2,750 research outputs found
Aggregation-based Multilevel Methods for Lattice QCD
In Lattice QCD computations a substantial amount of work is spent in solving
the Dirac equation. In the recent past it has been observed that conventional
Krylov solvers tend to critically slow down for large lattices and small quark
masses. We present a Schwarz alternating procedure (SAP) multilevel method as a
solver for the Clover improved Wilson discretization of the Dirac equation.
This approach combines two components (SAP and algebraic multigrid) that have
separately been used in lattice QCD before. In combination with a bootstrap
setup procedure we show that considerable speed-up over conventional Krylov
subspace methods for realistic configurations can be achieved.Comment: Talk presented at the XXIX International Symposium on Lattice Field
Theory, July 10-16, 2011, Lake Tahoe, Californi
Multilevel Methods for Uncertainty Quantification of Elliptic PDEs with Random Anisotropic Diffusion
We consider elliptic diffusion problems with a random anisotropic diffusion
coefficient, where, in a notable direction given by a random vector field, the
diffusion strength differs from the diffusion strength perpendicular to this
notable direction. The Karhunen-Lo\`eve expansion then yields a parametrisation
of the random vector field and, therefore, also of the solution of the elliptic
diffusion problem. We show that, given regularity of the elliptic diffusion
problem, the decay of the Karhunen-Lo\`eve expansion entirely determines the
regularity of the solution's dependence on the random parameter, also when
considering this higher spatial regularity. This result then implies that
multilevel collocation and multilevel quadrature methods may be used to lessen
the computation complexity when approximating quantities of interest, like the
solution's mean or its second moment, while still yielding the expected rates
of convergence. Numerical examples in three spatial dimensions are provided to
validate the presented theory
Multilevel Double Loop Monte Carlo and Stochastic Collocation Methods with Importance Sampling for Bayesian Optimal Experimental Design
An optimal experimental set-up maximizes the value of data for statistical
inferences and predictions. The efficiency of strategies for finding optimal
experimental set-ups is particularly important for experiments that are
time-consuming or expensive to perform. For instance, in the situation when the
experiments are modeled by Partial Differential Equations (PDEs), multilevel
methods have been proven to dramatically reduce the computational complexity of
their single-level counterparts when estimating expected values. For a setting
where PDEs can model experiments, we propose two multilevel methods for
estimating a popular design criterion known as the expected information gain in
simulation-based Bayesian optimal experimental design. The expected information
gain criterion is of a nested expectation form, and only a handful of
multilevel methods have been proposed for problems of such form. We propose a
Multilevel Double Loop Monte Carlo (MLDLMC), which is a multilevel strategy
with Double Loop Monte Carlo (DLMC), and a Multilevel Double Loop Stochastic
Collocation (MLDLSC), which performs a high-dimensional integration by
deterministic quadrature on sparse grids. For both methods, the Laplace
approximation is used for importance sampling that significantly reduces the
computational work of estimating inner expectations. The optimal values of the
method parameters are determined by minimizing the average computational work,
subject to satisfying the desired error tolerance. The computational
efficiencies of the methods are demonstrated by estimating the expected
information gain for Bayesian inference of the fiber orientation in composite
laminate materials from an electrical impedance tomography experiment. MLDLSC
performs better than MLDLMC when the regularity of the quantity of interest,
with respect to the additive noise and the unknown parameters, can be
exploited
Some multilevel methods on graded meshes
AbstractWe consider Yserentant's hierarchical basis method and multilevel diagonal scaling method on a class of refined meshes used in the numerical approximation of boundary value problems on polygonal domains in the presence of singularities. We show, as in the uniform case, that the stiffness matrix of the first method has a condition number bounded by (ln(1/h))2, where h is the meshsize of the triangulation. For the second method, we show that the condition number of the iteration operator is bounded by ln(1/h), which is worse than in the uniform case but better than the hierarchical basis method. As usual, we deduce that the condition number of the BPX iteration operator is bounded by ln(1/h). Finally, graded meshes fulfilling the general conditions are presented and numerical tests are given which confirm the theoretical bounds
Adaptive multilevel methods for obstacle problems
The authors consider the discretization of obstacle problems for second-order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by preconditioned conjugate gradient iterations. The proposed preconditioners are treated theoretically as abstract additive Schwarz methods and are implemented as truncated hierarchical basis preconditioners. To allow for local mesh refinement semilocal and local a posteriors error estimates are derived, providing lower and upper estimates for the discretization error. The theoretical results are illustrated by numerical computations
Algebraic Multilevel Methods for Markov Chains
A new algebraic multilevel algorithm for computing the second eigenvector of a column-stochastic matrix is presented. The method is based on a deflation approach in a multilevel aggregation framework. In particular a square and stretch approach, first introduced by Treister and Yavneh, is applied. The method is shown to yield good convergence properties for typical example problems
From rough path estimates to multilevel Monte Carlo
New classes of stochastic differential equations can now be studied using
rough path theory (e.g. Lyons et al. [LCL07] or Friz--Hairer [FH14]). In this
paper we investigate, from a numerical analysis point of view, stochastic
differential equations driven by Gaussian noise in the aforementioned sense.
Our focus lies on numerical implementations, and more specifically on the
saving possible via multilevel methods. Our analysis relies on a subtle
combination of pathwise estimates, Gaussian concentration, and multilevel
ideas. Numerical examples are given which both illustrate and confirm our
findings.Comment: 34 page
Implementing abstract multigrid or multilevel methods
Multigrid methods can be formulated as an algorithm for an abstract problem that is independent of the partial differential equation, domain, and discretization method. In such an abstract setting, problems not arising from partial differential equations can be treated. A general theory exists for linear problems. The general theory was motivated by a series of abstract solvers (Madpack). The latest version was motivated by the theory. Madpack now allows for a wide variety of iterative and direct solvers, preconditioners, and interpolation and projection schemes, including user callback ones. It allows for sparse, dense, and stencil matrices. Mildly nonlinear problems can be handled. Also, there is a fast, multigrid Poisson solver (two and three dimensions). The type of solvers and design decisions (including language, data structures, external library support, and callbacks) are discussed. Based on the author's experiences with two versions of Madpack, a better approach is proposed. This is based on a mixed language formulation (C and FORTRAN + preprocessor). Reasons for not using FORTRAN, C, or C++ (individually) are given. Implementing the proposed strategy is not difficult
Structural synthesis: Precursor and catalyst
More than twenty five years have elapsed since it was recognized that a rather general class of structural design optimization tasks could be properly posed as an inequality constrained minimization problem. It is suggested that, independent of primary discipline area, it will be useful to think about: (1) posing design problems in terms of an objective function and inequality constraints; (2) generating design oriented approximate analysis methods (giving special attention to behavior sensitivity analysis); (3) distinguishing between decisions that lead to an analysis model and those that lead to a design model; (4) finding ways to generate a sequence of approximate design optimization problems that capture the essential characteristics of the primary problem, while still having an explicit algebraic form that is matched to one or more of the established optimization algorithms; (5) examining the potential of optimum design sensitivity analysis to facilitate quantitative trade-off studies as well as participation in multilevel design activities. It should be kept in mind that multilevel methods are inherently well suited to a parallel mode of operation in computer terms or to a division of labor between task groups in organizational terms. Based on structural experience with multilevel methods general guidelines are suggested
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