The standard methodology handling nonlinear PDE's involves the two steps: numerical discretization to get a set of nonlinear algebraic equations, and then the application of the Newton iterative linearization or its variants to solve the nonlinear algebraic systems. Here we present an alternative strategy called direct linearization method (DLM). The DLM discretization algebraic equations of nonlinear PDE's is simply linear rather than nonlinear. The basic idea behind the DLM is that we see a nonlinear term as a new independent systematic variable and transfer a nonlinear PDE into a linear PDE with more than one independent variable. It is stressed that the DLM strategy can be applied combining any existing numerical discretization techniques. The resulting linear discretization equations can be either over-posed or well-posed. In particular, we also discuss how to create proper radial basis functions in conjunction with the DLM.
Previous papers have shown the impact of partial convergence of discretized PDE on the accuracy of tangent and adjoint linearizations. A series of papers suggested linearization of the fixed point iteration used in the solution process as a means of computing the sensitivities rather than linearizing the discretized PDE, as the lack of convergence of the nonlinear problem indicates that the discretized form of the governing equations has not been satisfied. These works showed that the accuracy of an approximate linearization depends in part on the convergence of the nonlinear system. This work…
Hybrid Gaussian-cubic radial basis functions for scattered data interpolation
Indian Institute of Technology Kharagpur（印度理工学院卡拉格普尔分校）；Hong Kong Baptist University（香港浸会大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）；Colorado School of Mines（科罗拉多矿业学院）
Fully implicit Runge-Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic framework for solving the nonlinear equations that arise from IRK methods (and discontinuous Galerkin discretizations in time) applied to nonlinear numerical PDEs, including PDEs with algebraic constraints. Several new linearizations of the nonlinear IRK equations are deve…
Sparse RBF Networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithms
Department of Mathematics, University of California, San Diego, CA 92093, United States（加州大学圣地亚哥分校数学系）；Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, United States（橡树岭国家实验室计算机科学与数学 division）
We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers fo…
Numerical reconstruction of Schrödinger equations with quadratic nonlinearities
Research Unit of Applied and Computational Mathematics, University of Oulu（奥卢大学应用与计算数学研究部）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）
Nonlinear multi-scale problems are ubiquitous in materials science and biology. Complicated interactions between nonlinearities and (nonseparable) multiple scales pose a major challenge for analysis and simulation. In this paper, we study the numerical homogenization for multi-scale elliptic PDEs with monotone nonlinearity, in particular the Leray-Lions problem (a prototypical example is the p-Laplacian equation), where the nonlinearity cannot be parameterized with low dimensional parameters, and the linearization error is non-negligible. We develop the iterated numerical homogenization scheme…
$L^1$ based optimization is widely used in image denoising, machine learning and related applications. One of the main features of such approach is that it naturally provide a sparse structure in the numerical solutions. In this paper, we study an $L^1$ based mixed DG method for second-order elliptic equations in the non-divergence form. The elliptic PDE in nondivergence form arises in the linearization of fully nonlinear PDEs. Due to the nature of the equations, classical finite element methods based on variational forms can not be employed directly. In this work, we propose a new optimizatio…
We present a method for solving linear and nonlinear PDEs based on the variable projection (VarPro) framework and artificial neural networks (ANN). For linear PDEs, enforcing the boundary/initial value problem on the collocation points leads to a separable nonlinear least squares problem about the network coefficients. We reformulate this problem by the VarPro approach to eliminate the linear output-layer coefficients, leading to a reduced problem about the hidden-layer coefficients only. The reduced problem is solved first by the nonlinear least squares method to determine the hidden-layer co…
We propose an Exponential DG approach for numerically solving partial differential equations (PDEs). The idea is to decompose the governing PDE operators into linear (fast dynamics extracted by linearization) and nonlinear (the remaining after removing the former) parts, on which we apply the discontinuous Galerkin (DG) spatial discretization. The resulting semi-discrete system is then integrated using exponential time-integrators: exact for the former and approximate for the latter. By construction, our approach i) is stable with a large Courant number (Cr > 1); ii) supports high-order soluti…
In this paper, we study the preferential stiffness and the crack-tip fields for an elastic porous solid of which material properties are dependent upon the density. Such a description is necessary to describe the failure that can be caused by damaged pores in many porous bodies such as ceramics, concrete and human bones. To that end, we revisit a new class of implicit constitutive relations under the assumption of small deformation. Although the constitutive relationship \textit{appears linear} in both the Cauchy stress and linearized strain, the governing equation bestowed from the balance of…
Partial-differential-algebraic equations of nonlinear dynamics by Physics-Informed Neural-Network: (I) Operator splitting and framework assessment
Aerospace Engineering, University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校航空航天工程系）；Institute of Technical Mechanics, Johannes Kepler University（约翰尼斯·开普勒大学技术力学研究所）
