3 research outputs found
Efficient implementations of the Multivariate Decomposition Method for approximating infinite-variate integrals
In this paper we focus on efficient implementations of the Multivariate
Decomposition Method (MDM) for approximating integrals of -variate
functions. Such -variate integrals occur for example as expectations in
uncertainty quantification. Starting with the anchored decomposition , where the sum is over all
finite subsets of and each depends only on the
variables with , our MDM algorithm approximates the
integral of by first truncating the sum to some `active set' and then
approximating the integral of the remaining functions
term-by-term using Smolyak or (randomized) quasi-Monte Carlo (QMC) quadratures.
The anchored decomposition allows us to compute explicitly by
function evaluations of . Given the specification of the active set and
theoretically derived parameters of the quadrature rules, we exploit structures
in both the formula for computing and the quadrature rules to
develop computationally efficient strategies to implement the MDM in various
scenarios. In particular, we avoid repeated function evaluations at the same
point. We provide numerical results for a test function to demonstrate the
effectiveness of the algorithm
Grouped Transformations and Regularization in High-Dimensional Explainable ANOVA Approximation
In this paper we propose a tool for high-dimensional approximation based on
trigonometric polynomials where we allow only low-dimensional interactions of
variables. In a general high-dimensional setting, it is already possible to
deal with special sampling sets such as sparse grids or rank-1 lattices. This
requires black-box access to the function, i.e., the ability to evaluate it at
any point. Here, we focus on scattered data points and grouped frequency index
sets along the dimensions. From there we propose a fast matrix-vector
multiplication, the grouped Fourier transform, for high-dimensional grouped
index sets. Those transformations can be used in the application of the
previously introduced method of approximating functions with low superposition
dimension based on the analysis of variance (ANOVA) decomposition where there
is a one-to-one correspondence from the ANOVA terms to our proposed groups. The
method is able to dynamically detected important sets of ANOVA terms in the
approximation. In this paper, we consider the involved least-squares problem
and add different forms of regularization: Classical Tikhonov-regularization,
namely, regularized least squares and the technique of group lasso, which
promotes sparsity in the groups. As for the latter, there are no explicit
solution formulas which is why we applied the fast iterative
shrinking-thresholding algorithm to obtain the minimizer. Moreover, we discuss
the possibility of incorporating smoothness information into the least-squares
problem. Numerical experiments in under-, overdetermined, and noisy settings
indicate the applicability of our algorithms. While we consider periodic
functions, the idea can be directly generalized to non-periodic functions as
well
Approximation of high-dimensional periodic functions with Fourier-based methods
In this paper we propose an approximation method for high-dimensional
-periodic functions based on the multivariate ANOVA decomposition. We
provide an analysis on the classical ANOVA decomposition on the torus and prove
some important properties such as the inheritance of smoothness for Sobolev
type spaces and the weighted Wiener algebra. We exploit special kinds of
sparsity in the ANOVA decomposition with the aim to approximate a function in a
scattered data or black-box approximation scenario. This method allows us to
simultaneously achieve an importance ranking on dimensions and dimension
interactions which is referred to as attribute ranking in some applications. In
scattered data approximation we rely on a special algorithm based on the
non-equispaced fast Fourier transform (or NFFT) for fast multiplication with
arising Fourier matrices. For black-box approximation we choose the well-known
rank-1 lattices as sampling schemes and show properties of the appearing
special lattices