967 research outputs found
A Dynamically Adaptive Sparse Grid Method for Quasi-Optimal Interpolation of Multidimensional Analytic Functions
In this work we develop a dynamically adaptive sparse grids (SG) method for
quasi-optimal interpolation of multidimensional analytic functions defined over
a product of one dimensional bounded domains. The goal of such approach is to
construct an interpolant in space that corresponds to the "best -terms"
based on sharp a priori estimate of polynomial coefficients. In the past, SG
methods have been successful in achieving this, with a traditional construction
that relies on the solution to a Knapsack problem: only the most profitable
hierarchical surpluses are added to the SG. However, this approach requires
additional sharp estimates related to the size of the analytic region and the
norm of the interpolation operator, i.e., the Lebesgue constant. Instead, we
present an iterative SG procedure that adaptively refines an estimate of the
region and accounts for the effects of the Lebesgue constant. Our approach does
not require any a priori knowledge of the analyticity or operator norm, is
easily generalized to both affine and non-affine analytic functions, and can be
applied to sparse grids build from one dimensional rules with arbitrary growth
of the number of nodes. In several numerical examples, we utilize our
dynamically adaptive SG to interpolate quantities of interest related to the
solutions of parametrized elliptic and hyperbolic PDEs, and compare the
performance of our quasi-optimal interpolant to several alternative SG schemes
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
A Non-Local Structure Tensor Based Approach for Multicomponent Image Recovery Problems
Non-Local Total Variation (NLTV) has emerged as a useful tool in variational
methods for image recovery problems. In this paper, we extend the NLTV-based
regularization to multicomponent images by taking advantage of the Structure
Tensor (ST) resulting from the gradient of a multicomponent image. The proposed
approach allows us to penalize the non-local variations, jointly for the
different components, through various matrix norms with .
To facilitate the choice of the hyper-parameters, we adopt a constrained convex
optimization approach in which we minimize the data fidelity term subject to a
constraint involving the ST-NLTV regularization. The resulting convex
optimization problem is solved with a novel epigraphical projection method.
This formulation can be efficiently implemented thanks to the flexibility
offered by recent primal-dual proximal algorithms. Experiments are carried out
for multispectral and hyperspectral images. The results demonstrate the
interest of introducing a non-local structure tensor regularization and show
that the proposed approach leads to significant improvements in terms of
convergence speed over current state-of-the-art methods
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