan Radu
We consider the convergence of iterative solvers for problems of nonlinear magnetostatics. Using the equivalence to an underlying minimization problem, we can establish global linear convergence of a large class of methods, including the damped Newton-method, fixed-point iteration, and the Kacanov iteration, which can all be interpreted as generalized gradient descent methods. Armijo backtracking isconsidered
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Computational decomposition and composition technique for approximate solution of nonstationary problems arXiv.cs.NA Pub Date : 2024-03-27 P. N. Vabishchevich
Stable computational algorithms for the approximate solution of the Cauchy problem for nonstationary problems are based on implicit time approximations. Computational costs for boundary value problems for systems of coupled multidimensional equations can be reduced by additive decomposition of the problem operator(s) and composition of the approximate solution using particular explicit-implicit time
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Fractional variational integrators based on convolution quadrature arXiv.cs.NA Pub Date : 2024-03-27 Khaled Hariz, Fernando Jiménez, Sina Ober-Blöbaum
Fractional dissipation is a powerful tool to study non-local physical phenomena such as damping models. The design of geometric, in particular, variational integrators for the numerical simulation of such systems relies on a variational formulation of the model. In [19], a new approach is proposed to deal with dissipative systems including fractionally damped systems in a variational way for both,
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Frozen Gaussian approximation for the fractional Schrödinger equation arXiv.cs.NA Pub Date : 2024-03-27 Lihui Chai, Hengzhun Chen, Xu Yang
We develop the frozen Gaussian approximation (FGA) for the fractional Schr\"odinger equation in the semi-classical regime, where the solution is highly oscillatory when the scaled Planck constant $\varepsilon$ is small. This method approximates the solution to the Schr\"odinger equation by an integral representation based on asymptotic analysis and provides a highly efficient computational method for
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Stability and convergence of the penalty formulation for nonlinear magnetostatics arXiv.cs.NA Pub Date : 2024-03-27 Herbert Egger, Felix Engertsberger, Klaus Roppert
The magnetostatic field distribution in a nonlinear medium amounts to the unique minimizer of the magnetic coenergy over all fields that can be generated by the same current. This is a nonlinear saddlepoint problem whose numerical solution can in principle be achieved by mixed finite element methods and appropriate nonlinear solvers. The saddlepoint structure, however, makes the solution cumbersome
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Mixed Variational Formulation of Coupled Plates arXiv.cs.NA Pub Date : 2024-03-27 Jun Hu, Zhen Liu, Rui Ma, Ruishu Wang
This paper proposes a mixed variational formulation for the problem of two coupled plates with a rigid {junction}. The proposed mixed {formulation} introduces {the union of} stresses and moments as {an auxiliary variable}, which {are} commonly of great interest in practical applications. The primary challenge lies in determining a suitable {space involving} both boundary and junction conditions of
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Higher order multi-dimension reduction methods via Einstein-product arXiv.cs.NA Pub Date : 2024-03-27 Alaeddine Zahir, Khalide Jbilou, Ahmed Ratnani
This paper explores the extension of dimension reduction (DR) techniques to the multi-dimension case by using the Einstein product. Our focus lies on graph-based methods, encompassing both linear and nonlinear approaches, within both supervised and unsupervised learning paradigms. Additionally, we investigate variants such as repulsion graphs and kernel methods for linear approaches. Furthermore, we
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Discretize first, filter next: learning divergence-consistent closure models for large-eddy simulation arXiv.cs.NA Pub Date : 2024-03-26 Syver Døving Agdestein, Benjamin Sanderse
We propose a new neural network based large eddy simulation framework for the incompressible Navier-Stokes equations based on the paradigm "discretize first, filter and close next". This leads to full model-data consistency and allows for employing neural closure models in the same environment as where they have been trained. Since the LES discretization error is included in the learning process, the
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Nonsingularity of unsymmetric Kansa matrices: random collocation by MultiQuadrics and Inverse MultiQuadrics arXiv.cs.NA Pub Date : 2024-03-26 R. Cavoretto, F. Dell'Accio, A. De Rossi, A. Sommariva, M. Vianello
Unisolvence of unsymmetric Kansa collocation is still a substantially open problem. We prove that Kansa matrices with MultiQuadrics and Inverse MultiQuadrics for the Dirichlet problem of the Poisson equation are almost surely nonsingular, when the collocation points are chosen by any continuous random distribution in the domain interior and arbitrarily on its boundary.
