266 research outputs found
Real Analysis, Harmonic Analysis and Applications to PDE
There have been important developments in the last few years in the point-of-view and methods of harmonic analysis, and at the same time significant concurrent progress in the application of these to partial differential equations and related subjects. The conference brought together experts and young scientists working in these two directions, with the objective of furthering these important interactions
Real Analysis, Harmonic Analysis and Applications
[no abstract available
Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format
We apply the Tensor Train (TT) decomposition to construct the tensor product
Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic
elliptic diffusion PDE with the stochastic Galerkin discretization, and to
compute some quantities of interest (mean, variance, exceedance probabilities).
We assume that the random diffusion coefficient is given as a smooth
transformation of a Gaussian random field. In this case, the PCE is delivered
by a complicated formula, which lacks an analytic TT representation. To
construct its TT approximation numerically, we develop the new block TT cross
algorithm, a method that computes the whole TT decomposition from a few
evaluations of the PCE formula. The new method is conceptually similar to the
adaptive cross approximation in the TT format, but is more efficient when
several tensors must be stored in the same TT representation, which is the case
for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin
matrix and to compute the solution of the elliptic equation and its
post-processing, staying in the TT format.
We compare our technique with the traditional sparse polynomial chaos and the
Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial
degree is bounded for each random variable independently. This provides higher
accuracy than the sparse polynomial set or the Monte Carlo method, but the
cardinality of the tensor product set grows exponentially with the number of
random variables. However, when the PCE coefficients are implicitly
approximated in the TT format, the computations with the full tensor product
polynomial set become possible. In the numerical experiments, we confirm that
the new methodology is competitive in a wide range of parameters, especially
where high accuracy and high polynomial degrees are required.Comment: This is a major revision of the manuscript arXiv:1406.2816 with
significantly extended numerical experiments. Some unused material is remove
Learning Theory and Approximation
Learning theory studies data structures from samples and aims at understanding unknown function relations behind them. This leads to interesting theoretical problems which can be often attacked with methods from Approximation Theory. This workshop - the second one of this type at the MFO - has concentrated on the following recent topics: Learning of manifolds and the geometry of data; sparsity and dimension reduction; error analysis and algorithmic aspects, including kernel based methods for regression and classification; application of multiscale aspects and of refinement algorithms to learning
Stability estimates for the expected utility in Bayesian optimal experimental design
We study stability properties of the expected utility function in Bayesian
optimal experimental design. We provide a framework for this problem in a
non-parametric setting and prove a convergence rate of the expected utility
with respect to a likelihood perturbation. This rate is uniform over the design
space and its sharpness in the general setting is demonstrated by proving a
lower bound in a special case. To make the problem more concrete we proceed by
considering non-linear Bayesian inverse problems with Gaussian likelihood and
prove that the assumptions set out for the general case are satisfied and
regain the stability of the expected utility with respect to perturbations to
the observation map. Theoretical convergence rates are demonstrated numerically
in three different examples.Comment: 20 pages; 6 figure
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