394 research outputs found
Sparse Quadrature for High-Dimensional Integration with Gaussian Measure
In this work we analyze the dimension-independent convergence property of an
abstract sparse quadrature scheme for numerical integration of functions of
high-dimensional parameters with Gaussian measure. Under certain assumptions of
the exactness and the boundedness of univariate quadrature rules as well as the
regularity of the parametric functions with respect to the parameters, we
obtain the convergence rate , where is the number of indices,
and is independent of the number of the parameter dimensions. Moreover, we
propose both an a-priori and an a-posteriori schemes for the construction of a
practical sparse quadrature rule and perform numerical experiments to
demonstrate their dimension-independent convergence rates
Multilevel Representations of Isotropic Gaussian Random Fields on the Sphere
Series expansions of isotropic Gaussian random fields on with
independent Gaussian coefficients and localized basis functions are
constructed. Such representations provide an alternative to the standard
Karhunen-Lo\`eve expansions of isotropic random fields in terms of spherical
harmonics. Their multilevel localized structure of basis functions is
especially useful in adaptive algorithms. The basis functions are obtained by
applying the square root of the covariance operator to spherical needlets.
Localization of the resulting covariance-dependent multilevel basis is shown
under decay conditions on the angular power spectrum of the random field. In
addition, numerical illustrations are given and an application to random
elliptic PDEs on the sphere is analyzed
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
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