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$L^\infty$-error bounds for approximations of the Koopman operator by kernel extended dynamic mode decomposition arXiv.cs.NA Pub Date : 2024-03-27 Frederik Köhne, Friedrich M. Philipp, Manuel Schaller, Anton Schiela, Karl Worthmann
Extended dynamic mode decomposition (EDMD) is a well-established method to generate a data-driven approximation of the Koopman operator for analysis and prediction of nonlinear dynamical systems. Recently, kernel EDMD (kEDMD) has gained popularity due to its ability to resolve the challenging task of choosing a suitable dictionary by defining data-based observables. In this paper, we provide the first
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Nonlinear model reduction for operator learning arXiv.cs.NA Pub Date : 2024-03-27 Hamidreza Eivazi, Stefan Wittek, Andreas Rausch
Operator learning provides methods to approximate mappings between infinite-dimensional function spaces. Deep operator networks (DeepONets) are a notable architecture in this field. Recently, an extension of DeepONet based on model reduction and neural networks, proper orthogonal decomposition (POD)-DeepONet, has been able to outperform other architectures in terms of accuracy for several benchmark
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Efficient Algorithms for Regularized Nonnegative Scale-invariant Low-rank Approximation Models arXiv.cs.NA Pub Date : 2024-03-27 Jeremy E. Cohen, Valentin Leplat
Regularized nonnegative low-rank approximations such as sparse Nonnegative Matrix Factorization or sparse Nonnegative Tucker Decomposition are an important branch of dimensionality reduction models with enhanced interpretability. However, from a practical perspective, the choice of regularizers and regularization coefficients, as well as the design of efficient algorithms, is challenging because of
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A thermodynamically consistent physics-informed deep learning material model for short fiber/polymer nanocomposites arXiv.cs.NA Pub Date : 2024-03-27 Betim Bahtiri, Behrouz Arash, Sven Scheffler, Maximilian Jux, Raimund Rolfes
This work proposes a physics-informed deep learning (PIDL)-based constitutive model for investigating the viscoelastic-viscoplastic behavior of short fiber-reinforced nanoparticle-filled epoxies under various ambient conditions. The deep-learning model is trained to enforce thermodynamic principles, leading to a thermodynamically consistent constitutive model. To accomplish this, a long short-term
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An inexact infeasible arc-search interior-point method for linear programming problems arXiv.cs.NA Pub Date : 2024-03-26 Einosuke Iida, Makoto Yamashita
Inexact interior-point methods (IPMs) are a type of interior-point methods that inexactly solve the linear equation system for obtaining the search direction. On the other hand,arc-search IPMs approximate the central path with an ellipsoidal arc obtained by solving two linear equation systems in each iteration, while conventional line-search IPMs solve one linear system, therefore, the improvement
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Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback design arXiv.cs.NA Pub Date : 2024-03-26 Jan Heiland, Yongho Kim, Steffen W. R. Werner
Polytopic autoencoders provide low-dimensional parametrizations of states in a polytope. For nonlinear PDEs, this is readily applied to low-dimensional linear parameter-varying (LPV) approximations as they have been exploited for efficient nonlinear controller design via series expansions of the solution to the state-dependent Riccati equation. In this work, we develop a polytopic autoencoder for control
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Divergence conforming DG method for the optimal control of the Oseen equation with variable viscosity arXiv.cs.NA Pub Date : 2024-03-23 Harpal Singh, Arbaz Khan
This study introduces the divergence-conforming discontinuous Galerkin finite element method (DGFEM) for numerically approximating optimal control problems with distributed constraints, specifically those governed by stationary generalized Oseen equations. We provide optimal a priori error estimates in energy norms for such problems using the divergence-conforming DGFEM approach. Moreover, we thoroughly
