arXiv subjects

Maciej Paszynski

Publications and source records attributed to Maciej Paszynski.

14 recordsLinked to original sources

arXiv2024

Robust Variational Physics-Informed Neural Networks

We introduce a Robust version of the Variational Physics-Informed Neural Networks method (RVPINNs). As in VPINNs, we define the quadratic loss functional in terms of a Petrov-Galerkin-type variational formulation of the PDE problem: the trial space is a (Deep) Neural Network (DNN) manifold, while the test space is a finite-dimensional vector space. Whereas the VPINN's loss depends upon the selected basis functions of a given test space, herein, we minimize a loss based on the discrete dual norm of the residual. The main advantage of such a loss definition is that it provides a reliable and efficient estimator of the true error in the energy norm under the assumption of the existence of a local Fortin operator. We test the performance and robustness of our algorithm in several advection-diffusion problems. These numerical results perfectly align with our theoretical findings, showing that our estimates are sharp.

math.NA↗

arXiv2023

Physics Informed Neural Networks with strong and weak residuals for advection-dominated diffusion problems

This paper deals with the following important research questions. Is it possible to solve challenging advection-dominated diffusion problems in one and two dimensions using Physics Informed Neural Networks (PINN) and Variational Physics Informed Neural Networks (VPINN)? How does it compare to the higher-order and continuity Finite Element Method (FEM)? How to define the loss functions for PINN and VPINN so they converge to the correct solutions? How to select points or test functions for training of PINN and VPINN? We focus on the one-dimensional advection-dominated diffusion problem and the two-dimensional Eriksson-Johnson model problem. We show that the standard Galerkin method for FEM cannot solve this problem. We discuss the stabilization of the advection-dominated diffusion problem with the Petrov-Galerkin (PG) formulation and present the FEM solution obtained with the PG method. We employ PINN and VPINN methods, defining several strong and weak loss functions. We compare the training and solutions of PINN and VPINN methods with higher-order FEM methods.

math.NA↗

arXiv2022

Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices

Petrov-Galerkin formulations with optimal test functions allow for the stabilization of finite element simulations. In particular, given a discrete trial space, the optimal test space induces a numerical scheme delivering the best approximation in terms of a problem-dependent energy norm. This ideal approach has two shortcomings: first, we need to explicitly know the set of optimal test functions; and second, the optimal test functions may have large supports inducing expensive dense linear systems. Nevertheless, parametric families of PDEs are an example where it is worth investing some (offline) computational effort to obtain stabilized linear systems that can be solved efficiently, for a given set of parameters, in an online stage. Therefore, as a remedy for the first shortcoming, we explicitly compute (offline) a function mapping any PDE-parameter, to the matrix of coefficients of optimal test functions (in a basis expansion) associated with that PDE-parameter. Next, as a remedy for the second shortcoming, we use the low-rank approximation to hierarchically compress the (non-square) matrix of coefficients of optimal test functions. In order to accelerate this process, we train a neural network to learn a critical bottleneck of the compression algorithm (for a given set of PDE-parameters). When solving online the resulting (compressed) Petrov-Galerkin formulation, we employ a GMRES iterative solver with inexpensive matrix-vector multiplications thanks to the low-rank features of the compressed matrix. We perform experiments showing that the full online procedure as fast as the original (unstable) Galerkin approach. In other words, we get the stabilization with hierarchical matrices and neural networks practically for free. We illustrate our findings by means of 2D Eriksson-Johnson and Hemholtz model problems.

math.NA↗

arXiv2022

Quasi-optimal $hp$-finite element refinements towards singularities via deep neural network prediction

We show how to construct the deep neural network (DNN) expert to predict quasi-optimal $hp$-refinements for a given computational problem. The main idea is to train the DNN expert during executing the self-adaptive $hp$-finite element method ($hp$-FEM) algorithm and use it later to predict further $hp$ refinements. For the training, we use a two-grid paradigm self-adaptive $hp$-FEM algorithm. It employs the fine mesh to provide the optimal $hp$ refinements for coarse mesh elements. We aim to construct the DNN expert to identify quasi-optimal $hp$ refinements of the coarse mesh elements. During the training phase, we use the direct solver to obtain the solution for the fine mesh to guide the optimal refinements over the coarse mesh element. After training, we turn off the self-adaptive $hp$-FEM algorithm and continue with quasi-optimal refinements as proposed by the DNN expert trained. We test our method on three-dimensional Fichera and two-dimensional L-shaped domain problems. We verify the convergence of the numerical accuracy with respect to the mesh size. We show that the exponential convergence delivered by the self-adaptive $hp$-FEM can be preserved if we continue refinements with a properly trained DNN expert. Thus, in this paper, we show that from the self-adaptive $hp$-FEM it is possible to train the DNN expert the location of the singularities, and continue with the selection of the quasi-optimal $hp$ refinements, preserving the exponential convergence of the method.

