314 research outputs found
Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
Elliptic partial differential equations with diffusion coefficients of
lognormal form, that is , where is a Gaussian random field, are
considered. We study the summability properties of the Hermite
polynomial expansion of the solution in terms of the countably many scalar
parameters appearing in a given representation of . These summability
results have direct consequences on the approximation rates of best -term
truncated Hermite expansions. Our results significantly improve on the state of
the art estimates available for this problem. In particular, they take into
account the support properties of the basis functions involved in the
representation of , in addition to the size of these functions. One
interesting conclusion from our analysis is that in certain relevant cases, the
Karhunen-Lo\`eve representation of may not be the best choice concerning
the resulting sparsity and approximability of the Hermite expansion
Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes
This paper is concerned with the design, analysis and implementation of
preconditioning concepts for spectral Discontinuous Galerkin discretizations of
elliptic boundary value problems. While presently known techniques realize a
growth of the condition numbers that is logarithmic in the polynomial degrees
when all degrees are equal and quadratic otherwise, our main objective is to
realize full robustness with respect to arbitrarily large locally varying
polynomial degrees degrees, i.e., under mild grading constraints condition
numbers stay uniformly bounded with respect to the mesh size and variable
degrees. The conceptual foundation of the envisaged preconditioners is the
auxiliary space method. The main conceptual ingredients that will be shown in
this framework to yield "optimal" preconditioners in the above sense are
Legendre-Gauss-Lobatto grids in connection with certain associated anisotropic
nested dyadic grids as well as specially adapted wavelet preconditioners for
the resulting low order auxiliary problems. Moreover, the preconditioners have
a modular form that facilitates somewhat simplified partial realizations. One
of the components can, for instance, be conveniently combined with domain
decomposition, at the expense though of a logarithmic growth of condition
numbers. Our analysis is complemented by quantitative experimental studies of
the main components.Comment: 41 pages, 11 figures; Major revision: rearrangement of the contents
for better readability, part on wavelet preconditioner adde
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New Discretization Methods for the Numerical Approximation of PDEs
The construction and mathematical analysis of numerical methods for PDEs is a fundamental area of modern applied mathematics. Among the various techniques that have been proposed in the past, some – in particular, finite element methods, – have been exceptionally successful in a range of applications. There are however a number of important challenges that remain, including the optimal adaptive finite element approximation of solutions to transport-dominated diffusion problems, the efficient numerical approximation of parametrized families of PDEs, and the efficient numerical approximation of high-dimensional partial differential equations (that arise from stochastic analysis and statistical physics, for example, in the form of a backward Kolmogorov equation, which, unlike its formal adjoint, the forward Kolmogorov equation, is not in divergence form, and therefore not directly amenable to finite element approximation, even when the spatial dimension is low). In recent years several original and conceptionally new ideas have emerged in order to tackle these open problems.
The goal of this workshop was to discuss and compare a number of novel approaches, to study their potential and applicability, and to formulate the strategic goals and directions of research in this field for the next five years
Metric based up-scaling
We consider divergence form elliptic operators in dimension with
coefficients. Although solutions of these operators are only
H\"{o}lder continuous, we show that they are differentiable ()
with respect to harmonic coordinates. It follows that numerical homogenization
can be extended to situations where the medium has no ergodicity at small
scales and is characterized by a continuum of scales by transferring a new
metric in addition to traditional averaged (homogenized) quantities from
subgrid scales into computational scales and error bounds can be given. This
numerical homogenization method can also be used as a compression tool for
differential operators.Comment: Final version. Accepted for publication in Communications on Pure and
Applied Mathematics. Presented at CIMMS (March 2005), Socams 2005 (April),
Oberwolfach, MPI Leipzig (May 2005), CIRM (July 2005). Higher resolution
figures are available at http://www.acm.caltech.edu/~owhadi
A posteriori error estimation for hp -version time-stepping methods for parabolic partial differential equations
The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiment
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