119 research outputs found
Fast Bayesian Optimal Experimental Design for Seismic Source Inversion
We develop a fast method for optimally designing experiments in the context
of statistical seismic source inversion. In particular, we efficiently compute
the optimal number and locations of the receivers or seismographs. The seismic
source is modeled by a point moment tensor multiplied by a time-dependent
function. The parameters include the source location, moment tensor components,
and start time and frequency in the time function. The forward problem is
modeled by elastodynamic wave equations. We show that the Hessian of the cost
functional, which is usually defined as the square of the weighted L2 norm of
the difference between the experimental data and the simulated data, is
proportional to the measurement time and the number of receivers. Consequently,
the posterior distribution of the parameters, in a Bayesian setting,
concentrates around the "true" parameters, and we can employ Laplace
approximation and speed up the estimation of the expected Kullback-Leibler
divergence (expected information gain), the optimality criterion in the
experimental design procedure. Since the source parameters span several
magnitudes, we use a scaling matrix for efficient control of the condition
number of the original Hessian matrix. We use a second-order accurate finite
difference method to compute the Hessian matrix and either sparse quadrature or
Monte Carlo sampling to carry out numerical integration. We demonstrate the
efficiency, accuracy, and applicability of our method on a two-dimensional
seismic source inversion problem
Smoothing the payoff for efficient computation of Basket option prices
We consider the problem of pricing basket options in a multivariate Black
Scholes or Variance Gamma model. From a numerical point of view, pricing such
options corresponds to moderate and high dimensional numerical integration
problems with non-smooth integrands. Due to this lack of regularity, higher
order numerical integration techniques may not be directly available, requiring
the use of methods like Monte Carlo specifically designed to work for
non-regular problems. We propose to use the inherent smoothing property of the
density of the underlying in the above models to mollify the payoff function by
means of an exact conditional expectation. The resulting conditional
expectation is unbiased and yields a smooth integrand, which is amenable to the
efficient use of adaptive sparse grid cubature. Numerical examples indicate
that the high-order method may perform orders of magnitude faster compared to
Monte Carlo or Quasi Monte Carlo in dimensions up to 35
Sparse approximation of multilinear problems with applications to kernel-based methods in UQ
We provide a framework for the sparse approximation of multilinear problems
and show that several problems in uncertainty quantification fit within this
framework. In these problems, the value of a multilinear map has to be
approximated using approximations of different accuracy and computational work
of the arguments of this map. We propose and analyze a generalized version of
Smolyak's algorithm, which provides sparse approximation formulas with
convergence rates that mitigate the curse of dimension that appears in
multilinear approximation problems with a large number of arguments. We apply
the general framework to response surface approximation and optimization under
uncertainty for parametric partial differential equations using kernel-based
approximation. The theoretical results are supplemented by numerical
experiments
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