# Partial inversion of elliptic operator to speed up computation of likelihood in Bayesian inference

Handle URI:
http://hdl.handle.net/10754/625309
Title:
Partial inversion of elliptic operator to speed up computation of likelihood in Bayesian inference
Authors:
Litvinenko, Alexander ( 0000-0001-5427-3598 )
Abstract:
In this paper, we speed up the solution of inverse problems in Bayesian settings. By computing the likelihood, the most expensive part of the Bayesian formula, one compares the available measurement data with the simulated data. To get simulated data, repeated solution of the forward problem is required. This could be a great challenge. Often, the available measurement is a functional $F(u)$ of the solution $u$ or a small part of $u$. Typical examples of $F(u)$ are the solution in a point, solution on a coarser grid, in a small subdomain, the mean value in a subdomain. It is a waste of computational resources to evaluate, first, the whole solution and then compute a part of it. In this work, we compute the functional $F(u)$ direct, without computing the full inverse operator and without computing the whole solution $u$. The main ingredients of the developed approach are the hierarchical domain decomposition technique, the finite element method and the Schur complements. To speed up computations and to reduce the storage cost, we approximate the forward operator and the Schur complement in the hierarchical matrix format. Applying the hierarchical matrix technique, we reduced the computing cost to $\mathcal{O}(k^2n \log^2 n)$, where $k\ll n$ and $n$ is the number of degrees of freedom. Up to the $\H$-matrix accuracy, the computation of the functional $F(u)$ is exact. To reduce the computational resources further, we can approximate $F(u)$ on, for instance, multiple coarse meshes. The offered method is well suited for solving multiscale problems. A disadvantage of this method is the assumption that one has to have access to the discretisation and to the procedure of assembling the Galerkin matrix.
KAUST Department:
Extreme Computing Research Center; SRI Center for Uncertainty Quantification in Computational Science and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
Issue Date:
9-Aug-2017
Type:
Technical Report
ECRC, SRI UQ KAUST
Appears in Collections:
Technical Reports

DC FieldValue Language
dc.contributor.authorLitvinenko, Alexanderen
dc.date.accessioned2017-08-10T06:36:44Z-
dc.date.available2017-08-10T06:36:44Z-
dc.date.issued2017-08-09-
dc.identifier.urihttp://hdl.handle.net/10754/625309-
dc.description.abstractIn this paper, we speed up the solution of inverse problems in Bayesian settings. By computing the likelihood, the most expensive part of the Bayesian formula, one compares the available measurement data with the simulated data. To get simulated data, repeated solution of the forward problem is required. This could be a great challenge. Often, the available measurement is a functional $F(u)$ of the solution $u$ or a small part of $u$. Typical examples of $F(u)$ are the solution in a point, solution on a coarser grid, in a small subdomain, the mean value in a subdomain. It is a waste of computational resources to evaluate, first, the whole solution and then compute a part of it. In this work, we compute the functional $F(u)$ direct, without computing the full inverse operator and without computing the whole solution $u$. The main ingredients of the developed approach are the hierarchical domain decomposition technique, the finite element method and the Schur complements. To speed up computations and to reduce the storage cost, we approximate the forward operator and the Schur complement in the hierarchical matrix format. Applying the hierarchical matrix technique, we reduced the computing cost to $\mathcal{O}(k^2n \log^2 n)$, where $k\ll n$ and $n$ is the number of degrees of freedom. Up to the $\H$-matrix accuracy, the computation of the functional $F(u)$ is exact. To reduce the computational resources further, we can approximate $F(u)$ on, for instance, multiple coarse meshes. The offered method is well suited for solving multiscale problems. A disadvantage of this method is the assumption that one has to have access to the discretisation and to the procedure of assembling the Galerkin matrix.en
dc.subjectUncertainty Quantificationen
dc.subjectBayesian estimationen
dc.subjectpartial inverseen
dc.subjectStochastic PDEsen
dc.subjectDomain Decompositionen
dc.subjecthierarchical matricesen
dc.subjectmultiscale methodsen
dc.titlePartial inversion of elliptic operator to speed up computation of likelihood in Bayesian inferenceen
dc.typeTechnical Reporten
dc.contributor.departmentExtreme Computing Research Centeren
dc.contributor.departmentSRI Center for Uncertainty Quantification in Computational Science and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabiaen
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Partial inversion of elliptic operator to speed up computation of likelihood in Bayesian inference