28 research outputs found
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
Exponential Rosenbrock methods without order reduction when integrating nonlinear initial value problems
A technique is described in this paper to avoid order reduction when
integrating reaction-diffusion initial boundary value problems with explicit
exponential Rosenbrock methods. The technique is valid for any Rosenbrock
method, without having to impose any stiff order conditions, and for general
time-dependent boundary values. An analysis on the global error is thoroughly
performed and some numerical experiments are shown which corroborate the
theoretical results, and in which a big gain in efficiency with respect to
applying the standard method of lines can be observed.Comment: 27 pages, 2 figure
A robust spectral method for pricing of American put options on zero-coupon bonds
American put options on a zero-coupon bond problem is reformulated as a
linear complementarity problem of the option value and approximated by a nonlinear
partial differential equation. The equation is solved by an exponential time differencing
method combined with a barycentric Legendre interpolation and the Krylov projection
algorithm. Numerical examples shows the stability and good accuracy of the method. A bond is a financial instrument which allows an investor to loan money to an entity
(a corporate or governmental) that borrows the funds for a period of time at a fixed interest rate (the coupon) and agrees to pay a fixed amount (the principal) to the investor
at maturity. A zero-coupon bond is a bond that makes no periodic interest payments
Adaptive rational Krylov methods for exponential Runge--Kutta integrators
We consider the solution of large stiff systems of ordinary differential
equations with explicit exponential Runge--Kutta integrators. These problems
arise from semi-discretized semi-linear parabolic partial differential
equations on continuous domains or on inherently discrete graph domains. A
series of results reduces the requirement of computing linear combinations of
-functions in exponential integrators to the approximation of the
action of a smaller number of matrix exponentials on certain vectors.
State-of-the-art computational methods use polynomial Krylov subspaces of
adaptive size for this task. They have the drawback that the required Krylov
subspace iteration numbers to obtain a desired tolerance increase drastically
with the spectral radius of the discrete linear differential operator, e.g.,
the problem size. We present an approach that leverages rational Krylov
subspace methods promising superior approximation qualities. We prove a novel
a-posteriori error estimate of rational Krylov approximations to the action of
the matrix exponential on vectors for single time points, which allows for an
adaptive approach similar to existing polynomial Krylov techniques. We discuss
pole selection and the efficient solution of the arising sequences of shifted
linear systems by direct and preconditioned iterative solvers. Numerical
experiments show that our method outperforms the state of the art for
sufficiently large spectral radii of the discrete linear differential
operators. The key to this are approximately constant rational Krylov iteration
numbers, which enable a near-linear scaling of the runtime with respect to the
problem size