math.NA↗

arXiv2022

Complexity of direct and iterative solvers on space-time formulations versus time--marching schemes for h-refined grids towards singularities

We study computational complexity aspects for Finite Element formulations considering hypercubic space--time full and time--marching discretization schemes for $h$--refined grids towards singularities. We perform a relatively comprehensive study of comparing the computational time via time complexities of direct and iterative solvers. We focus on the space-time formulation with refined computational grids and on the corresponding time slabs, namely, computational grids obtained by taking the "cross-sections" of the refined space-time mesh. We compare the computational complexity of the space-time formulation and the corresponding time--marching scheme. Our consideration concerns the computational complexity of the multi-frontal solvers, the iterative solvers, as well as the static condensation. Numerical experiments with Octave confirm our theoretical findings.

math.NA↗

arXiv2021

Alternating directions implicit higher-order finite element method for simulations of time-dependent electromagnetic wave propagation in non-regular biological tissues

We focus on non-stationary Maxwell equations defined on a regular patch of elements as considered in the isogeometric analysis (IGA). We apply the time-integration scheme following the ideas developed by the finite difference community [M. Hochbruck, T. Jahnke, R. Schnaubelt, Convergence of an ADI splitting for Maxwell's equations, Numerishe Mathematik, 2015] to derive a weak formulation resulting in discretization with Kronecker product matrices. We take the tensor product structure of the computational patch of elements from the IGA framework as an advantage, allowing for linear computational cost factorization in every time step. We design our solver to target simulations of electromagnetic waves propagations in non-regular biological tissues. We show that the linear cost of the alternating direction solver is preserved when we arbitrarily vary material data coefficients across the computational domain. We verify the solver using the manufactured solution and the problem of propagation of electromagnetic waves on the human head.

math.NA↗

arXiv2021

Isogeometric Residual Minimization Method (iGRM) with Direction Splitting for Non-Stationary Advection-Diffusion Problems

In this paper, we propose a novel computational implicit method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. The method mixes the benefits resulting from isogeometric analysis, implicit dynamics, residual minimization, and alternating direction solver. We utilize tensor product B-spline basis functions in space, implicit second order time integration schemes, residual minimization in every time step, and we exploit Kronecker product structure of the matrix to employ linear computational cost alternating direction solver. We implement an implicit time integration scheme and apply, for each space-direction, a stabilized mixed method based on residual minimization. We show that the resulting system of linear equations has a Kronecker product structure, which results in a linear computational cost of the direct solver, even using implicit time integration schemes together with the stabilized mixed formulation. We test our method on three advection-diffusion computational examples, including model ``membrane'' problem, the circular wind problem, and the simulations modeling pollution propagating from a chimney.

math.NA↗

arXiv2021

Supermodeling of tumor dynamics with parallel isogeometric analysis solver

Supermodeling is a modern, model-ensembling paradigm that integrates several self-synchronized imperfect sub-models by controlling a few meta-parameters to generate more accurate predictions of complex systems' dynamics. Continual synchronization between sub-models allows for trajectory predictions with superior accuracy compared to a single model or a classical ensemble of independent models whose decision fusion is based on the majority voting or averaging the outcomes. However, it comes out from numerous observations that the supermodeling procedure's convergence depends on a few principal factors such as (1) the number of sub-models, (2) their proper selection, and (3) the choice of the convergent optimization procedure, which assimilates the supermodel meta-parameters to data. Herein, we focus on modeling the evolution of the system described by a set of PDEs. We prove that supermodeling is conditionally convergent to a fixed-point attractor regarding only the supermodel meta-parameters. We investigate the formal conditions of the convergence of the supermodeling scheme theoretically. We employ the Banach fixed point theorem for the supermodeling correction operator, updating the synchronization constants' values iteratively. The "nudging" of the supermodel to the ground truth should be well balanced because both too small and too large attraction to data cause the supermodel desynchronization. The time-step size can control the convergence of the training procedure, by balancing the Lipshitz continuity constant of the PDE operator. All the sub-models have to be close to the ground-truth along the training trajectory but still sufficiently diverse to explore the phase space better. As an example, we discuss the three-dimensional supermodel of tumor evolution to demonstrate the supermodel's perfect fit to artificial data generated based on real medical images.

cs.CE↗

arXiv2020

Isogeometric Residual Minimization (iGRM) for Non-Stationary Stokes and Navier-Stokes Problems

We show that it is possible to obtain a linear computational cost FEM-based solver for non-stationary Stokes and Navier-Stokes equations. Our method employs a technique developed by Guermond and Minev, which consists of singular perturbation plus a splitting scheme. While the time-integration schemes are implicit, we use finite elements to discretize the spatial counterparts. At each time-step, we solve a PDE having weak-derivatives in one direction only (which allows for the linear computational cost), at the expense of handling strong second-order derivatives of the previous time step solution, on the right-hand side of these PDEs. This motivates the use of smooth functions such as B-splines. For high Reynolds numbers, some of these PDEs become unstable. To deal robustly with these instabilities, we propose to use a residual minimization technique. We test our method on problems having manufactured solutions, as well as on the cavity flow problem.

