6,432 research outputs found
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
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
Some Preconditioning Techniques for Saddle Point Problems
Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constrained preconditioners.\ud
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The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336
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