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On the Optimal Solution of Large Linear Systems
The information-based study of the optimal solution of large linear systems is initiated by studying the case of Krylov information. Among the algorithms which use Krylov information are minimal residual, conjugate gradient, Chebyshev, and successive approximation algorithms. A "sharp" lower bound on the number of matrix-vector multiplications required to compute an E- approximation is obtained for any orthogonally invariant class of matrices. Examples of such classes include many of practical interest such as symmetric matrices, symmetric positive definite matrices, and matrices with bounded condition number. It is shown that the minimal residual algorithm is within at most one matrix-vector multiplication of the lower bound. A similar result is obtained for the generalized minimal residual algorithm. The lower bound is computed for certain classes of orthogonally invariant matrices. We show how the lack of certain properties (symmetry, positive definiteness) increases the lower bound. A conjecture and a number of open problems are stated
Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements
We consider the e�cient solution of the Cahn-Hilliard variational inequality using an implicit time discretization, which is formulated as an optimal control problem with pointwise constraints on the control. By applying a semi-smooth Newton method combined with a Moreau-Yosida regularization technique for handling the control constraints we show superlinear convergence in function space. At the heart of this method lies the solution of large and sparse linear systems for which we propose the use of preconditioned Krylov subspace solvers using an e�ective Schur complement approximation. Numerical results illustrate the competitiveness of this approach
Performance of algebraic multigrid methods for non-symmetric matrices arising in particle methods
Large linear systems with sparse, non-symmetric matrices arise in the
modeling of Markov chains or in the discretization of convection-diffusion
problems. Due to their potential to solve sparse linear systems with an effort
that is linear in the number of unknowns, algebraic multigrid (AMG) methods are
of fundamental interest for such systems. For symmetric positive definite
matrices, fundamental theoretical convergence results are established, and
efficient AMG solvers have been developed. In contrast, for non-symmetric
matrices, theoretical convergence results have been provided only recently. A
property that is sufficient for convergence is that the matrix be an M-matrix.
In this paper, we present how the simulation of incompressible fluid flows with
particle methods leads to large linear systems with sparse, non-symmetric
matrices. In each time step, the Poisson equation is approximated by meshfree
finite differences. While traditional least squares approaches do not guarantee
an M-matrix structure, an approach based on linear optimization yields
optimally sparse M-matrices. For both types of discretization approaches, we
investigate the performance of a classical AMG method, as well as an AMLI type
method. While in the considered test problems, the M-matrix structure turns out
not to be necessary for the convergence of AMG, problems can occur when it is
violated. In addition, the matrices obtained by the linear optimization
approach result in fast solution times due to their optimal sparsity.Comment: 16 pages, 7 figure
Linear Hamilton Jacobi Bellman Equations in High Dimensions
The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal
solution to large classes of control problems. Unfortunately, this generality
comes at a price, the calculation of such solutions is typically intractible
for systems with more than moderate state space size due to the curse of
dimensionality. This work combines recent results in the structure of the HJB,
and its reduction to a linear Partial Differential Equation (PDE), with methods
based on low rank tensor representations, known as a separated representations,
to address the curse of dimensionality. The result is an algorithm to solve
optimal control problems which scales linearly with the number of states in a
system, and is applicable to systems that are nonlinear with stochastic forcing
in finite-horizon, average cost, and first-exit settings. The method is
demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with
system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201
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