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Stochastic variance reduced gradient method for linear ill-posed inverse problems arXiv.cs.NA Pub Date : 2024-03-19 Qinian Jin, Liuhong Chen
In this paper we apply the stochastic variance reduced gradient (SVRG) method, which is a popular variance reduction method in optimization for accelerating the stochastic gradient method, to solve large scale linear ill-posed systems in Hilbert spaces. Under {\it a priori} choices of stopping indices, we derive a convergence rate result when the sought solution satisfies a benchmark source condition
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Phase transitions and minimal interfaces on manifolds with conical singularities arXiv.cs.NA Pub Date : 2024-03-11 Daniel Grieser, Sina Held, Hannes Uecker, Boris Vertman
Using $\Gamma$-convergence, we study the Cahn-Hilliard problem with interface width parameter $\varepsilon > 0$ for phase transitions on manifolds with conical singularities. We prove that minimizers of the corresponding energy functional exist and converge, as $\varepsilon \to 0$, to a function that takes only two values with an interface along a hypersurface that has minimal area among those satisfying
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A better compression driver? CutFEM 3D shape optimization taking viscothermal losses into account arXiv.cs.NA Pub Date : 2024-03-14 Martin Berggren, Anders Bernland, André Massing, Daniel Noreland, Eddie Wadbro
The compression driver, the standard sound source for midrange acoustic horns, contains a cylindrical compression chamber connected to the horn throat through a system of channels known as a phase plug. The main challenge in the design of the phase plug is to avoid resonance and interference phenomena. The complexity of these phenomena makes it difficult to carry out this design task manually, particularly
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Helmholtz preconditioning for the compressible Euler equations using mixed finite elements with Lorenz staggering arXiv.cs.NA Pub Date : 2024-03-06 David Lee, Alberto Martin, Kieran Ricardo
Implicit solvers for atmospheric models are often accelerated via the solution of a preconditioned system. For block preconditioners this typically involves the factorisation of the (approximate) Jacobian for the coupled system into a Helmholtz equation for some function of the pressure. Here we present a preconditioner for the compressible Euler equations with a flux form representation of the potential
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Application of Deep Learning Reduced-Order Modeling for Single-Phase Flow in Faulted Porous Media arXiv.cs.NA Pub Date : 2024-03-06 Enrico Ballini, Luca Formaggia, Alessio Fumagalli, Anna Scotti, Paolo Zunino
We apply reduced-order modeling (ROM) techniques to single-phase flow in faulted porous media, accounting for changing rock properties and fault geometry variations using a radial basis function mesh deformation method. This approach benefits from a mixed-dimensional framework that effectively manages the resulting non-conforming mesh. To streamline complex and repetitive calculations such as sensitivity
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Multi-Derivative Runge-Kutta Flux Reconstruction for hyperbolic conservation laws arXiv.cs.NA Pub Date : 2024-03-04 Arpit Babbar, Praveen Chandrashekar
We extend the fourth order, two stage Multi-Derivative Runge Kutta (MDRK) scheme of Li and Du to the Flux Reconstruction (FR) framework by writing both of the stages in terms of a time averaged flux and then use the approximate Lax-Wendroff procedure. Numerical flux is computed in each stage using D2 dissipation and EA flux, enhancing Fourier CFL stability and accuracy respectively. A subcell based
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Optimized Bayesian Framework for Inverse Heat Transfer Problems Using Reduced Order Methods arXiv.cs.NA Pub Date : 2024-02-29 Kabir Bakhshaei, Umberto Emil Morelli, Giovanni Stabile, Gianluigi Rozza
A stochastic inverse heat transfer problem is formulated to infer the transient heat flux, treated as an unknown Neumann boundary condition. Therefore, an Ensemble-based Simultaneous Input and State Filtering as a Data Assimilation technique is utilized for simultaneous temperature distribution prediction and heat flux estimation. This approach is incorporated with Radial Basis Functions not only to
