625 research outputs found
Factorizing the Stochastic Galerkin System
Recent work has explored solver strategies for the linear system of equations
arising from a spectral Galerkin approximation of the solution of PDEs with
parameterized (or stochastic) inputs. We consider the related problem of a
matrix equation whose matrix and right hand side depend on a set of parameters
(e.g. a PDE with stochastic inputs semidiscretized in space) and examine the
linear system arising from a similar Galerkin approximation of the solution. We
derive a useful factorization of this system of equations, which yields bounds
on the eigenvalues, clues to preconditioning, and a flexible implementation
method for a wide array of problems. We complement this analysis with (i) a
numerical study of preconditioners on a standard elliptic PDE test problem and
(ii) a fluids application using existing CFD codes; the MATLAB codes used in
the numerical studies are available online.Comment: 13 pages, 4 figures, 2 table
Approximation of the scattering amplitude
The simultaneous solution of Ax=b and ATy=g is required in a number of situations. Darmofal and Lu have proposed a method based on the Quasi-Minimal residual algorithm (QMR). We will introduce a technique for the same purpose based on the LSQR method and show how its performance can be improved when using the Generalized LSQR method. We further show how preconditioners can be introduced to enhance the speed of convergence and discuss different preconditioners that can be used. The scattering amplitude gTx, a widely used quantity in signal processing for example, has a close connection to the above problem since x represents the solution of the forward problem and g is the right hand side of the adjoint system. We show how this quantity can be efficiently approximated using Gauss quadrature and introduce a Block-Lanczos process that approximates the scattering amplitude and which can also be used with preconditioners
Efficient approximation of functions of some large matrices by partial fraction expansions
Some important applicative problems require the evaluation of functions
of large and sparse and/or \emph{localized} matrices . Popular and
interesting techniques for computing and , where
is a vector, are based on partial fraction expansions. However,
some of these techniques require solving several linear systems whose matrices
differ from by a complex multiple of the identity matrix for computing
or require inverting sequences of matrices with the same
characteristics for computing . Here we study the use and the
convergence of a recent technique for generating sequences of incomplete
factorizations of matrices in order to face with both these issues. The
solution of the sequences of linear systems and approximate matrix inversions
above can be computed efficiently provided that shows certain decay
properties. These strategies have good parallel potentialities. Our claims are
confirmed by numerical tests
Symmetric Stair Preconditioning of Linear Systems for Parallel Trajectory Optimization
There has been a growing interest in parallel strategies for solving
trajectory optimization problems. One key step in many algorithmic approaches
to trajectory optimization is the solution of moderately-large and sparse
linear systems. Iterative methods are particularly well-suited for parallel
solves of such systems. However, fast and stable convergence of iterative
methods is reliant on the application of a high-quality preconditioner that
reduces the spread and increase the clustering of the eigenvalues of the target
matrix. To improve the performance of these approaches, we present a new
parallel-friendly symmetric stair preconditioner. We prove that our
preconditioner has advantageous theoretical properties when used in conjunction
with iterative methods for trajectory optimization such as a more clustered
eigenvalue spectrum. Numerical experiments with typical trajectory optimization
problems reveal that as compared to the best alternative parallel
preconditioner from the literature, our symmetric stair preconditioner provides
up to a 34% reduction in condition number and up to a 25% reduction in the
number of resulting linear system solver iterations.Comment: Accepted to ICRA 2024, 8 pages, 3 figure
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