We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form $\m{A}x=λ\m{B}x$, where the matrices $\m{A}$ and/or $\m{B}$ may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilized formulations are used for the numerical approximation of partial differential equations. With the help of classical numerical examples we show that the presence of one (or both) parameters can produce unexpected results.
In this article we apply reduced order techniques for the approximation of parametric eigenvalue problems. The effect of the choice of sampling points is investigated. Here we use the standard proper orthogonal decomposition technique to obtain the basis of the reduced space and Galerking orthogonal technique is used to get the reduced problem. We present some numerical results and observe that the use of sparse sampling is a good idea for sampling if the dimension of parameter space is high.
We consider PDE eigenvalue problems as they occur in two-dimensional photonic crystal modeling. If the permittivity of the material is frequency-dependent, then the eigenvalue problem becomes nonlinear. In the lossless case, linearization techniques allow an equivalent reformulation as an extended but linear and Hermitian eigenvalue problem, which satisfies a Garding inequality. For this, known iterative schemes for the matrix case such as the inverse power or the Arnoldi method are extended to the infinite-dimensional case. We prove convergence of the inverse power method on operator level an…
In this article, we propose a data-driven reduced basis (RB) method for the approximation of parametric eigenvalue problems. The method is based on the offline and online paradigms. In the offline stage, we generate snapshots and construct the basis of the reduced space, using a POD approach. Gaussian process regressions (GPR) are used for approximating the eigenvalues and projection coefficients of the eigenvectors in the reduced space. All the GPR corresponding to the eigenvalues and projection coefficients are trained in the offline stage, using the data generated in the offline stage. The …
Sharp Error Bounds for Piecewise Polynomial Approximation: Revisit and Application to Elliptic PDE Eigenvalue Computation
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）；Beijing Computational Science Research Center（北京计算科学研究中心）；Department of Mathematics, Wayne State University（韦恩州立大学数学系）
Department of Mathematics, Stanford University（斯坦福大学数学系）；Department of Mathematics and ICME, Stanford University（斯坦福大学数学系与计算与数学工程研究所）
In this paper, we propose a semigroup method for solving high-dimensional elliptic partial differential equations (PDEs) and the associated eigenvalue problems based on neural networks. For the PDE problems, we reformulate the original equations as variational problems with the help of semigroup operators and then solve the variational problems with neural network (NN) parameterization. The main advantages are that no mixed second-order derivative computation is needed during the stochastic gradient descent training and that the boundary conditions are taken into account automatically by the s…
Laplacian preconditioning of elliptic PDEs: Localization of the eigenvalues of the discretized operator
Institute of Computer Sciences, Czech Academy of Science（捷克科学院计算机科学研究所）；Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院）；Department of Mathematics, University of Oslo（奥斯陆大学数学系）；Department of Numerical Analysis and Scientific Computing, Simula Research Laboratory（Simula研究实验室数值分析与科学计算部）；Faculty of Science and Technology, Norwegian University of Life Sciences（挪威生命科技大学理学院）
In this paper, a Parallel Direct Eigensolver for Sequences of Hermitian Eigenvalue Problems with no tridiagonalization is proposed, denoted by \texttt{PDESHEP}, and it combines direct methods with iterative methods. \texttt{PDESHEP} first reduces a Hermitian matrix to its banded form, then applies a spectrum slicing algorithm to the banded matrix, and finally computes the eigenvectors of the original matrix via backtransform. Therefore, compared with conventional direct eigensolvers, \texttt{PDESHEP} avoids tridiagonalization, which consists of many memory-bounded operations. In this work, the…
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Southern University of Sciences and Technology（南方科技大学）；Nanyang Technological University（南洋理工大学）；Hong Kong Baptist University（香港浸会大学）；Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）
We propose a preconditioner to accelerate the convergence of the GMRES iterative method for solving the system of linear equations obtained from discretize-then-optimize approach applied to optimal control problems constrained by a partial differential equation. Eigenvalue distribution of the preconditioned matrix as well as its eigenvectors are discussed. Numerical results of the proposed preconditioner are compared with several existing preconditioners to show its efficiency.
Linear evolution PDE $\partial_t u(x,t) = -\mathcal{L} u$, where $\mathcal{L}$ is a strongly elliptic operator independent of time, is studied as an example to show if one can superpose snapshots of a single (or a finite number of) solution(s) to construct an arbitrary solution. Our study shows that it depends on the growth rate of the eigenvalues, $μ_n$, of $\mathcal{L}$ in terms of $n$. When the statement is true, a simple data-driven approach for model reduction and approximation of an arbitrary solution of a PDE without knowing the underlying PDE is designed. Numerical experiments are pres…
Stochastic PDE eigenvalue problems often arise in the field of uncertainty quantification, whereby one seeks to quantify the uncertainty in an eigenvalue, or its eigenfunction. In this paper we present an efficient multilevel quasi-Monte Carlo (MLQMC) algorithm for computing the expectation of the smallest eigenvalue of an elliptic eigenvalue problem with stochastic coefficients. Each sample evaluation requires the solution of a PDE eigenvalue problem, and so tackling this problem in practice is notoriously computationally difficult. We speed up the approximation of this expectation in four wa…
We describe a temporal multiscale approach for the simulation of long-term processes with short-term influences involving partial differential equations. The specific problem under consideration is a growth process in blood vessels. The \emph{Candy Wrapper Process} describes a restenosis in a vessel that has previously be widened by inserting a stent. The development of a new stenosis takes place on a long time horizon (months) while the acting forces are mainly given by the pulsating blood flow. We describe a coupled pde model and a finite element simulation that is used as basis for our mult…
Spectral coarse spaces based on indefinite operators: the $H_k$-GenEO method
Inria（法国国家信息与自动化研究所）；Univ. Lille（里尔大学）；Laboratoire Paul Painlevé（保罗·潘勒韦实验室）；Eindhoven University of Technology（埃因霍温理工大学）；University of Strathclyde（斯特拉斯克莱德大学）；University of Bath（巴斯大学）
We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in $\mathbb{R}^3$, the restriction to rotationally symmetric domains is used to reduce shape optimization problems to a two-dimensional setting. For the optimization of an eigenvalue arising in a problem of optimal insulation, the existence of an optimal domain is proven. An algorithm is proposed that can be applied to general shape optimization problems under the geometric constraints of convexity and rotational symmetry. The approximated optimal domains for …
We consider a new algorithm in light of the min-max Collatz-Wielandt formalism to compute the principal eigenvalue and the eigenvector (eigen-function) for a class of positive Perron-Frobenius-like operators. Such operators are natural generalizations of the usual nonnegative primitive matrices. These have nontrivial applications in PDE problems such as computing the principal eigenvalue of Dirichlet Laplacian operators on general domains. We rigorously prove that for general initial data the corresponding numerical iterates converge globally to the unique principal eigenvalue with quadratic c…
A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions. Part II: Eigenvalue problems
University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；RWTH Aachen University（亚琛工业大学）；Forschungszentrum Jülich（于利希研究中心）
Two-level domain decomposition preconditioners lead to fast convergence and scalability of iterative solvers. However, for highly heterogeneous problems, where the coefficient function is varying rapidly on several possibly non-separated scales, the condition number of the preconditioned system generally depends on the contrast of the coefficient function leading to a deterioration of convergence. Enhancing the methods by coarse spaces constructed from suitable local eigenvalue problems, also denoted as adaptive or spectral coarse spaces, restores robust, contrast-independent convergence. Howe…