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Efficient quaternion CUR method for low-rank approximation to quaternion matrix arXiv.cs.NA Pub Date : 2024-02-29 Peng-Ling Wu, Kit Ian Kou, Hongmin Cai, Zhaoyuan Yu
The low-rank quaternion matrix approximation has been successfully applied in many applications involving signal processing and color image processing. However, the cost of quaternion models for generating low-rank quaternion matrix approximation is sometimes considerable due to the computation of the quaternion singular value decomposition (QSVD), which limits their application to real large-scale
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Weighted least $\ell_p$ approximation on compact Riemannian manifolds arXiv.cs.NA Pub Date : 2024-02-29 Jiansong Li, Yun Ling, Jiaxin Geng, Heping Wang
Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a compact space, Gr\"ochenig in \cite{G} discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all $1\le p\le\infty$, we develop weighted least $\ell_p$ approximation induced by a sequence of Marcinkiewicz-Zygmund inequalities in $L_p$ on a compact smooth Riemannian manifold $\Bbb M$
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Highly efficient Gauss's law-preserving spectral algorithms for Maxwell's double-curl source and eigenvalue problems based on eigen-decomposition arXiv.cs.NA Pub Date : 2024-02-29 Sen Lin, Huiyuan Li, Zhiguo Yang
In this paper, we present Gauss's law-preserving spectral methods and their efficient solution algorithms for curl-curl source and eigenvalue problems in two and three dimensions arising from Maxwell's equations. Arbitrary order $H(curl)$-conforming spectral basis functions in two and three dimensions are firstly proposed using compact combination of Legendre polynomials. A mixed formulation involving
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Recovering the Polytropic Exponent in the Porous Medium Equation: Asymptotic Approach arXiv.cs.NA Pub Date : 2024-02-29 Hagop Karakazian, Toni Sayah, Faouzi Triki
In this paper we consider the time dependent Porous Medium Equation, $u_t = \Delta u^\gamma$ with real polytropic exponent $\gamma>1$, subject to a homogeneous Dirichlet boundary condition. We are interested in recovering $\gamma$ from the knowledge of the solution $u$ at a given large time $T$. Based on an asymptotic inequality satisfied by the solution $u(T)$, we propose a numerical algorithm allowing
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Fractional material derivative: pointwise representation and a finite volume numerical scheme arXiv.cs.NA Pub Date : 2024-02-29 Łukasz Płociniczak, Marek A. Teuerle
The fractional material derivative appears as the fractional operator that governs the dynamics of the scaling limits of L\'evy walks - a stochastic process that originates from the famous continuous-time random walks. It is usually defined as the Fourier-Laplace multiplier, therefore, it can be thought of as a pseudo-differential operator. In this paper, we show that there exists a local representation
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Splitting integrators for linear Vlasov equations with stochastic perturbations arXiv.cs.NA Pub Date : 2024-02-29 Charles-Edouard Bréhier, David Cohen
We consider a class of linear Vlasov partial differential equations driven by Wiener noise. Different types of stochastic perturbations are treated: additive noise, multiplicative It\^o and Stratonovich noise, and transport noise. We propose to employ splitting integrators for the temporal discretization of these stochastic partial differential equations. These integrators are designed in order to
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Equivalence of ADER and Lax-Wendroff in DG / FR framework for linear problems arXiv.cs.NA Pub Date : 2024-02-29 Arpit Babbar, Praveen Chandrashekar
ADER (Arbitrary high order by DERivatives) and Lax-Wendroff (LW) schemes are two high order single stage methods for solving time dependent partial differential equations. ADER is based on solving a locally implicit equation to obtain a space-time predictor solution while LW is based on an explicit Taylor's expansion in time. We cast the corrector step of ADER Discontinuous Galerkin (DG) scheme into
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Error estimation for finite element method on meshes that contain thin elements arXiv.cs.NA Pub Date : 2024-02-29 Kenta Kobayashi, Takuya Tsuchiya
