15,801 research outputs found
Recent Advances in Graph Partitioning
We survey recent trends in practical algorithms for balanced graph
partitioning together with applications and future research directions
Locally optimal unstructured finite element meshes in 3 dimensions
This paper investigates the adaptive finite element solution of a general class of variational problems in three dimensions using a combination of node movement, edge swapping, face swapping and node insertion. The adaptive strategy proposed is a generalization of previous work in two dimensions and is based upon the construction of a hierarchy of locally optimal meshes.
Results presented, both for a single equation and a system of coupled equations, suggest that this approach is able to produce better meshes of tetrahedra than those obtained by more conventional adaptive strategies and in a relatively efficient manner
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
Parallel implementation of an optimal two level additive Schwarz preconditioner for the 3-D finite element solution of elliptic partial differential equations
This paper presents a description of the extension and parallel implementation of a new two level additive Schwarz (AS) preconditioner for the solution of 3-D elliptic partial differential equations (PDEs). This preconditioner, introduced in Bank et al. (SIAM J. Sci. Comput. 2002; 23: 1818), is based upon the use of a novel form of overlap between the subdomains which makes use of a hierarchy of meshes: with just a single layer of overlapping elements at each level of the hierarchy. The generalization considered here is based upon the restricted AS approach reported in (SIAM J. Sci. Comput. 1999; 21: 792) and the parallel implementation is an extension of work in two dimensions (Concurrency Comput. Practice Experience 2001; 13: 327)
A Dimension-Adaptive Multi-Index Monte Carlo Method Applied to a Model of a Heat Exchanger
We present an adaptive version of the Multi-Index Monte Carlo method,
introduced by Haji-Ali, Nobile and Tempone (2016), for simulating PDEs with
coefficients that are random fields. A classical technique for sampling from
these random fields is the Karhunen-Lo\`eve expansion. Our adaptive algorithm
is based on the adaptive algorithm used in sparse grid cubature as introduced
by Gerstner and Griebel (2003), and automatically chooses the number of terms
needed in this expansion, as well as the required spatial discretizations of
the PDE model. We apply the method to a simplified model of a heat exchanger
with random insulator material, where the stochastic characteristics are
modeled as a lognormal random field, and we show consistent computational
savings
A multilevel approach for obtaining locally optimal finite element meshes
In this paper we consider the adaptive finite element solution of a general class of variational problems using a combination of node insertion, node movement and edge swapping. The adaptive strategy that is proposed is based upon the construction of a hierarchy of locally optimal meshes starting with a coarse grid for which the location and connectivity of the nodes is optimized.
This grid is then locally refined and the new mesh is optimized in the same manner. Results presented indicate that this approach is able to produce better meshes than those possible by more conventional adaptive strategies and in a relatively efficient manner
Optimization of mesh hierarchies in Multilevel Monte Carlo samplers
We perform a general optimization of the parameters in the Multilevel Monte
Carlo (MLMC) discretization hierarchy based on uniform discretization methods
with general approximation orders and computational costs. We optimize
hierarchies with geometric and non-geometric sequences of mesh sizes and show
that geometric hierarchies, when optimized, are nearly optimal and have the
same asymptotic computational complexity as non-geometric optimal hierarchies.
We discuss how enforcing constraints on parameters of MLMC hierarchies affects
the optimality of these hierarchies. These constraints include an upper and a
lower bound on the mesh size or enforcing that the number of samples and the
number of discretization elements are integers. We also discuss the optimal
tolerance splitting between the bias and the statistical error contributions
and its asymptotic behavior. To provide numerical grounds for our theoretical
results, we apply these optimized hierarchies together with the Continuation
MLMC Algorithm. The first example considers a three-dimensional elliptic
partial differential equation with random inputs. Its space discretization is
based on continuous piecewise trilinear finite elements and the corresponding
linear system is solved by either a direct or an iterative solver. The second
example considers a one-dimensional It\^o stochastic differential equation
discretized by a Milstein scheme
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