math.NA↗

arXiv2020

Isogeometric Residual Minimization Method (iGRM) with Direction Splitting Preconditoner for Stationary Advection-Diffusion Problems

In this paper, we introduce the isoGeometric Residual Minimization (iGRM) method. The method solves stationary advection-dominated diffusion problems. We stabilize the method via residual minimization. We discretize the problem using B-spline basis functions. We then seek to minimize the isogeometric residual over a spline space built on a tensor product mesh. We construct the solution over a smooth subspace of the residual. We can specify the solution subspace by reducing the polynomial order, by increasing the continuity, or by a combination of these. The Gramm matrix for the residual minimization method is approximated by a weighted H1 norm, which we can express as Kronecker products, due to the tensor-product structure of the approximations. We use the Gramm matrix as a preconditional which can be applied in a computational cost proportional to the number of degrees of freedom in 2D and 3D. Building on these approximations, we construct an iterative algorithm. We test the residual minimization method on several numerical examples, and we compare it to the Discontinuous Petrov-Galerkin (DPG) and the Streamline Upwind Petrov-Galerkin (SUPG) stabilization methods. The iGRM method delivers similar quality solutions as the DPG method, it uses smaller grids, it does not require breaking of the spaces, but it is limited to tensor-product meshes. The computational cost of the iGRM is higher than for SUPG, but it does not require the determination of problem specific parameters.

math.NA↗

arXiv2019

Fast isogeometric solvers for hyperbolic wave propagation problems

We use the alternating direction method to simulate implicit dynamics. ur spatial discretization uses isogeometric analysis. Namely, we simulate a (hyperbolic) wave propagation problem in which we use tensor-product B-splines in space and an implicit time marching method to fully discretize the problem. We approximate our discrete operator as a Kronecker product of one-dimensional mass and stiffness matrices. As a result of this algebraic transformation, we can factorize the resulting system of equations in linear (i.e., O(N)) time at each step of the implicit method. We demonstrate the performance of our method in the model P-wave propagation problem. We then extend it to simulate the linear elasticity problem once we decouple the vector problem using alternating triangular methods. We proof theoretically and experimentally the unconditional stability of both methods.

math.NA↗

arXiv2019

Isogeometric analysis with piece-wise constant test functions

We focus on the finite element method computations with higher-order C1 continuity basis functions that preserve the partition of unity. We show that the rows of the system of linear equations can be combined, and the test functions can be sum up to 1 using the partition of unity property at the quadrature points. Thus, the test functions in higher continuity IGA can be set to piece-wise constants. This formulation is equivalent to testing with piece-wise constant basis functions, with supports span over some parts of the domain. The resulting method is a Petrov-Galerkin formulation with piece-wise constant test functions. This observation has the following consequences. The numerical integration cost can be reduced because we do not need to evaluate the test functions since they are equal to 1. This observation is valid for any basis functions preserving the partition of unity property. It is independent of the problem dimension and geometry of the computational domain. It also can be used in time-dependent problems, e.g., in the explicit dynamics computations, where we can reduce the cost of generation of the right-hand side. This summation of test functions can be performed for an arbitrary linear differential operator resulting from the Galerkin method applied to a PDE where we discretize with C1 continuity basis functions. The resulting method is equivalent to a linear combination of the collocations at points and with weights resulting from applied quadrature over the spans defined by supports of the piece-wise constant test functions.

math.NA↗

arXiv2012

A direct solver with reutilization of previously-computed LU factorizations for h-adaptive finite element grids with point singularities

This paper describes a direct solver algorithm for a sequence of finite element meshes that are h-refined towards one or several point singularities. For such a sequence of grids, the solver delivers linear computational cost O(N) in terms of CPU time and memory with respect to the number of unknowns N. The linear computational cost is achieved by utilizing the recursive structure provided by the sequence of h-adaptive grids with a special construction of the elimination tree that allows for reutilization of previously computed partial LU factorizations over the entire unrefined part of the computational mesh. The reutilization technique reduces the computational cost of the entire sequence of h-refined grids from O(N^2) down to O(N). Theoretical estimates are illustrated with numerical results on two- and three-dimensional model problems exhibiting one or several point singularities.

math.NA↗

arXiv2012

Computational complexity and memory usage for multi-frontal direct solvers in structured mesh finite elements

The multi-frontal direct solver is the state-of-the-art algorithm for the direct solution of sparse linear systems. This paper provides computational complexity and memory usage estimates for the application of the multi-frontal direct solver algorithm on linear systems resulting from B-spline-based isogeometric finite elements, where the mesh is a structured grid. Specifically we provide the estimates for systems resulting from $C^{p-1}$ polynomial B-spline spaces and compare them to those obtained using $C^0$ spaces.

math.NA↗