In an error estimation of finite element solutions to the Poisson equation, we usually impose the shape regularity assumption on the meshes to be used. In this paper, we show that even if the shape regularity condition is violated, the standard error estimation can be obtained if "bad" elements (elements that violate the shape regularity or maximum angle condition) are covered virtually by "good" simplices
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An Adaptive Orthogonal Basis Method for Computing Multiple Solutions of Differential Equations with polynomial nonlinearities arXiv.cs.NA Pub Date : 2024-02-29 Lin Li, Yangyi Ye, Wenrui Hao, Huiyuan Li
This paper presents an innovative approach, the Adaptive Orthogonal Basis Method, tailored for computing multiple solutions to differential equations characterized by polynomial nonlinearities. Departing from conventional practices of predefining candidate basis pools, our novel method adaptively computes bases, considering the equation's nature and structural characteristics of the solution. It further
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Sixth-order parabolic equation on an interval: Eigenfunction expansion, Green's function, and intermediate asymptotics for a finite thin film with elastic resistance arXiv.cs.NA Pub Date : 2024-02-28 Nectarios C. Papanicolaou, Ivan C. Christov
A linear sixth-order partial differential equation (PDE) of ``parabolic'' type describes the dynamics of thin liquid films beneath surfaces with elastic bending resistance when deflections from the equilibrium film height are small. On a finite domain, the associated sixth-order Sturm--Liouville eigenvalue value problem is self-adjoint for the boundary conditions corresponding to a thin film in a closed
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A collocation method for nonlinear tensor differential equations on low-rank manifolds arXiv.cs.NA Pub Date : 2024-02-28 Alec Dektor
We present a new method to compute the solution to a nonlinear tensor differential equation with dynamical low-rank approximation. The idea of dynamical low-rank approximation is to project the differential equation onto the tangent space of a low-rank tensor manifold at each time. Traditionally, an orthogonal projection onto the tangent space is employed, which is challenging to compute for nonlinear
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Versatile mixed methods for compressible flows arXiv.cs.NA Pub Date : 2024-02-28 Edward A. Miller, David M. Williams
Versatile mixed finite element methods were originally developed by Chen and Williams for isothermal incompressible flows in "Versatile mixed methods for the incompressible Navier-Stokes equations," Computers & Mathematics with Applications, Volume 80, 2020. Thereafter, these methods were extended by Miller, Chen, and Williams to non-isothermal incompressible flows in "Versatile mixed methods for non-isothermal
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Derivative-enhanced Deep Operator Network arXiv.cs.NA Pub Date : 2024-02-29 Yuan Qiu, Nolan Bridges, Peng Chen
Deep operator networks (DeepONets), a class of neural operators that learn mappings between function spaces, have recently been developed as surrogate models for parametric partial differential equations (PDEs). In this work we propose a derivative-enhanced deep operator network (DE-DeepONet), which leverages the derivative information to enhance the prediction accuracy, and provide a more accurate
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High multiplicity of positive solutions in a superlinear problem of Moore-Nehari type arXiv.cs.NA Pub Date : 2024-02-29 Pablo Cubillos, Julián López-Gómez, Andrea Tellini
In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in [43]. Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number $\kappa\geq 1$ of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending
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Image-To-Mesh Conversion for Biomedical Simulations arXiv.cs.NA Pub Date : 2024-02-27 Fotis Drakopoulos, Kevin Garner, Christopher Rector, Nikos Chrisochoides
Converting a three-dimensional medical image into a 3D mesh that satisfies both the quality and fidelity constraints of predictive simulations and image-guided surgical procedures remains a critical problem. Presented is an image-to-mesh conversion method called CBC3D. It first discretizes a segmented image by generating an adaptive Body-Centered Cubic (BCC) mesh of high-quality elements. Next, the
