Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

This study discusses generalized Rayleigh quotient and high efficiency finite element discretization schemes. Some results are as follows: 1) Rayleigh quotient accelerate technique is extended to nonselfadjoint problems. Generalized Rayleigh quotients of operator form and weak form are defined and the basic relationship between approximate eigenfunction and its generalized Rayleigh quotient is established. 2) New error estimates are obtained by replacing the ascent of exact eigenvalue with the ascent of finite element approximate eigenvalue. 3) Based on the work of Xu Jinchao and Zhou Aihui, finite element two-grid discretization schemes are established to solve nonselfadjoint elliptic differential operator eigenvalue problems and these schemes are used in both conforming finite element and nonconforming finite element. Besides, the efficiency of the schemes is proved by both theoretical analysis and numerical experiments. 4) Iterated Galerkin method, interpolated correction method and gradient recovery for selfadjoint elliptic differential operator eigenvalue problems are extended to nonselfadjoint elliptic differential operator eigenvalue problems.

We consider the solution of elliptic partial differential equations with random input data. In particular, we are interested in the mean and the correlation of the solution of these partial differential equations. Once the correlation is available, the variance can be computed from its diagonal. If the dependence of the solution on the random input data is linear and the mean of the input data is available, the mean of the solution can be computed whenever the corresponding partial differential equation can be solved. When the correlation of the input data is available, the correlation of the solution is given as the solution of a higher dimensional partial differential equation in the product domain. A Galerkin discretization yields a matrix equation with a typically densely populated right-hand side. This and the higher dimension of the problem makes the equation prohibitively expensive to solve. Since existing approaches to these correlation equations are known to struggle for roughly correlated input data, we use hierarchical matrices and their corresponding arithmetic to represent and solve the matrix equation. The regularity assumptions required for hierarchical matrices can be justified on sufficiently smooth domains. The feasibility of the approach is illustrated for finite element and boundary element discretizations and several specialities of hierarchical matrices for finite element discretizations are discussed. An almost linear scaling with respect to the dimension of the used finite element spaces is verified. The nonlinear dependence of the solution on the random input data is exemplarily discussed for the case of random domains. Extending previous perturbation approaches, we can linearize the problem and can compute the mean and the correlation up to third order accuracy in almost linear time. We discuss that these spaces can become even fourth order accurate under certain circumstances. A full convergence analysis is presented, which shows that higher order boundary elements are required for the solution of the problem. Therefore, we develop a fast multipole method for higher order boundary element methods on parametric surfaces.

A comparison is made between the probabilistic domain decomposition (DD) method and a certain deterministic DD method for solving linear elliptic boundary-value problems. Since in the deterministic approach the CPU time is affected by intercommunications among the processors, it turns out that the probabilistic method performs better, especially when the number of subdomains (hence, of processors) is increased. This fact is clearly illustrated by some examples. The probabilistic DD algorithm has been implemented in an MPI environment, in order to exploit distributed computer architectures. Scalability and fault-tolerance of the probabilistic DD algorithm are emphasized.

We consider elliptic PDE eigenvalue problems on a tensorized domain, discretized such that the resulting matrix eigenvalue problem Ax=λx exhibits Kronecker product structure. In particular, we are concerned with the case of high dimensions, where standard approaches to the solution of matrix eigenvalue problems fail due to the exponentially growing degrees of freedom. Recent work shows that this curse of dimensionality can in many cases be addressed by approximating the desired solution vector x in a low-rank tensor format. In this paper, we use the hierarchical Tucker decomposition to develop a low-rank variant of LOBPCG, a classical preconditioned eigenvalue solver. We also show how the ALS and MALS (DMRG) methods known from computational quantum physics can be adapted to the hierarchical Tucker decomposition. Finally, a combination of ALS and MALS with LOBPCG and with our low-rank variant is proposed. A number of numerical experiments indicate that such combinations represent the methods of choice.

This dissertation focuses on novel computational method for eigenvalue problems. In Chapter 1, preliminaries of functional analysis related to eigenvalue problems are presented. Some classical methods for matrix eigenvalue problems are discussed. Several PDE eigenvalue problems are covered. The chapter is concluded with a summary of the contributions. In Chapter 2, a novel recursive contour integral method (RIM) for matrix eigenvalue problem is proposed. This method can effectively find all eigenvalues in a region on the complex plane with no a priori spectrum information. Regions that contain eigenvalues are subdivided and tested recursively until the size of region reaches specified precision. The method is robust, which is demonstrated using various examples. In Chapter 3, we propose an improved version of RIM for non-Hermitian eigenvalue problems, called SIM-M. By incorporating Cayley transformation and Arnoldi’s method, the main computation cost of solving linear systems is reduced significantly. The numerical experiments demonstrate that RIM-M gains significant speed-up over RIM. In Chapter 4, we propose a multilevel spectral indicator method (SIM-M) to address the memory requirement for large sparse matrices. We modify the indicator of RIM-M such that it requires much less memory. Matrices from University of Florida Sparse Matrix Collection are tested, suggesting that a parallel version of SIM-M has the potential to be efficient. In Chapter 5, we develop a novel method to solve the elliptic PDE eigenvalue problem. We construct a multi-wavelet basis with Riesz stability in H1 0 ( ). By incorporating multi-grid discretization scheme and sparse grids, the method retains the optimal convergence rate for the smallest eigenvalue with much less computational cost.

