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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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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
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Generalised Soft Finite Element Method for Elliptic Eigenvalue Problems arXiv.cs.NA Pub Date : 2024-02-25 Jipei Chen, Victor M. Calo, Quanling Deng
The recently proposed soft finite element method (SoftFEM) reduces the stiffness (condition numbers), consequently improving the overall approximation accuracy. The method subtracts a least-square term that penalizes the gradient jumps across mesh interfaces from the FEM stiffness bilinear form while maintaining the system's coercivity. Herein, we present two generalizations for SoftFEM that aim to
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Bayesian D-Optimal Experimental Designs via Column Subset Selection: The Power of Reweighted Sensors arXiv.cs.NA Pub Date : 2024-02-25 Srinivas Eswar, Vishwas Rao, Arvind K. Saibaba
This paper tackles optimal sensor placement for Bayesian linear inverse problems, a popular version of the more general Optimal Experiment Design (OED) problem, using the D-optimality criterion. This is done by establishing connections between sensor placement and Column Subset Selection Problem (CSSP), which is a well-studied problem in Numerical Linear Algebra (NLA). In particular, we use the Golub-Klema-Stewart
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Data-driven micromorphic mechanics for materials with strain localization arXiv.cs.NA Pub Date : 2024-02-25 Jacinto Ulloa, Laurent Stainier, Michael Ortiz, José E. Andrade
This paper explores the role of generalized continuum mechanics, and the feasibility of model-free data-driven computing approaches thereof, in solids undergoing failure by strain localization. Specifically, we set forth a methodology for capturing material instabilities using data-driven mechanics without prior information regarding the failure mode. We show numerically that, in problems involving
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Implementing Recycling Methods for Linear Systems in Python with an Application to Multiple Objective Optimization arXiv.cs.NA Pub Date : 2024-02-25 Ainara Garcia, Sihong Xie, Arielle Carr
Sequences of linear systems arise in the predictor-corrector method when computing the Pareto front for multi-objective optimization. Rather than discarding information generated when solving one system, it may be advantageous to recycle information for subsequent systems. To accomplish this, we seek to reduce the overall cost of computation when solving linear systems using common recycling methods
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Interpolation-based immersogeometric analysis methods for multi-material and multi-physics problems arXiv.cs.NA Pub Date : 2024-02-24 Jennifer E. Fromm, Nils Wunsch, Kurt Maute, John A. Evans, Jiun-Shyan Chen
Immersed boundary methods are high-order accurate computational tools used to model geometrically complex problems in computational mechanics. While traditional finite element methods require the construction of high-quality boundary-fitted meshes, immersed boundary methods instead embed the computational domain in a background grid. Interpolation-based immersed boundary methods augment existing finite
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A Finite Element Model for Hydro-thermal Convective Flow in a Porous Medium: Effects of Hydraulic Resistivity and Thermal Diffusivity arXiv.cs.NA Pub Date : 2024-02-24 S. M. Mallikarjunaiah, Dambaru Bhatta
In this article, a finite element model is implemented to analyze hydro-thermal convective flow in a porous medium. The mathematical model encompasses Darcy's law for incompressible fluid behavior, which is coupled with a convection-diffusion-type energy equation to characterize the temperature in the porous medium. The current investigation presents an efficient, stable, and accurate finite element
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Stable Liftings of Polynomial Traces on Tetrahedra arXiv.cs.NA Pub Date : 2024-02-24 Charles Parker, Endre Süli
On the reference tetrahedron $K$, we construct, for each $k \in \mathbb{N}_0$, a right inverse for the trace operator $u \mapsto (u, \partial_{n} u, \ldots, \partial_{n}^k u)|_{\partial K}$. The operator is stable as a mapping from the trace space of $W^{s, p}(K)$ to $W^{s, p}(K)$ for all $p \in (1, \infty)$ and $s \in (k+1/p, \infty)$. Moreover, if the data is the trace of a polynomial of degree $N
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An optimization based limiter for enforcing positivity in a semi-implicit discontinuous Galerkin scheme for compressible Navier-Stokes equations arXiv.cs.NA Pub Date : 2024-02-23 Chen Liu, Xiangxiong Zhang
We consider an optimization based limiter for enforcing positivity of internal energy in a semi-implicit scheme for solving gas dynamics equations. With Strang splitting, the compressible Navier-Stokes system is splitted into the compressible Euler equations, solved by the positivity-preserving Runge-Kutta discontinuous Galerkin (DG) method, and the parabolic subproblem, solved by Crank-Nicolson method
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The LC Method: A parallelizable numerical method for approximating the roots of single-variable polynomials arXiv.cs.NA Pub Date : 2024-02-23 Daniel Alba-Cuellar