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Numerical Analysis on Neural Network Projected Schemes for Approximating One Dimensional Wasserstein Gradient Flows arXiv.cs.NA Pub Date : 2024-02-26 Xinzhe Zuo, Jiaxi Zhao, Shu Liu, Stanley Osher, Wuchen Li
We provide a numerical analysis and computation of neural network projected schemes for approximating one dimensional Wasserstein gradient flows. We approximate the Lagrangian mapping functions of gradient flows by the class of two-layer neural network functions with ReLU (rectified linear unit) activation functions. The numerical scheme is based on a projected gradient method, namely the Wasserstein
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A stochastic perturbation approach to nonlinear bifurcating problems arXiv.cs.NA Pub Date : 2024-02-26 Isabella Carla Gonnella, Moaad Khamlich, Federico Pichi, Gianluigi Rozza
Incorporating probabilistic terms in mathematical models is crucial for capturing and quantifying uncertainties in real-world systems. Indeed, randomness can have a significant impact on the behavior of the problem's solution, and a deeper analysis is needed to obtain more realistic and informative results. On the other hand, the investigation of stochastic models may require great computational resources
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Finite element schemes with tangential motion for fourth order geometric curve evolutions in arbitrary codimension arXiv.cs.NA Pub Date : 2024-02-26 Klaus Deckelnick, Robert Nürnberg
We introduce novel finite element schemes for curve diffusion and elastic flow in arbitrary codimension. The schemes are based on a variational form of a system that includes a specifically chosen tangential motion. We derive optimal $L^2$- and $H^1$-error bounds for continuous-in-time semidiscrete finite element approximations that use piecewise linear elements. In addition, we consider fully discrete
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Asymptotic-preserving and energy stable dynamical low-rank approximation for thermal radiative transfer equations arXiv.cs.NA Pub Date : 2024-02-26 Chinmay Patwardhan, Martin Frank, Jonas Kusch
The thermal radiative transfer equations model temperature evolution through a background medium as a result of radiation. When a large number of particles are absorbed in a short time scale, the dynamics tend to a non-linear diffusion-type equation called the Rosseland approximation. The main challenges for constructing numerical schemes that exhibit the correct limiting behavior are posed by the
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Mathematical foundations of spectral methods for time-dependent PDEs arXiv.cs.NA Pub Date : 2024-02-26 Arieh Iserles
The contention of this paper is that a spectral method for time-dependent PDEs is basically no more than a choice of an orthonormal basis of the underlying Hilbert space. This choice is governed by a long list of considerations: stability, speed of convergence, geometric numerical integration, fast approximation and efficient linear algebra. We subject different choices of orthonormal bases, focussing
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On bundle closures of matrix pencils and matrix polynomials arXiv.cs.NA Pub Date : 2024-02-26 Fernando De Terán, Froilán M. Dopico, Vadym Koval, Patryk Pagacz
Bundles of matrix polynomials are sets of matrix polynomials with the same size and grade and the same eigenstructure up to the specific values of the eigenvalues. It is known that the closure of the bundle of a pencil $L$ (namely, a matrix polynomial of grade $1$), denoted by $\mathcal{B}(L)$, is the union of $\mathcal{B}(L)$ itself with a finite number of other bundles. The first main contribution
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Structure-Preserving Numerical Methods for Two Nonlinear Systems of Dispersive Wave Equations arXiv.cs.NA Pub Date : 2024-02-26 Joshua Lampert, Hendrik Ranocha
We use the general framework of summation by parts operators to construct conservative, entropy-stable and well-balanced semidiscretizations of two different nonlinear systems of dispersive shallow water equations with varying bathymetry: (i) a variant of the coupled Benjamin-Bona-Mahony (BBM) equations and (ii) a recently proposed model by Sv\"ard and Kalisch (2023) with enhanced dispersive behavior
