Related papers: Overlapping Schwarz methods with GenEO coarse spac…
We analyse two-level Schwarz domain-decomposition GMRES preconditioners -- both the classic additive Schwarz preconditioner and a hybrid variant -- for finite-element discretisations of the Helmholtz equation with wavenumber $k$, where the…
Accelerating iterative eigenvalue algorithms is often achieved by employing a spectral shifting strategy. Unfortunately, improved shifting typically leads to a smaller eigenvalue for the resulting shifted operator, which in turn results in…
A strategy to construct physics-based local surrogate models for parametric Stokes flows and coupled Stokes-Darcy systems is presented. The methodology relies on the proper generalized decomposition (PGD) method to reduce the dimensionality…
Latent diffusion models have established a new state-of-the-art in high-resolution visual generation. Integrating Vision Foundation Model priors improves generative efficiency, yet existing latent designs remain largely heuristic. These…
Variational Autoencoder (VAE) and its variations are classic generative models by learning a low-dimensional latent representation to satisfy some prior distribution (e.g., Gaussian distribution). Their advantages over GAN are that they can…
GMRES is one of the most popular iterative methods for the solution of large linear systems of equations that arise from the discretization of linear well-posed problems, such as Dirichlet boundary value problems for elliptic partial…
Vertebral fractures are a consequence of osteoporosis, with significant health implications for affected patients. Unfortunately, grading their severity using CT exams is hard and subjective, motivating automated grading methods. However,…
Two-level domain decomposition (DD) methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level…
The enduring inability of image generative models to recreate intricate geometric features, such as those present in human hands and fingers has been an ongoing problem in image generation for nearly a decade. While strides have been made…
In this paper we give new results on domain decomposition preconditioners for GMRES when computing piecewise-linear finite-element approximations of the Helmholtz equation $-\Delta u - (k^2+ {\rm i} \varepsilon)u = f$, with absorption…
Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a…
Gaussian processes (GPs), implemented through multivariate Gaussian distributions for a finite collection of data, are the most popular approach in small-area spatial statistical modelling. In this context they are used to encode…
Large-scale overlapping problems are prevalent in practical engineering applications, and the optimization challenge is significantly amplified due to the existence of shared variables. Decomposition-based cooperative coevolution (CC)…
This paper is concerned with error estimates of the fully discrete generalized finite element method (GFEM) with optimal local approximation spaces for solving elliptic problems with heterogeneous coefficients. The local approximation…
Engineering problems frequently require solution of governing equations with spatially-varying discontinuous coefficients. Even for linear elliptic problems, mapping large ensembles of coefficient fields to solutions can become a major…
We consider a new coarse space for the ASM and RAS preconditioners to solve elliptic partial differential equations on perforated domains, where the numerous polygonal perforations represent structures such as walls and buildings in urban…
Neural operators (NOs) struggle with high-contrast multiscale partial differential equations (PDEs), where fine-scale heterogeneities cause large errors. To address this, we use the Generalized Multiscale Finite Element Method (GMsFEM) that…
Normal and anomalous diffusion are ubiquitous in many complex systems [1] . Here, we define a time and space generalized diffusion equation (GDE), which uses fractional-time derivatives and transformed d-path Laplacian operators on…
In this work, we propose and analyze two two-level hybrid Schwarz preconditioners for solving the Helmholtz equation with high wave number in two and three dimensions. Both preconditioners are defined over a set of overlapping subdomains,…
Diffusion-based methods represented as stochastic differential equations on a continuous-time domain have recently proven successful as a non-adversarial generative model. Training such models relies on denoising score matching, which can…