Related papers: Parallel Element-based Algebraic Multigrid for H(c…
Due to the wide separation of time scales in geophysical fluid dynamics, semi-implicit time integrators are commonly used in operational atmospheric forecast models. They guarantee the stable treatment of fast (acoustic and gravity) waves,…
We propose a path cover adaptive algebraic multigrid (PC-$\alpha$AMG) method for solving linear systems of weighted graph Laplacians and can also be applied to discretized second order elliptic partial differential equations. The…
In this paper, the authors constructed an auxiliary space multigrid preconditioner for the weak Galerkin finite element method for second-order diffusion equations, discretized on simplicial 2D or 3D meshes. The idea of the auxiliary space…
Large language models (LLMs) commonly struggle with specialized or emerging topics which are rarely seen in the training corpus. Graph-based retrieval-augmented generation (GraphRAG) addresses this by structuring domain knowledge as a graph…
We introduce a family of implementations of low order, additive, geometric multilevel solvers for systems of Helmholtz equations. Both grid spacing and arithmetics may comprise complex numbers and we thus can apply complex scaling…
A Multigrid Full Approximation Storage algorithm for solving Deep Residual Networks is developed to enable neural network parallelized layer-wise training and concurrent computational kernel execution on GPUs. This work demonstrates a 10.2x…
Modern high performance computers are massively parallel; for many PDE applications spatial parallelism saturates long before the computer's capability is reached. Parallel-in-time methods enable further speedup beyond spatial saturation by…
Efficient numerical solvers for sparse linear systems are crucial in science and engineering. One of the fastest methods for solving large-scale sparse linear systems is algebraic multigrid (AMG). The main challenge in the construction of…
The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE…
We investigate a novel monolithic algebraic multigrid (AMG) preconditioner for the Taylor-Hood ($\pmb{\mathbb{P}}_2/\mathbb{P}_1$) and Scott-Vogelius ($\pmb{\mathbb{P}}_2/\mathbb{P}_1^{disc}$) discretizations of the Stokes equations. The…
The aim of this paper is to develop an algebraic multigrid method to solve eigenvalue problems based on the combination of the multilevel correction scheme and the algebraic multigrid method for linear equations. Our approach uses the…
The two-level overlapping additive Schwarz method offers a robust and scalable preconditioner for various linear systems resulting from elliptic problems. One of the key to these properties is the construction of the coarse space used to…
This work is related to PHG (Parallel Hierarchical Grid). PHG is a toolbox for developing parallel adaptive finite element programs, which is under active development at the State Key Laboratory of Scientific and Engineering Computing. The…
This paper compares the performance of seven different element agglomeration algorithms on unstructured triangular/tetrahedral meshes when used as part of a geometric multigrid. Five of these algorithms come from the literature on AMGe…
This paper proposes some efficient and accurate adaptive two-grid (ATG) finite element algorithms for linear and nonlinear partial differential equations (PDEs). The main idea of these algorithms is to utilize the solutions on the $k$-th…
This article discusses the uncertainty quantification (UQ) for time-independent linear and nonlinear partial differential equation (PDE)-based systems with random model parameters carried out using sampling-free intrusive stochastic…
We study the solution of block-structured linear algebra systems arising in optimization by using iterative solution techniques. These systems are the core computational bottleneck of many problems of interest such as parameter estimation,…
Retrieval-Augmented Generation (RAG) has demonstrated considerable effectiveness in open-domain question answering. However, when applied to heterogeneous documents, comprising both textual and tabular components, existing RAG approaches…
This paper addresses the efficient solution of linear systems arising from curl-conforming finite element discretizations of $H(\mathrm{curl})$ elliptic problems with heterogeneous coefficients. We first employ the discrete form of a…
Retrieval-Augmented Generation (RAG) integrates non-parametric knowledge into Large Language Models (LLMs), typically from unstructured texts and structured graphs. While recent progress has advanced text-based RAG to multi-turn reasoning…