Related papers: Learning Preconditioner for Conjugate Gradient PDE…
Rapid and reliable solvers for parametric partial differential equations (PDEs) are needed in many scientific and engineering disciplines. For example, there is a growing demand for composites and architected materials with heterogeneous…
Conjugate gradient is an efficient algorithm for solving large sparse linear systems. It has been utilized to accelerate the computation in Bayesian analysis for many large-scale problems. This article discusses the applications of…
Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…
Radial basis functions provide highly useful and flexible interpolants to multivariate functions. Further, they are beginning to be used in the numerical solution of partial differential equations. Unfortunately, their construction requires…
We present a lightweighted neural PDE representation to discover the hidden structure and predict the solution of different nonlinear PDEs. Our key idea is to leverage the prior of ``translational similarity'' of numerical PDE differential…
A new polynomial preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to Conjugate Orthogonal Conjugate Gradient…
We explore a scaled spectral preconditioner for the efficient solution of sequences of symmetric and positive-definite linear systems. We design the scaled preconditioner not only as an approximation of the inverse of the linear system but…
We have presented a fast method for solving a specific type of block four-by-four saddlepoint problem arising from the finite element discretization of the generalized 3D Stokes problem. We analyze the eigenvalue distribution and the…
We present a novel linear solver that works well for large systems obtained from discretizing PDEs. It is robust and, for the examples we studied, the computational effort scales linearly with the number of equations. The algorithm is based…
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,…
Natural Gradient Descent (NGD) has emerged as a promising optimization algorithm for training neural network-based solvers for partial differential equations (PDEs), such as Physics-Informed Neural Networks (PINNs). However, its practical…
In this paper, we focus on solving a sequence of linear systems with an identical (or similar) coefficient matrix. For this type of problems, we investigate the subspace correction and deflation methods, which use an auxiliary matrix…
In this note we exploit polynomial preconditioners for the Conjugate Gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix…
The conjugate gradient method is a widely used algorithm for the numerical solution of a system of linear equations. It is particularly attractive because it allows one to take advantage of sparse matrices and produces (in case of infinite…
Nonlinear conjugate gradient (NLCG) based optimizers have shown superior loss convergence properties compared to gradient descent based optimizers for traditional optimization problems. However, in Deep Neural Network (DNN) training, the…
The efficient solution of discretisations of coupled systems of partial differential equations (PDEs) is at the core of much of numerical simulation. Significant effort has been expended on scalable algorithms to precondition Krylov…
Elliptic partial differential equations (PDEs) frequently arise in continuum descriptions of physical processes relevant to science and engineering. Multilevel preconditioners represent a family of scalable techniques for solving discrete…
The primal-dual hybrid gradient (PDHG) algorithm is popular in solving min-max problems which are being widely used in a variety of areas. To improve the applicability and efficiency of PDHG for different application scenarios, we focus on…
Recent advances in the field of machine learning open a new era in high performance computing. Applications of machine learning algorithms for the development of accurate and cost-efficient surrogates of complex problems have already…
Solving sparse linear systems from discretized PDEs is challenging. Direct solvers have in many cases quadratic complexity (depending on geometry), while iterative solvers require problem dependent preconditioners to be robust and…