
hIPPYlib: An Extensible Software Framework for LargeScale Inverse Problems Governed by PDEs; Part I: Deterministic Inversion and Linearized Bayesian Inference
We present an extensible software framework, hIPPYlib, for solution of l...
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hIPPYlibMUQ: A Bayesian Inference Software Framework for Integration of Data with Complex Predictive Models under Uncertainty
Bayesian inference provides a systematic means of quantifying uncertaint...
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Novel Deep neural networks for solving Bayesian statistical inverse
We consider the simulation of Bayesian statistical inverse problems gove...
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A polarization tensor approximation for the Hessian in iterative solvers for nonlinear inverse problems
For many inverse parameter problems for partial differential equations i...
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Hierarchical adaptive lowrank format with applications to discretized PDEs
A novel compressed matrix format is proposed that combines an adaptive h...
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Randomization for the Efficient Computation of Parametric Reduced Order Models for Inversion
Nonlinear parametric inverse problems appear in many applications. Here,...
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GraphInduced Rank Structures and their Representations
A new framework is proposed to study rankstructured matrices arising fr...
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Hierarchical Matrix Approximations of Hessians Arising in Inverse Problems Governed by PDEs
Hessian operators arising in inverse problems governed by partial differential equations (PDEs) play a critical role in delivering efficient, dimensionindependent convergence for both Newton solution of deterministic inverse problems, as well as Markov chain Monte Carlo sampling of posteriors in the Bayesian setting. These methods require the ability to repeatedly perform such operations on the Hessian as multiplication with arbitrary vectors, solving linear systems, inversion, and (inverse) square root. Unfortunately, the Hessian is a (formally) dense, implicitlydefined operator that is intractable to form explicitly for practical inverse problems, requiring as many PDE solves as inversion parameters. Low rank approximations are effective when the data contain limited information about the parameters, but become prohibitive as the data become more informative. However, the Hessians for many inverse problems arising in practical applications can be well approximated by matrices that have hierarchically low rank structure. Hierarchical matrix representations promise to overcome the high complexity of dense representations and provide effective data structures and matrix operations that have only loglinear complexity. In this work, we describe algorithms for constructing and updating hierarchical matrix approximations of Hessians, and illustrate them on a number of representative inverse problems involving timedependent diffusion, advectiondominated transport, frequency domain acoustic wave propagation, and low frequency Maxwell equations, demonstrating up to an order of magnitude speedup compared to globally low rank approximations.
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