294 research outputs found
A Preconditioned Iteration Method for Solving Sylvester Equations
A preconditioned gradient-based iterative method is derived by judicious selection of two auxil- iary matrices. The strategy is based on the Newton’s iteration method and can be regarded as a
generalization of the splitting iterative method for system of linear equations. We analyze the convergence of the method and illustrate that the approach is able to considerably accelerate the convergence of the gradient-based iterative method
Isogeometric preconditioners based on fast solvers for the Sylvester equation
We consider large linear systems arising from the isogeometric discretization
of the Poisson problem on a single-patch domain. The numerical solution of such
systems is considered a challenging task, particularly when the degree of the
splines employed as basis functions is high. We consider a preconditioning
strategy which is based on the solution of a Sylvester-like equation at each
step of an iterative solver. We show that this strategy, which fully exploits
the tensor structure that underlies isogeometric problems, is robust with
respect to both mesh size and spline degree, although it may suffer from the
presence of complicated geometry or coefficients. We consider two popular
solvers for the Sylvester equation, a direct one and an iterative one, and we
discuss in detail their implementation and efficiency for 2D and 3D problems on
single-patch or conforming multi-patch NURBS geometries. Numerical experiments
for problems with different domain geometries are presented, which demonstrate
the potential of this approach
Matrix-equation-based strategies for convection-diffusion equations
We are interested in the numerical solution of nonsymmetric linear systems
arising from the discretization of convection-diffusion partial differential
equations with separable coefficients and dominant convection. Preconditioners
based on the matrix equation formulation of the problem are proposed, which
naturally approximate the original discretized problem. For certain types of
convection coefficients, we show that the explicit solution of the matrix
equation can effectively replace the linear system solution. Numerical
experiments with data stemming from two and three dimensional problems are
reported, illustrating the potential of the proposed methodology
Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure
The numerical solution of partial differential equations on high-dimensional
domains gives rise to computationally challenging linear systems. When using
standard discretization techniques, the size of the linear system grows
exponentially with the number of dimensions, making the use of classic
iterative solvers infeasible. During the last few years, low-rank tensor
approaches have been developed that allow to mitigate this curse of
dimensionality by exploiting the underlying structure of the linear operator.
In this work, we focus on tensors represented in the Tucker and tensor train
formats. We propose two preconditioned gradient methods on the corresponding
low-rank tensor manifolds: A Riemannian version of the preconditioned
Richardson method as well as an approximate Newton scheme based on the
Riemannian Hessian. For the latter, considerable attention is given to the
efficient solution of the resulting Newton equation. In numerical experiments,
we compare the efficiency of our Riemannian algorithms with other established
tensor-based approaches such as a truncated preconditioned Richardson method
and the alternating linear scheme. The results show that our approximate
Riemannian Newton scheme is significantly faster in cases when the application
of the linear operator is expensive.Comment: 24 pages, 8 figure
Efficient Approaches for Enclosing the United Solution Set of the Interval Generalized Sylvester Matrix Equation
In this work, we investigate the interval generalized Sylvester matrix
equation and develop some
techniques for obtaining outer estimations for the so-called united solution
set of this interval system. First, we propose a modified variant of the
Krawczyk operator which causes reducing computational complexity to cubic,
compared to Kronecker product form. We then propose an iterative technique for
enclosing the solution set. These approaches are based on spectral
decompositions of the midpoints of , , and
and in both of them we suppose that the midpoints of and
are simultaneously diagonalizable as well as for the midpoints of
the matrices and . Some numerical experiments are given to
illustrate the performance of the proposed methods
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
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