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We present a deep learning-based iterative approach to solve the discrete heterogeneous Helmholtz equation for high wavenumbers. Combining classical iterative multigrid solvers and convolutional neural networks (CNNs) via preconditioning,…
The use of the Preconditioned Conjugate Gradient (PCG) method for computing the Generalized Least Squares (GLS) estimator of the General Linear Model (GLM) is considered. The GLS estimator is expressed in terms of the solution of an…
Preconditioners are generally essential for fast convergence in the iterative solution of linear systems of equations. However, the computation of a good preconditioner can be expensive. So, while solving a sequence of many linear systems,…
Domain decomposition methods are widely used and effective in the approximation of solutions to partial differential equations. Yet the optimal construction of these methods requires tedious analysis and is often available only in…
Convolutional neural networks (CNNs) have achieved great success on grid-like data such as images, but face tremendous challenges in learning from more generic data such as graphs. In CNNs, the trainable local filters enable the automatic…
Despite hundreds of papers on preconditioned linear systems of equations, there remains a significant lack of comprehensive performance benchmarks comparing various preconditioners for solving symmetric positive definite (SPD) systems. In…
The paper focuses on developing and studying efficient block preconditioners based on classical algebraic multigrid for the large-scale sparse linear systems arising from the fully coupled and implicitly cell-centered finite volume…
Uncovering rationales behind predictions of graph neural networks (GNNs) has received increasing attention over the years. Existing literature mainly focus on selecting a subgraph, through combinatorial optimization, to provide faithful…
We apply preconditioning, which is widely used in classical solvers for linear systems $A\textbf{x}=\textbf{b}$, to the variational quantum linear solver. By utilizing incomplete LU factorization as a preconditioner for linear equations…
Incomplete LU factorizations of sparse matrices are widely used as preconditioners in Krylov subspace methods to speed up solving linear systems. Unfortunately, computing the preconditioner itself can be time-consuming and sensitive to…
Graph deep learning methods have become popular tools to process collections of correlated time series. Unlike traditional multivariate forecasting methods, graph-based predictors leverage pairwise relationships by conditioning forecasts on…
This paper studies the solution of nonsymmetric linear systems by preconditioned Krylov methods based on the normal equations, LSQR in particular. On some examples, preconditioned LSQR is seen to produce errors many orders of magnitude…
Graph Neural Networks (GNNs) are deep-learning architectures designed for graph-type data, where understanding relationships among individual observations is crucial. However, achieving promising GNN performance, especially on unseen data,…
A primary computational problem in kernel regression is solution of a dense linear system with the $N\times N$ kernel matrix. Because a direct solution has an O($N^3$) cost, iterative Krylov methods are often used with fast matrix-vector…
Developing suitable approximate models for analyzing and simulating complex nonlinear systems is practically important. This paper aims at exploring the skill of a rich class of nonlinear stochastic models, known as the conditional Gaussian…
Within high-performance computing (HPC), solving large sparse linear systems efficiently remains paramount, with iterative methods being the predominant choice. However, the performance of these methods is tightly coupled to the aptness of…
The optimization of multi-user multi-input multi-output (MU-MIMO) precoders is a widely recognized challenging problem. Existing work has demonstrated the potential of graph neural networks (GNNs) in learning precoding policies. However,…
This work develops neural-network--based preconditioners to accelerate solution of the Wilson-Dirac normal equation in lattice quantum field theories. The approach is implemented for the two-flavor lattice Schwinger model near the critical…
Neural networks often struggle with high-dimensional but small sample-size tabular datasets. One reason is that current weight initialisation methods assume independence between weights, which can be problematic when there are insufficient…
Convolution-type integral equations arise from various fields, \textit{e.g.}, finite impulse response filters in signal processing and deblurring problems in image processing. When solving these equations, conventional numerical methods,…