3,205 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
Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients
We consider elliptic partial differential equations with diffusion
coefficients that depend affinely on countably many parameters. We study the
summability properties of polynomial expansions of the function mapping
parameter values to solutions of the PDE, considering both Taylor and Legendre
series. Our results considerably improve on previously known estimates of this
type, in particular taking into account structural features of the affine
parametrization of the coefficient. Moreover, the results carry over to more
general Jacobi polynomial expansions. We demonstrate that the new bounds are
sharp in certain model cases and we illustrate them by numerical experiments
Multi-index Stochastic Collocation convergence rates for random PDEs with parametric regularity
We analyze the recent Multi-index Stochastic Collocation (MISC) method for
computing statistics of the solution of a partial differential equation (PDEs)
with random data, where the random coefficient is parametrized by means of a
countable sequence of terms in a suitable expansion. MISC is a combination
technique based on mixed differences of spatial approximations and quadratures
over the space of random data and, naturally, the error analysis uses the joint
regularity of the solution with respect to both the variables in the physical
domain and parametric variables. In MISC, the number of problem solutions
performed at each discretization level is not determined by balancing the
spatial and stochastic components of the error, but rather by suitably
extending the knapsack-problem approach employed in the construction of the
quasi-optimal sparse-grids and Multi-index Monte Carlo methods. We use a greedy
optimization procedure to select the most effective mixed differences to
include in the MISC estimator. We apply our theoretical estimates to a linear
elliptic PDEs in which the log-diffusion coefficient is modeled as a random
field, with a covariance similar to a Mat\'ern model, whose realizations have
spatial regularity determined by a scalar parameter. We conduct a complexity
analysis based on a summability argument showing algebraic rates of convergence
with respect to the overall computational work. The rate of convergence depends
on the smoothness parameter, the physical dimensionality and the efficiency of
the linear solver. Numerical experiments show the effectiveness of MISC in this
infinite-dimensional setting compared with the Multi-index Monte Carlo method
and compare the convergence rate against the rates predicted in our theoretical
analysis
Constructive Tensor Field Theory
We provide an up-to-date review of the recent constructive program for field
theories of the vector, matrix and tensor type, focusing not on the models
themselves but on the mathematical tools used.Comment: arXiv admin note: text overlap with arXiv:1401.500
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