Sparse generalized Fourier transforms

Block-diagonalization of sparse equivariant discretization matrices is studied. Such matrices typically arise when partial differential equations that evolve in symmetric geometries are discretized via the finite element method or via finite differences. By considering sparse equivariant matrices as equivariant graphs, we identify a condition for when block-diagonalization via a sparse variant of a generalized Fourier transform (GFT) becomes particularly simple and fast. Characterizations for finite element triangulations of a symmetric domain are given, and formulas for assembling the block-diagonalized matrix directly are presented. It is emphasized that the GFT preserves symmetric (Hermitian) properties of an equivariant matrix. By simulating the heat equation at the surface of a sphere discretized by an icosahedral grid, it is demonstrated that the block-diagonalization pays off. The gain is significant for a direct method, and modest for an iterative method. A comparison with a block-diagonalization approach based upon the continuous formulation is made. It is argued that the sparse GFT method is an appropriate way to discretize the resulting continuous subsystems, since the spectrum and the symmetry are preserved

Generic programming aspects of symmetry exploiting numerical software

The use of the generalized Fourier transform as a means to diagonalize certain types of equivariant matrices, and thus speeding up the solution of numerical systems, is discussed. Such matrices may arise in various applications with geometrical symmetries, for example when the boundary element method is used to solve an electrostatic problem in the exterior of a symmetric object. The method is described in detail for an object with a triangular symmetry, and the feasibility of the method is confirmed by numerical experiments. The design of numerical software for this kind of applications is a challenge. It is argued that generic programming is very suitable in this context, mainly because it is type safe and promotes polymorphism capabilities in link time. A generic C++ design of important mathematical abstractions such as groups, vector spaces, and group algebras, is outlined, illustrating the potential provided by generative programming techniques. The integration of explicit support for various data layouts for efficiency tuning purposes is discussed.Note: To appear in the proceedings of the mini-symposium "Software Concepts and Free Software for PDEs" of the ECCOMAS 2004 congress, Jyväskylä, Finland, 24-28 July 2004</p

On applications of the generalized Fourier transform in numerical linear algebra

Matrices equivariant under a group of permutation matrices are considered. Such matrices typically arise in numerical applications where the computational domain exhibits geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform. This technique yields substantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions. The theory for applying the Generalized Fourier Transform is explained, building upon the familiar special (finite commutative) case of circulant matrices being diagonalized with the Discrete Fourier Transform. The classical convolution theorem and diagonalization results are generalized to the non-commutative case of block diagonalizing equivariant matrices. Our presentation stresses the connection between multiplication with an equivariant matrices and the application of a convolution. This approach highlights the role of the underlying mathematical structures such as the group algebra, and it also simplifies the application of fast Generalized Fourier Transforms. The theory is illustrated with a selection of numerical examples

Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform

We show that the generalized Fourier transform can be used for reducing the computational cost and memory requirements of radial basis function methods for multi-dimensional option pricing. We derive a general algorithm, including a transformation of the Black-Scholes equation into the heat equation, that can be used in any number of dimensions. Numerical experiments in two and three dimensions show that the gain is substantial even for small problem sizes. Furthermore, the gain increases with the number of dimensions

A hybrid method for the wave equation

Hybrid finite element/finite difference simulation of the wave equation is studied. The simulation method is hybrid in the sense that different numerical methods, finite elements and finite differences, are used in different subdomains. The purpose is to combine the flexibility of finite elements with the efficiency of finite differences. The construction of proper geometry discretisations is important for the hybrid approach. A decomposition of the computational domain is described, which yields simple communication between structured and unstructured subdomains. An explicit hybrid method for the wave equation is constructed where the explicit finite difference schemes and finite element schemes coincide for structured subdomains. These schemes are used in the hybrid approach, keeping finite differences on the structured subdomains and applying finite elements on the unstructured domains. As a consequence of the discretisation strategy, the resulting hybrid scheme can be regarded as a pure finite element scheme. Any numerical difficulties such as instabilities at the interfaces are thus avoided. The feasibility of the hybrid approach is illustrated by numerous wave equation simulations in two and three space dimensions. In particular, the approach can easily be used for implementing absorbing boundary conditions. The efficiency of different approaches is a key issue of the current study. For our test cases, the hybrid approach is about 5 times faster than a corresponding highly optimised finite element method. It is concluded that the hybrid approach may be an important tool to reduce the execution time and memory requirement for this kind of large scale computations.Also available as Preprint 2001-14 in Chalmers Finite Element Center Preprint series</p