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Generalized sparsity-promoting solvers for Bayesian inverse problems: Versatile sparsifying transforms and unknown noise variances arXiv.cs.NA Pub Date : 2024-02-26 Jonathan Lindbloom, Jan Glaubitz, Anne Gelb
Bayesian hierarchical models can provide efficient algorithms for finding sparse solutions to ill-posed inverse problems. The models typically comprise a conditionally Gaussian prior model for the unknown which is augmented by a generalized gamma hyper-prior model for variance hyper-parameters. This investigation generalizes these models and their efficient maximum a posterior (MAP) estimation using
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Structure-Preserving Operator Learning: Modeling the Collision Operator of Kinetic Equations arXiv.cs.NA Pub Date : 2024-02-26 Jae Yong Lee, Steffen Schotthöfer, Tianbai Xiao, Sebastian Krumscheid, Martin Frank
This work explores the application of deep operator learning principles to a problem in statistical physics. Specifically, we consider the linear kinetic equation, consisting of a differential advection operator and an integral collision operator, which is a powerful yet expensive mathematical model for interacting particle systems with ample applications, e.g., in radiation transport. We investigate
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Convergence analysis for a fully-discrete finite element approximation of the unsteady $p(\cdot,\cdot)$-Navier-Stokes equations arXiv.cs.NA Pub Date : 2024-02-26 Luigi C. Berselli, Alex Kaltenbach
In the present paper, we establish the well-posedness, stability, and (weak) convergence of a fully-discrete approximation of the unsteady $p(\cdot,\cdot)$-Navier-Stokes equations employing an implicit Euler step in time and a discretely inf-sup-stable finite element approximation in space. Moreover, numerical experiments are carried out that supplement the theoretical findings.
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A randomized algorithm for simultaneously diagonalizing symmetric matrices by congruence arXiv.cs.NA Pub Date : 2024-02-26 Haoze He, Daniel Kressner
A family of symmetric matrices $A_1,\ldots, A_d$ is SDC (simultaneous diagonalization by congruence) if there is an invertible matrix $X$ such that every $X^T A_k X$ is diagonal. In this work, a novel randomized SDC (RSDC) algorithm is proposed that reduces SDC to a generalized eigenvalue problem by considering two (random) linear combinations of the family. We establish exact recovery: RSDC achieves
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Point collocation with mollified piecewise polynomial approximants for high-order partial differential equations arXiv.cs.NA Pub Date : 2024-02-26 Dewangga Alfarisy, Lavi Zuhal, Michael Ortiz, Fehmi Cirak, Eky Febrianto
The solution approximation for partial differential equations (PDEs) can be substantially improved using smooth basis functions. The recently introduced mollified basis functions are constructed through mollification, or convolution, of cell-wise defined piecewise polynomials with a smooth mollifier of certain characteristics. The properties of the mollified basis functions are governed by the order
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Discovering Artificial Viscosity Models for Discontinuous Galerkin Approximation of Conservation Laws using Physics-Informed Machine Learning arXiv.cs.NA Pub Date : 2024-02-26 Matteo Caldana, Paola F. Antonietti, Luca Dede'
Finite element-based high-order solvers of conservation laws offer large accuracy but face challenges near discontinuities due to the Gibbs phenomenon. Artificial viscosity is a popular and effective solution to this problem based on physical insight. In this work, we present a physics-informed machine learning algorithm to automate the discovery of artificial viscosity models in a non-supervised paradigm
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To be, or not to be, that is the Question: Exploring the pseudorandom generation of texts to write Hamlet from the perspective of the Infinite Monkey Theorem arXiv.cs.NA Pub Date : 2024-02-26 Ergon Cugler de Moraes Silva
This article explores the theoretical and computational aspects of the Infinite Monkey Theorem, investigating the number of attempts and the time required for a set of pseudorandom characters to assemble and recite Hamlets iconic phrase, To be, or not to be, that is the Question. Drawing inspiration from Emile Borels original concept (1913), the study delves into the practical implications of pseudorandomness