53 research outputs found
Fast computation of the matrix exponential for a Toeplitz matrix
The computation of the matrix exponential is a ubiquitous operation in
numerical mathematics, and for a general, unstructured matrix it
can be computed in operations. An interesting problem arises
if the input matrix is a Toeplitz matrix, for example as the result of
discretizing integral equations with a time invariant kernel. In this case it
is not obvious how to take advantage of the Toeplitz structure, as the
exponential of a Toeplitz matrix is, in general, not a Toeplitz matrix itself.
The main contribution of this work are fast algorithms for the computation of
the Toeplitz matrix exponential. The algorithms have provable quadratic
complexity if the spectrum is real, or sectorial, or more generally, if the
imaginary parts of the rightmost eigenvalues do not vary too much. They may be
efficient even outside these spectral constraints. They are based on the
scaling and squaring framework, and their analysis connects classical results
from rational approximation theory to matrices of low displacement rank. As an
example, the developed methods are applied to Merton's jump-diffusion model for
option pricing
Computing the Exponential of Large Block-Triangular Block-Toeplitz Matrices Encountered in Fluid Queues
The Erlangian approximation of Markovian fluid queues leads to the problem of
computing the matrix exponential of a subgenerator having a block-triangular,
block-Toeplitz structure. To this end, we propose some algorithms which exploit
the Toeplitz structure and the properties of generators. Such algorithms allow
to compute the exponential of very large matrices, which would otherwise be
untreatable with standard methods. We also prove interesting decay properties
of the exponential of a generator having a block-triangular, block-Toeplitz
structure
A fast two-level Strang splitting method for multi-dimensional spatial fractional Allen-Cahn equations with discrete maximum principle
In this paper, we study the numerical solutions of the multi-dimensional
spatial fractional Allen-Cahn equations. After semi-discretization for the
spatial fractional Riesz derivative, a system of nonlinear ordinary
differential equations with Toeplitz structure is obtained. For the sake of
reducing the computational complexity, a two-level Strang splitting method is
proposed, where the Toeplitz matrix in the system is split into the sum of a
circulant matrix and a skew-circulant matrix. Therefore, the proposed method
can be quickly implemented by the fast Fourier transform, substituting to
calculate the expensive Toeplitz matrix exponential. Theoretically, the
discrete maximum principle of our method is unconditionally preserved.
Moreover, the analysis of error in the infinite norm with second-order accuracy
is conducted in both time and space. Finally, numerical tests are given to
corroborate our theoretical conclusions and the efficiency of the proposed
method
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