The LC method described in this work seeks to approximate the roots of polynomial equations in one variable. This book allows you to explore the LC method, which uses geometric structures of Lines L and Circumferences C in the plane of complex numbers, based on polynomial coefficients. These structures depend on the inclination angle of a line with fixed point that seeks to contain one of the roots;
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Neural Operators with Localized Integral and Differential Kernels arXiv.cs.NA Pub Date : 2024-02-26 Miguel Liu-Schiaffini, Julius Berner, Boris Bonev, Thorsten Kurth, Kamyar Azizzadenesheli, Anima Anandkumar
Neural operators learn mappings between function spaces, which is practical for learning solution operators of PDEs and other scientific modeling applications. Among them, the Fourier neural operator (FNO) is a popular architecture that performs global convolutions in the Fourier space. However, such global operations are often prone to over-smoothing and may fail to capture local details. In contrast
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Approximation and perturbations of stable solutions to a stationary mean field game system arXiv.cs.NA Pub Date : 2024-02-26 Jules BerryIRMAR, INSA Rennes, UR, Olivier LeyIRMAR, Francisco J SilvaXLIM
This work introduces a new general approach for the numerical analysis of stable equilibria to second order mean field games systems in cases where the uniqueness of solutions may fail. For the sake of simplicity, we focus on a simple stationary case. We propose an abstract framework to study these solutions by reformulating the mean field game system as an abstract equation in a Banach space. In this
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Discrete Fourier Transform Approximations Based on the Cooley-Tukey Radix-2 Algorithm arXiv.cs.NA Pub Date : 2024-02-25 D. F. G. Coelho, R. J. Cintra
This report elaborates on approximations for the discrete Fourier transform by means of replacing the exact Cooley-Tukey algorithm twiddle-factors by low-complexity integers, such as $0, \pm \frac{1}{2}, \pm 1$.
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New approach method for solving nonlinear differential equations of blood flow with nanoparticle in presence of magnetic field arXiv.cs.NA Pub Date : 2024-02-25 Seyed Morteza Hamzeh Pahnehkolaei, Amirreza Kachabi, Milad Heydari Sipey, Davood Domiri Ganji
In this paper, effect of physical parameters in presence of magnetic field on heat transfer and flow of third grade non-Newtonian Nanofluid in a porous medium with annular cross sectional analytically has been investigated. The viscosity of Nanofluid categorized in 3 model include constant model and variable models with temperature that in variable category Reynolds Model and Vogel's Model has been
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Direct and Inverse scattering in a three-dimensional planar waveguide arXiv.cs.NA Pub Date : 2024-02-25 Yan Chang, Yukun Guo, Yue Zhao
In this paper, we study the direct and inverse scattering of the Schr\"odinger equation in a three-dimensional planar waveguide. For the direct problem, we derive a resonance-free region and resolvent estimates for the resolvent of the Schr\"odinger operator in such a geometry. Based on the analysis of the resolvent, several inverse problems are investigated. First, given the potential function, we
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Criticality measure-based error estimates for infinite dimensional optimization arXiv.cs.NA Pub Date : 2024-02-25 Danlin Li, Johannes Milz
Motivated by optimization with differential equations, we consider optimization problems with Hilbert spaces as decision spaces. As a consequence of their infinite dimensionality, the numerical solution necessitates finite dimensional approximations and discretizations. We develop an approximation framework and demonstrate criticality measure-based error estimates. We consider criticality measures
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High-performance finite elements with MFEM arXiv.cs.NA Pub Date : 2024-02-25 Julian Andrej, Nabil Atallah, Jan-Phillip Bäcker, John Camier, Dylan Copeland, Veselin Dobrev, Yohann Dudouit, Tobias Duswald, Brendan Keith, Dohyun Kim, Tzanio Kolev, Boyan Lazarov, Ketan Mittal, Will Pazner, Socratis Petrides, Syun'ichi Shiraiwa, Mark Stowell, Vladimir Tomov
The MFEM (Modular Finite Element Methods) library is a high-performance C++ library for finite element discretizations. MFEM supports numerous types of finite element methods and is the discretization engine powering many computational physics and engineering applications across a number of domains. This paper describes some of the recent research and development in MFEM, focusing on performance portability
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Grid Peeling of Parabolas arXiv.cs.NA Pub Date : 2024-02-24 Günter Rote, Moritz Rüber, Morteza Saghafian
Grid peeling is the process of repeatedly removing the convex hull vertices of the grid-points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical
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Design, Implementation and Analysis of a Compressed Sensing Photoacoustic Projection Imaging System arXiv.cs.NA Pub Date : 2024-02-24 Markus Haltmeier, Matthias Ye, Karoline Felbermayer, Florian Hinterleitner, Peter Burgholzer
