Related papers: Robust and efficient solvers for nonlinear partial…
We describe a three precision variant of Newton's method for nonlinear equations. We evaluate the nonlinear residual in double precision, store the Jacobian matrix in single precision, and solve the equation for the Newton step with…
The relaxed physical factorization (RPF) preconditioner is a recent algorithm allowing for the efficient and robust solution to the block linear systems arising from the three-field displacement-velocity-pressure formulation of coupled…
Randomized neural network (RaNN) methods have been proposed for solving various partial differential equations (PDEs), demonstrating high accuracy and efficiency. However, initializing the fixed parameters remains challenging. Additionally,…
Internal flows inside gravitationally stable astrophysical objects, such as the Sun, stars and compact stars are compressed and extremely subsonic. Such low Mach number flows are usually encountered when studying for example dynamo action…
In this paper we propose an efficiently preconditioned Newton method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners based on the BFGS update formula is…
This work introduces a new approach to reduce the computational cost of solving partial differential equations (PDEs) with convection-dominated solutions: model reduction with implicit feature tracking. Traditional model reduction…
Challenging aspects of large-scale turbulent edge simulations in plasma physics include robust nonlinear solvers and efficient preconditioners. This paper presents recent advances in the scalable solution of nonlinear partial differential…
Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs). Such models, widely used in the random feature method…
Solving nonlinear partial differential equations (PDEs) with multiple solutions using neural networks has found widespread applications in various fields such as physics, biology, and engineering. However, classical neural network methods…
Time-dependent partial differential equations (PDEs) often develop sharp fronts, localized peaks, and other moving structures that occupy only a small portion of the space--time domain but dominate the approximation error. This makes fixed…
We present novel model reduction methods for rapid solution of parametrized nonlinear partial differential equations (PDEs) in real-time or many-query contexts. Our approach combines reduced basis (RB) space for rapidly convergent…
We propose a Pretrained Finite Element Method (PFEM),a physics driven framework that bridges the efficiency of neural operator learning with the accuracy and robustness of classical finite element methods (FEM). PFEM consists of a physics…
We propose a new framework to implement interior point method (IPM) to solve very large linear programs (LP). Traditional IPMs typically use Newton's method to approximately solve a subproblem that aims to minimize a log-barrier penalty…
We consider solving equality-constrained nonlinear, nonconvex optimization problems. This class of problems appears widely in a variety of applications in machine learning and engineering, ranging from constrained deep neural networks, to…
Recently, collocation based radial basis function (RBF) partition of unity methods (PUM) for solving partial differential equations have been formulated and investigated numerically and theoretically. When combined with stable evaluation…
Linear programming (LP) is an extremely useful tool which has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such…
We present a meshless Schwarz-type non-overlapping domain decomposition method based on artificial neural networks for solving forward and inverse problems involving partial differential equations (PDEs). To ensure the consistency of…
We consider the problem of computing dense correspondences between non-rigid shapes with potentially significant partiality. Existing formulations tackle this problem through heavy manifold optimization in the spectral domain, given…
The focus in this work is on interior-point methods for inequality-constrained quadratic programs, and particularly on the system of nonlinear equations to be solved for each value of the barrier parameter. Newton iterations give high…
This work blends the inexact Newton method with iterative combined approximations (ICA) for solving topology optimization problems under the assumption of geometric nonlinearity. The density-based problem formulation is solved using a…