1,577 research outputs found
A block Krylov subspace time-exact solution method for linear ODE systems
We propose a time-exact Krylov-subspace-based method for solving linear ODE (ordinary differential equation) systems of the form and , where is the unknown function. The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the source term , constructed with the help of the truncated SVD (singular value decomposition). The second stage is a special residual-based block Krylov subspace method. The accuracy of the method is only restricted by the accuracy of the piecewise polynomial approximation and by the error of the block Krylov process. Since both errors can, in principle, be made arbitrarily small, this yields, at some costs, a time-exact method. Numerical experiments are presented to demonstrate efficiency of the new method, as compared to an exponential time integrator with Krylov subspace matrix function evaluations
Krylov projection methods for linear Hamiltonian systems
We study geometric properties of Krylov projection methods for large and
sparse linear Hamiltonian systems. We consider in particular energy
preservation. We discuss the connection to structure preserving model
reduction. We illustrate the performance of the methods by applying them to
Hamiltonian PDEs.Comment: 16 pages, 17 figure
Order reduction methods for solving large-scale differential matrix Riccati equations
We consider the numerical solution of large-scale symmetric differential
matrix Riccati equations. Under certain hypotheses on the data, reduced order
methods have recently arisen as a promising class of solution strategies, by
forming low-rank approximations to the sought after solution at selected
timesteps. We show that great computational and memory savings are obtained by
a reduction process onto rational Krylov subspaces, as opposed to current
approaches. By specifically addressing the solution of the reduced differential
equation and reliable stopping criteria, we are able to obtain accurate final
approximations at low memory and computational requirements. This is obtained
by employing a two-phase strategy that separately enhances the accuracy of the
algebraic approximation and the time integration. The new method allows us to
numerically solve much larger problems than in the current literature.
Numerical experiments on benchmark problems illustrate the effectiveness of the
procedure with respect to existing solvers
A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators
We develop an algorithm for computing the solution of a large system of
linear ordinary differential equations (ODEs) with polynomial inhomogeneity.
This is equivalent to computing the action of a certain matrix function on the
vector representing the initial condition. The matrix function is a linear
combination of the matrix exponential and other functions related to the
exponential (the so-called phi-functions). Such computations are the major
computational burden in the implementation of exponential integrators, which
can solve general ODEs. Our approach is to compute the action of the matrix
function by constructing a Krylov subspace using Arnoldi or Lanczos iteration
and projecting the function on this subspace. This is combined with
time-stepping to prevent the Krylov subspace from growing too large. The
algorithm is fully adaptive: it varies both the size of the time steps and the
dimension of the Krylov subspace to reach the required accuracy. We implement
this algorithm in the Matlab function phipm and we give instructions on how to
obtain and use this function. Various numerical experiments show that the phipm
function is often significantly more efficient than the state-of-the-art.Comment: 20 pages, 3 colour figures, code available from
http://www.maths.leeds.ac.uk/~jitse/software.html . v2: Various changes to
improve presentation as suggested by the refere
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