Significance: Compressed sensing (CS) uses special measurement designs combined with powerful mathematical algorithms to reduce the amount of data to be collected while maintaining image quality. This is relevant to almost any imaging modality, and in this paper we focus on CS in photoacoustic projection imaging (PAPI) with integrating line detectors (ILDs). Aim: Our previous research involved rather
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Operator Learning: Algorithms and Analysis arXiv.cs.NA Pub Date : 2024-02-24 Nikola B. Kovachki, Samuel Lanthaler, Andrew M. Stuart
Operator learning refers to the application of ideas from machine learning to approximate (typically nonlinear) operators mapping between Banach spaces of functions. Such operators often arise from physical models expressed in terms of partial differential equations (PDEs). In this context, such approximate operators hold great potential as efficient surrogate models to complement traditional numerical
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Learning Semilinear Neural Operators : A Unified Recursive Framework For Prediction And Data Assimilation arXiv.cs.NA Pub Date : 2024-02-24 Ashutosh Singh, Ricardo Augusto Borsoi, Deniz Erdogmus, Tales Imbiriba
Recent advances in the theory of Neural Operators (NOs) have enabled fast and accurate computation of the solutions to complex systems described by partial differential equations (PDEs). Despite their great success, current NO-based solutions face important challenges when dealing with spatio-temporal PDEs over long time scales. Specifically, the current theory of NOs does not present a systematic
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Smooth and Sparse Latent Dynamics in Operator Learning with Jerk Regularization arXiv.cs.NA Pub Date : 2024-02-23 Xiaoyu Xie, Saviz Mowlavi, Mouhacine Benosman
Spatiotemporal modeling is critical for understanding complex systems across various scientific and engineering disciplines, but governing equations are often not fully known or computationally intractable due to inherent system complexity. Data-driven reduced-order models (ROMs) offer a promising approach for fast and accurate spatiotemporal forecasting by computing solutions in a compressed latent
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Attached and separated rotating flow over a finite height ridge arXiv.cs.NA Pub Date : 2024-02-23 Stefan Frei, Erik Burman, Edward R Johnson
This paper discusses the effect of rotation on the boundary layer in high Reynolds number flow over a ridge using a numerical method based on stabilised finite elements that captures steady solutions up to Reynolds number of order $10^6$. The results are validated against boundary layer computations in shallow flows and for deep flows against experimental observations reported in Machicoane et al.
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A generalized formulation for gradient schemes in unstructured finite volume method arXiv.cs.NA Pub Date : 2024-02-09 Mandeep Deka, Ashwani Assam, Ganesh Natarajan
We present a generic framework for gradient reconstruction schemes on unstructured meshes using the notion of a dyadic sum-vector product. The proposed formulation reconstructs centroidal gradients of a scalar from its directional derivatives along specific directions in a suitably defined neighbourhood. We show that existing gradient reconstruction schemes can be encompassed within this framework
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HAMLET: Graph Transformer Neural Operator for Partial Differential Equations arXiv.cs.NA Pub Date : 2024-02-05 Andrey Bryutkin, Jiahao Huang, Zhongying Deng, Guang Yang, Carola-Bibiane Schönlieb, Angelica Aviles-Rivero
We present a novel graph transformer framework, HAMLET, designed to address the challenges in solving partial differential equations (PDEs) using neural networks. The framework uses graph transformers with modular input encoders to directly incorporate differential equation information into the solution process. This modularity enhances parameter correspondence control, making HAMLET adaptable to PDEs
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Detailed Error Analysis of the HHL Algorithm arXiv.cs.NA Pub Date : 2024-01-30 Xinbo Li Christopher Phillips
We reiterate the contribution made by Harrow, Hassidim, and Llyod to the quantum matrix equation solver with the emphasis on the algorithm description and the error analysis derivation details. Moreover, the behavior of the amplitudes of the phase register on the completion of the Quantum Phase Estimation is studied. This study is beneficial for the comprehension of the choice of the phase register
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Robust fully discrete error bounds for the Kuznetsov equation in the inviscid limit arXiv.cs.NA Pub Date : 2024-01-12 Benjamin Dörich, Vanja Nikolić
The Kuznetsov equation is a classical wave model of acoustics that incorporates quadratic gradient nonlinearities. When its strong damping vanishes, it undergoes a singular behavior change, switching from a parabolic-like to a hyperbolic quasilinear evolution. In this work, we establish for the first time the optimal error bounds for its finite element approximation as well as a semi-implicit fully