数值分析

We study an inverse random obstacle scattering problems in $\mathbb{R}^2$ where the scatterer is formulated by a Gaussian process defined on the angular parameter domain. Equipped with a modified covariance function which is mathematically…

The low-rank approximation properties of Randomly Pivoted LU (RPLU), a variant of Gaussian elimination where pivots are sampled proportional to the squared entries of the Schur complement, are analyzed. It is shown that the RPLU iterates…

We study the discrete constrained saddle dynamics and their momentum variants for locating saddle points on manifolds. Under the assumption of exact unstable eigenvectors, we establish a local linear convergence of the discrete constrained…

In this paper, we investigate Durrmeyer-type generalizations of maximum-minimum neural network operators. The primary objective of this study is to establish the convergence of these operators in the $L^{p}$ norm for functions $f\in…

Large language models (LLMs) process and predict sequences containing text to answer questions, and address tasks including document summarization, providing recommendations, writing software and solving quantitative problems. We provide a…

We consider a (sub)diffusion equation with a nonlinearity of the form $pf(u)-qu$, where $p$ and $q$ are space dependent functions. Prominent examples are the Fisher-KPP, the Frank-Kamenetskii-Zeldovich and the Allen-Cahn equations. We…

This contribution provides numerical experiments for a finite volume scheme for an approximation of the stochastic Allen-Cahn equation with homogeneous Neumann boundary conditions. The approximation is done by a Yosida approximation of the…

The Radon cumulative distribution transform (R-CDT) exploits one-dimensional Wasserstein transport and the Radon transform to represent prominent features in images. It is closely related to the sliced Wasserstein distance and facilitates…

In this work, we propose data-integrated neural networks (DataInNet) for solving partial differential equations (PDEs), offering a novel approach to leveraging data (e.g., source terms, initial conditions, and boundary conditions). The core…

In this work, we propose a generalized alternating Anderson acceleration method, a periodic scheme composed of $t$ fixed-point iteration steps, interleaved with $s$ steps of Anderson acceleration with window size $m$, to solve linear and…

In this paper, we present a NURBS-enhanced finite element method that integrates the NURBS-based boundary representation of a geometric domain into a standard finite element framework for hexahedral meshes. We decompose an open, bounded,…

This paper analyses the classical mixed finite element method (FEM) and a pressure-robust variant with divergence-free reconstruction operators for the coupled Stokes-Darcy problem. Its main contribution is to provide viscosity-explicit a…

We propose a novel variational approach for computing neural Signed Distance Fields (SDF) from unoriented point clouds. To this end, we replace the commonly used eikonal equation with the heat method, carrying over to the neural domain what…

We study a regularization framework that combines a convex fidelity term with multiple $\ell_1$-based regularizers, each linked to a distinct linear transform. This multi-penalty model enhances flexibility in promoting structured sparsity.…

We consider a setting in which an evolving surface is implicitly characterized as the zero level of a level set function. Such an implicit surface does not encode any information about the path of a single point on the evolving surface. In…

The hierarchical matrix framework partitions matrices into subblocks that are either small or of low numerical rank, enabling linear storage complexity and efficient matrix-vector multiplication. This work focuses on the $H^2$-matrix format…

Differential geometries derived from tensor decompositions have been extensively studied and provided the foundations for a variety of efficient numerical methods. Despite the practical success of the tensor ring (TR) decomposition, its…

We present a novel Krylov subspace method for approximating $L_f(A, E) \vc{b}$, the matrix-vector product of the Fr\'echet derivative $L_f(A, E)$ of a large-scale matrix function $f(A)$ in direction $E$, a task that arises naturally in the…

In this work, we present a POD-greedy reduced basis method for parabolic partial differential equations (PDEs), based on the least squares space-time formulation proposed in [Hinze, Kahle, Stahl, A least-squares space-time approach for…

Kernel methods approximate nonlinear maps in a data-driven manner by projecting the target map onto a finite-dimensional Hilbert space called the solution space. Traditionally, this space is a subspace of a fixed ambient reproducing kernel…