86 research outputs found
From low-rank approximation to an efficient rational Krylov subspace method for the Lyapunov equation
We propose a new method for the approximate solution of the Lyapunov equation
with rank- right-hand side, which is based on extended rational Krylov
subspace approximation with adaptively computed shifts. The shift selection is
obtained from the connection between the Lyapunov equation, solution of systems
of linear ODEs and alternating least squares method for low-rank approximation.
The numerical experiments confirm the effectiveness of our approach.Comment: 17 pages, 1 figure
Deflated GMRES for Systems with Multiple Shifts and Multiple Right-Hand Sides
We consider solution of multiply shifted systems of nonsymmetric linear
equations, possibly also with multiple right-hand sides. First, for a single
right-hand side, the matrix is shifted by several multiples of the identity.
Such problems arise in a number of applications, including lattice quantum
chromodynamics where the matrices are complex and non-Hermitian. Some Krylov
iterative methods such as GMRES and BiCGStab have been used to solve multiply
shifted systems for about the cost of solving just one system. Restarted GMRES
can be improved by deflating eigenvalues for matrices that have a few small
eigenvalues. We show that a particular deflated method, GMRES-DR, can be
applied to multiply shifted systems. In quantum chromodynamics, it is common to
have multiple right-hand sides with multiple shifts for each right-hand side.
We develop a method that efficiently solves the multiple right-hand sides by
using a deflated version of GMRES and yet keeps costs for all of the multiply
shifted systems close to those for one shift. An example is given showing this
can be extremely effective with a quantum chromodynamics matrix.Comment: 19 pages, 9 figure
Matrix Equation Techniques for Certain Evolutionary Partial Differential Equations
We show that the discrete operator stemming from time-space discretization of evolutionary partial differential equations can be represented in terms of a single Sylvester matrix equation. A novel solution strategy that combines projection techniques with the full exploitation of the entry-wise structure of the involved coefficient matrices is proposed. The resulting scheme is able to efficiently solve problems with a tremendous number of degrees of freedom while maintaining a low storage demand as illustrated in several numerical examples
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