arXiv subjects

Arvind K. Saibaba

Publications and source records attributed to Arvind K. Saibaba.

58 records · Page 4Linked to original sources

arXiv2015

Fast computation of uncertainty quantification measures in the geostatistical approach to solve inverse problems

We consider the computational challenges associated with uncertainty quantification involved in parameter estimation such as seismic slowness and hydraulic transmissivity fields. The reconstruction of these parameters can be mathematically described as Inverse Problems which we tackle using the Geostatistical approach. The quantification of uncertainty in the Geostatistical approach involves computing the posterior covariance matrix which is prohibitively expensive to fully compute and store. We consider an efficient representation of the posterior covariance matrix at the maximum a posteriori (MAP) point as the sum of the prior covariance matrix and a low-rank update that contains information from the dominant generalized eigenmodes of the data misfit part of the Hessian and the inverse covariance matrix. The rank of the low-rank update is typically independent of the dimension of the unknown parameter. The cost of our method scales as $\bigO(m\log m)$ where $m $ dimension of unknown parameter vector space. Furthermore, we show how to efficiently compute measures of uncertainty that are based on scalar functions of the posterior covariance matrix. The performance of our algorithms is demonstrated by application to model problems in synthetic travel-time tomography and steady-state hydraulic tomography. We explore the accuracy of the posterior covariance on different experimental parameters and show that the cost of approximating the posterior covariance matrix depends on the problem size and is not sensitive to other experimental parameters.

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arXiv2015

Fast Kalman Filter using Hierarchical-matrices and low-rank perturbative approach

We develop a fast algorithm for Kalman Filter applied to the random walk forecast model. The key idea is an efficient representation of the estimate covariance matrix at each time-step as a weighted sum of two contributions - the process noise covariance matrix and a low rank term computed from a generalized eigenvalue problem, which combines information from the noise covariance matrix and the data. We describe an efficient algorithm to update the weights of the above terms and the computation of eigenmodes of the generalized eigenvalue problem (GEP). The resulting algorithm for the Kalman filter with a random walk forecast model scales as $\bigO(N)$ in memory and $\bigO(N \log N)$ in computational cost, where $N$ is the number of grid points. We show how to efficiently compute measures of uncertainty and conditional realizations from the state distribution at each time step. An extension to the case with nonlinear measurement operators is also discussed. Numerical experiments demonstrate the performance of our algorithms, which are applied to a synthetic example from monitoring CO$_2$ in the subsurface using travel time tomography.

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arXiv2014

Fast algorithms for hyperspectral Diffuse Optical Tomography

The image reconstruction of chromophore concentrations using Diffuse Optical Tomography (DOT) data can be described mathematically as an ill-posed inverse problem. Recent work has shown that the use of hyperspectral DOT data, as opposed to data sets comprising of a single or, at most, a dozen wavelengths, has the potential for improving the quality of the reconstructions. The use of hyperspectral diffuse optical data in the formulation and solution of the inverse problem poses a significant computational burden. The forward operator is, in actuality, nonlinear. However, under certain assumptions, a linear approximation, called the Born approximation, provides a suitable surrogate for the forward operator, and we assume this to be true in the present work. Computation of the Born matrix requires the solution of thousands of large scale discrete PDEs and the reconstruction problem, requires matrix-vector products with the (dense) Born matrix. In this paper, we address both of these difficulties, thus making the Born approach a computational viable approach for hyDOT reconstruction. In this paper, we assume that the images we wish to reconstruct are anomalies of unknown shape and constant value, described using a parametric level set approach, (PaLS) on a constant background. Specifically, to address the issue of the PDE solves, we develop a novel recycling-based Krylov subspace approach that leverages certain system similarities across wavelengths. To address expense of using the Born operator in the inversion, we present a fast algorithm for compressing the Born operator that locally compresses across wavelengths for a given source-detector set and then recursively combines the low-rank factors to provide a global low-rank approximation. This low-rank approximation can be used implicitly to speed up the recovery of the shape parameters and the chromophore concentrations.

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arXiv2014

A Flexible Krylov Solver for Shifted Systems with Application to Oscillatory Hydraulic Tomography

We discuss efficient solutions to systems of shifted linear systems arising in computations for oscillatory hydraulic tomography (OHT). The reconstruction of hydrogeological parameters such as hydraulic conductivity and specific storage using limited discrete measurements of pressure (head) obtained from sequential oscillatory pumping tests, leads to a nonlinear inverse problem. We tackle this using the quasi-linear geostatistical approach \cite{kitanidis1995quasi}. This method requires repeated solution of the forward (and adjoint) problem for multiple frequencies, for which we use flexible preconditioned Krylov subspace solvers specifically designed for shifted systems based on ideas in \cite{gu2007flexible}. The solvers allow the preconditioner to change at each iteration. We analyze the convergence of the solver and perform an error analysis when an iterative solver is used for inverting the preconditioner matrices. Finally, we apply our algorithm to a challenging application taken from oscillatory hydraulic tomography to demonstrate the computational gains by using the resulting method.

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