Likelihood Approximation With Hierarchical Matrices For Large Spatial Datase

Abstract:

We use available measurements to estimate the unknown parameters (variance, smoothness parameter, and covariance length) of a covariance function by maximizing the joint Gaussian log-likelihood function. To overcome cubic complexity in the linear algebra, we approximate the discretized covariance function in the hierarchical (H-) matrix format. The H-matrix format has a log-linear computational cost and storage O(kn log n), where the rank k is a small integer and n is the number of locations. The H-matrix technique allows us to work with general covariance matrices in an ecient way, since H-matrices can approximate inhomogeneous covariance functions, with a fairly general mesh that is not necessarily axes-parallel, and neither the covariance matrix itself nor its inverse have to be sparse. We demonstrate our method with Monte Carlo simulations and an application to soil moisture data. The C, C++ codes and data are freely available.

Biography:

Alexander Litvinenko joined the Extreme Computing Research Center at KAUST, directed by David Keyes, in 2015. Before that he was a research scientist at the Uncertainty Quantification Center also at KAUST. He specializes in efficient numerical methods for stochastic PDEs, uncertainty quantification, and multi-linear algebra.  He is involved in Bayesian update methods for solving inverse problems, with the goal of reducing the complexity both the stochastic forward problem as well as the Bayesian update by a low-rank (sparse) tensor data approximation. Applications include aerodynamics, subsurface flow, and spatial statistics. Alexander earned B.S. (2000) and M.S. (2002) degrees in mathematics at Novosibirsk State University, and his PhD (2006) in the group of Prof. Hackbusch at Max-Planck-Institut in Leipzig, on the combination of domain decomposition methods and hierarchical matrices for solving elliptic PDEs with jumping and oscillatory coeff icients. From 2007-2013 he was a Postdoctoral Research Fellow at the TU Braunschweig in Germany.