arXiv subjects

Arvind Saibaba

Publications and source records attributed to Arvind Saibaba.

4 recordsLinked to original sources

arXiv2022

Hyper-differential sensitivity analysis in the context of Bayesian inference applied to ice-sheet problems

Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model which must be estimated. Although the Bayesian formulation is attractive for such problems, computational cost and high dimensionality frequently prohibit a thorough exploration of the parametric uncertainty. A common approach is to reduce the dimension by fixing some parameters (which we will call auxiliary parameters) to a best estimate and use techniques from PDE-constrained optimization to approximate properties of the Bayesian posterior distribution. For instance, the maximum a posteriori probability (MAP) and the Laplace approximation of the posterior covariance can be computed. In this article, we propose using hyper-differential sensitivity analysis (HDSA) to assess the sensitivity of the MAP point to changes in the auxiliary parameters. We establish an interpretation of HDSA as correlations in the posterior distribution. Our proposed framework is demonstrated on the inversion of bedrock topography for the Greenland ice sheet with uncertainties arising from the basal friction coefficient and climate forcing (ice accumulation rate)

stat.ME↗

arXiv2022

Parametric Level-sets Enhanced To Improve Reconstruction (PaLEnTIR)

We introduce PaLEnTIR, a significantly enhanced parametric level-set (PaLS) method addressing the restoration and reconstruction of piecewise constant objects. Our key contribution involves a unique PaLS formulation utilizing a single level-set function to restore scenes containing multi-contrast piecewise-constant objects without requiring knowledge of the number of objects or their contrasts. Unlike standard PaLS methods employing radial basis functions (RBFs), our model integrates anisotropic basis functions (ABFs), thereby expanding its capacity to represent a wider class of shapes. Furthermore, PaLEnTIR improves the conditioning of the Jacobian matrix, required as part of the parameter identification process, and consequently accelerates optimization methods. We validate PaLEnTIR's efficacy through diverse experiments encompassing sparse and limited angle of view X-ray computed tomography (2D and 3D), nonlinear diffuse optical tomography (DOT), denoising, and deconvolution tasks using both real and simulated data sets.

math.NA↗

arXiv2022

Monte Carlo Methods for Estimating the Diagonal of a Real Symmetric Matrix

For real symmetric matrices that are accessible only through matrix vector products, we present Monte Carlo estimators for computing the diagonal elements. Our probabilistic bounds for normwise absolute and relative errors apply to Monte Carlo estimators based on random Rademacher, sparse Rademacher, normalized and unnormalized Gaussian vectors, and to vectors with bounded fourth moments. The novel use of matrix concentration inequalities in our proofs represents a systematic model for future analyses. Our bounds mostly do not depend on the matrix dimension, target different error measures than existing work, and imply that the accuracy of the estimators increases with the diagonal dominance of the matrix. An application to derivative-based global sensitivity metrics corroborates this, as do numerical experiments on synthetic test matrices. We recommend against the use in practice of sparse Rademacher vectors, which are the basis for many randomized sketching and sampling algorithms, because they tend to deliver barely a digit of accuracy even under large sampling amounts.

math.NA↗

arXiv2016

Low Rank Independence Samplers in Bayesian Inverse Problems

In Bayesian inverse problems, the posterior distribution is used to quantify uncertainty about the reconstructed solution. In practice, Markov chain Monte Carlo algorithms often are used to draw samples from the posterior distribution. However, implementations of such algorithms can be computationally expensive. We present a computationally efficient scheme for sampling high-dimensional Gaussian distributions in ill-posed Bayesian linear inverse problems. Our approach uses Metropolis-Hastings independence sampling with a proposal distribution based on a low-rank approximation of the prior-preconditioned Hessian. We show the dependence of the acceptance rate on the number of eigenvalues retained and discuss conditions under which the acceptance rate is high. We demonstrate our proposed sampler by using it with Metropolis-Hastings-within-Gibbs sampling in numerical experiments in image deblurring, computerized tomography, and NMR relaxometry.

math.NA↗