A second-order continuity domain decomposition technique based on integrated Chebyshev polynomials for two-dimensional elliptic problems

In this paper we present a residual-based a posteriori error estimator for hp-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems. In particular, we use as a model problem the Laplace eigenvalue problem on bounded domains in ℝd, d = 2, 3, with homogeneous Dirichlet boundary conditions. Analogous error estimators can be easily obtained for more complicated elliptic eigenvalue problems. We prove the reliability and efficiency of the residual-based error estimator also for non-convex domains and use numerical experiments to show that, under an hp-adaptation strategy driven by the error estimator, exponential convergence can be achieved, even for non-smooth eigenfunctions.

Domain decomposition for finite elements and their implementation on the IBM 3090/600 VF

Domain decomposition methods were first developed for elliptic problems, taking advantage of the strong regularity of their solutions. In the last two decades, many investigations have been devoted to improve the performance of these methods for elliptic and parabolic problems. The situation is less clear for hyperbolic problems with possible singular solutions. In this paper, we will discuss a nonoverlapping domain decomposition method for nonlinear hyperbolic problems. We use the finite volume method and an implicit version of the Roe approximate Riemann solver, and propose a new interface variable inspired by Dolean and Lanteri [1]. The new variable makes the Schur complement approach simpler and allows the treatment of diffusion terms. Numerical results for the compressible Navier-Stokes equations in various 2D and 3D configurations such as the Sod shock tube problem or the lid driven cavity problem show that our method is robust and efficient. Comparisons of performances on parallel computers with up to 512 processors are also reported.

A non-intrusive proper generalized decomposition (PGD) strategy, coupled with an overlapping domain decomposition (DD) method, is proposed to efficiently construct surrogate models of parametric linear elliptic problems. A parametric multi-domain formulation is presented, with local subproblems featuring arbitrary Dirichlet interface conditions represented through the traces of the finite element functions used for spatial discretization at the subdomain level, with no need for additional auxiliary basis functions. The linearity of the operator is exploited to devise low-dimensional problems with only few active boundary parameters. An overlapping Schwarz method is used to glue the local surrogate models, solving a linear system for the nodal values of the parametric solution at the interfaces, without introducing Lagrange multipliers to enforce the continuity in the overlapping region. The proposed DD-PGD methodology relies on a fully algebraic formulation allowing for real-time computation based on the efficient interpolation of the local surrogate models in the parametric space, with no additional problems to be solved during the execution of the Schwarz algorithm. Numerical results for parametric diffusion and convection–diffusion problems are presented to showcase the accuracy of the DD-PGD approach, its robustness in different regimes and its superior performance with respect to standard high-fidelity DD methods.

Preconditioning for boundary element methods in domain decomposition

A multiscale domain decomposition approach for chemical vapor deposition

For the numerical solution of linear systems that arise from discretized linear partial differential equations, multigrid and domain decomposition methods are well established. Multigrid methods are known to have optimal complexity and domain decomposition methods are in particular useful for parallelization of the implemented algorithm. For linear random operator equations, the classical theory is not directly applicable, since condition numbers of system matrices may be close to degenerate due to non-uniform random input. It is shown that iterative methods converge in the strong, i.e. $$L^p$$, sense if the random input satisfies certain integrability conditions. As a main result, standard multigrid and domain decomposition methods are applicable in the case of linear elliptic partial differential equations with lognormal diffusion coefficients and converge strongly with deterministic bounds on the computational work which are essentially optimal. This enables the application of multilevel Monte Carlo methods with rigorous, deterministic bounds on the computational work.

We sketch a non‐overlapping domain decomposition method (DDM) for a linear quadratic optimal control problem governed by the Oseen equations. The DDM is applied to the system of necessary and sufficient optimality conditions. The approach extends balanced Neumann Neumann DDMs from single partial differential equations (PDEs) to the optimization control context, and it extends previous work on balanced Neumann Neumann DDMs for the optimal control of scalar elliptic PDEs to the optimal control of the Oseen equations. This extension requires a careful handling of the incompressibility constraint and resulting compatibility conditions, as well as a careful handling of the advection term. The DDM is used to parallelize the matrix‐vector operations for the optimality system, as well as to parallelize the preconditioner. We present two approaches. One tackles the global optimality system directly, the other forms the Schur complement corresponding to variables on the subdomain interfaces. We present numerical experiments which clearly show the potential of the approaches. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Spatial diffusion and advection are important factors affecting population dynamics in aquatic environments. This paper presents a reaction–diffusion–advection model that incorporates spatial heterogeneity to study the interactions between populations and toxicants in contaminated ecosystems. We analyze the existence and uniqueness of the semi-trivial steady state of the model using the eigenvalue theory of elliptic systems and establish the well-posedness through an inductive method. By studying the corresponding elliptic eigenvalue problem, we derive the variational expression for the basic reproduction number [Formula: see text] and investigate its role as a threshold parameter for population persistence or extinction. Furthermore, we discuss the monotonicity and asymptotic profiles of [Formula: see text] with respect to the diffusion rate and advection rate. In addition, the habitat is classified into lightly polluted area and heavily polluted area, and the spatial dynamics of the model are analyzed through the level set classification of [Formula: see text]. Finally, numerical simulations validate and complement our theoretical analysis results, illustrating the impact of spatial heterogeneity on population persistence. This work provides new insights into the mechanisms of population-toxicant interactions and offers a framework for ecological protection in polluted aquatic ecosystems.