Shift-invert Arnoldi Method Research Articles

Double-shift-invert Arnoldi method for computing the matrix exponential

Computing the matrix exponential for large sparse matrices is essential for solving evolution equations numerically. Conventionally, the shift-invert Arnoldi method is used to compute a matrix exponential and a vector product. This method transforms the original matrix using a shift, and generates the Krylov subspace of the transformed matrix for approximation. The transformation makes the approximation converge faster. In this paper, a new method called the double-shift-invert Arnoldi method is explored. This method uses two shifts for transforming the matrix, and this transformation results in a faster convergence than the shift-invert Arnoldi method.

Fast numerical solution for fractional diffusion equations by exponential quadrature rule

After spatial discretization to the fractional diffusion equation by the shifted Grünwald formula, it leads to a system of ordinary differential equations, where the resulting coefficient matrix possesses the Toeplitz-like structure. An exponential quadrature rule is employed to solve such a system of ordinary differential equations. The convergence by the proposed method is theoretically studied. In practical computation, the product of a Toeplitz-like matrix exponential and a vector is calculated by the shift-invert Arnoldi method. Meanwhile, the coefficient matrix satisfies a condition that guarantees the fast approximation by the shift-invert Arnoldi method. Numerical results are given to demonstrate the efficiency of the proposed method.

Fast Exponential Time Integration for Pricing Options in Stochastic Volatility Jump Diffusion Models

AbstractThe stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

Shift-Invert Arnoldi's Method with Preconditioned Iterative Solves

We consider the computation of a few eigenvectors and corresponding eigenvalues of a large sparse nonsymmetric matrix using shift-invert Arnoldi's method with and without implicit restarts. For the inner iterations we use preconditioned GMRES as the inexact iterative solver. The costs of the solves are measured by the number of inner iterations needed by the iterative solver at each outer step of the algorithm. We first extend the relaxation strategy developed by Simoncini [SIAM J. Numer. Anal., 43 (2005), pp. 1155-1174] to implicitly restarted Arnoldi's method, which yields an improvement in the overall costs of the method. Secondly, we apply a new preconditioning strategy to the inner solver. We show that small rank changes to the preconditioner can produce significant savings in the total number of iterations. The combination of the new preconditioner with the relaxation strategy in implicitly restarted Arnoldi produces enhancement in the overall costs of around 50 percent in the examples considered here. Numerical experiments illustrate the theory throughout the paper.

Shift-Invert Arnoldi Approximation to the Toeplitz Matrix Exponential

The shift-invert Arnoldi method is employed to generate an orthonormal basis from the Krylov subspace corresponding to a real Toeplitz matrix and an initial vector. The vectors and recurrence coefficients produced by this method are exploited to approximate the Toeplitz matrix exponential. Toeplitz matrix inversion formula and rapid Toeplitz matrix-vector multiplications are utilized to lower the computational costs. For convergence analysis, a sufficient condition is established to guarantee that the error bound is independent of the norm of the matrix. Numerical results are given to demonstrate the efficiency of the method.

Shift-Invert Arnoldi's Method with Preconditioned Iterative Solves

We consider the computation of a few eigenvectors and corresponding eigenvalues of a large sparse nonsymmetric matrix using shift-invert Arnoldi's method with and without implicit restarts. For the...

A parallel adaptive unstructured finite volume method for linear stability (normal mode) analysis of viscoelastic fluid flows

A parallel unstructured finite volume method is presented for analysis of the stability of two-dimensional steady Oldroyd-B fluid flows to small amplitude three-dimensional perturbations. A semi-staggered dilation-free finite volume discretization with Newton’s method is used to compute steady base flows. The linear stability problem is treated as a generalized eigenvalue problem (GEVP) in which the rightmost eigenvalue determines the stability of the base flow. The rightmost eigenvalues associated with the most dangerous eigenfunctions are computed through the use of the shift-invert Arnoldi method. To avoid fine meshing in the regions where the flow variables are changing slowly, a local mesh refinement technique is used in order to increase numerical accuracy with a lower computational cost. The CUBIT mesh generation environment has been used to refine the quadrilateral meshes locally. In order to achieve higher performance on parallel machines the algebraic systems of equations resulting from the steady problem and the GEVP have been solved by implementing the MUltifrontal Massively Parallel Solver ( MUMPS). The proposed method is applied to the linear stability analysis of the flow of an Oldroyd-B fluid past a linear periodic array of circular cylinders in a channel and a linear array of circular half cylinders placed on channel walls. Two different leading eigenfunctions are identified for close and wide cylinder spacing for the periodic array of cylinders. The numerical results indicate good agreement with the numerical and experimental results available in the literature.