Fast Structured Spectral Methods

Yingwei Wang, Purdue University


Spectral methods, including Galerkin, Petrov-Galerkin, collocation and tau formulations, are a class of numerical methods for solving differential equations. One of the fascinating merits of spectral methods is that they enjoy high accuracy, and even exponential rates of convergence in many cases, thanks to the fact that high-order orthogonal polynomials or trigonometric functions are employed as basis sets. However, spectral methods always suffer from many difficulties, notably, dense linear algebra, solutions with low regularities, high-dimensional problems and so on. The theme of this dissertation is to design, analyze and implement novel strategies to break these bottlenecks in classical spectral methods. At first, we propose fast structured numerical algorithms for dense matrices, which are frequently encountered in spectral methods. The core idea is that although these matrices are dense, they enjoy a hidden nice property, named low-rank property, which means their off-diagonal blocks have small or even bounded numerical ranks for a given tolerance. This property could be exploited to dramatically reduce the computational cost and give birth to fast direct solvers and transforms with nearly optimal complexity and memory, thanks to the hierarchically semiseparable representation for structured matrices. A well designed framework of fast structured spectral methods with nearly optimal complexity and memory requirements has been established to numerically solve various differential equations, including fast structured Galerkin methods, fast structured collocation methods and fast structured Jacobi transforms. Numerical tests for several important but notoriously difficult problems show the superior efficiency and accuracy of our proposed fast structured solvers and transforms. Second, we develop a new kind of spectral method involving singular basis, called Müntz-Galerkin method. The significant advantage of this method is that the exponential rates of convergence could be recovered for the problems with singularities. The key to the success of this method is that the specially tuned Müntz polynomials are employed here to deal with the singular behaviors of the underlying problems. By exploring the relations between Jacobi polynomials and Müntz polynomials, we provide efficient implementation procedures for the Müntz-Galerkin method and perform the optimal spectral-type error estimates. As examples of applications, we consider the Poisson equation with mixed Dirichlet-Neumann boundary conditions, whose solution behaves like O(r1/2) near the singular point, and demonstrate that the Müntz-Galerkin method greatly improves the rates of convergence of the usual spectral method. Third, we propose to use spectral sparse grid methods based on hyperbolic cross to reduced the computational complexity in high dimensional approximations. We have already applied our method in solving high-dimensional electronic Schrödinger equations, resulting from Born-Oppenheimer approximation to the Schrödinger equation for a system of electrons and nuclei, which is one of the core problems in the theory of quantum mechanics and calculation of electronic structures. We develop two efficient spectral-element methods for this problem, based on Legendre and Laguerre polynomials, respectively. Numerical experiments show that our methods enjoy exponential convergence rate for the one electron case, and for multi-electron cases, can lead to a target accuracy with significantly fewer number of unknowns than other approaches. This work received support from NSF grants DMS-1217066, MS-1620262 and AFOSR FA9550-16-1-0102.




Shen, Purdue University.

Subject Area

Computer science

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