351 research outputs found
Restarted Hessenberg method for solving shifted nonsymmetric linear systems
It is known that the restarted full orthogonalization method (FOM)
outperforms the restarted generalized minimum residual (GMRES) method in
several circumstances for solving shifted linear systems when the shifts are
handled simultaneously. Many variants of them have been proposed to enhance
their performance. We show that another restarted method, the restarted
Hessenberg method [M. Heyouni, M\'ethode de Hessenberg G\'en\'eralis\'ee et
Applications, Ph.D. Thesis, Universit\'e des Sciences et Technologies de Lille,
France, 1996] based on Hessenberg procedure, can effectively be employed, which
can provide accelerating convergence rate with respect to the number of
restarts. Theoretical analysis shows that the new residual of shifted restarted
Hessenberg method is still collinear with each other. In these cases where the
proposed algorithm needs less enough CPU time elapsed to converge than the
earlier established restarted shifted FOM, weighted restarted shifted FOM, and
some other popular shifted iterative solvers based on the short-term vector
recurrence, as shown via extensive numerical experiments involving the recent
popular applications of handling the time fractional differential equations.Comment: 19 pages, 7 tables. Some corrections for updating the reference
A low-rank matrix equation method for solving PDE-constrained optimization problems
PDE-constrained optimization problems arise in a broad number of applications such as hyperthermia cancer treatment and blood flow simulation. Discretization of the optimization problem and using a Lagrangian approach result in a large-scale saddle-point system, which is challenging to solve, and acquiring a full space-time solution is often infeasible. We present a new framework to efficiently compute a low-rank approximation to the solution by reformulating the KKT system into a Sylvester-like matrix equation. This matrix equation is subsequently projected onto a small subspace via an iterative rational Krylov method, and we obtain a reduced problem by imposing a Galerkin condition on its residual. In our work we discuss implementation details and dependence on the various problem parameters. Numerical experiments illustrate the performance of the new strategy also when compared to other low-rank approaches
Numerical Methods for the Nonlocal Wave Equation of the Peridynamics
In this paper we will consider the peridynamic equation of motion which is
described by a second order in time partial integro-differential equation. This
equation has recently received great attention in several fields of Engineering
because seems to provide an effective approach to modeling mechanical systems
avoiding spatial discontinuous derivatives and body singularities. In
particular, we will consider the linear model of peridynamics in a
one-dimensional spatial domain. Here we will review some numerical techniques
to solve this equation and propose some new computational methods of higher
order in space; moreover we will see how to apply the methods studied for the
linear model to the nonlinear one. Also a spectral method for the spatial
discretization of the linear problem will be discussed. Several numerical tests
will be given in order to validate our results
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