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AuthorLitvinenko, Alexander (3)Matthies, Hermann G. (2)Dolgov, Sergey (1)Khoromskij, Boris N. (1)Department

Extreme Computing Research Center (3)

Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ) (2)SRI Uncertainty Quantification Center (1)SubjectBayesian estimation (1)Bayesian update PCE (1)Bayesian update surrogate (1)conditional expectation (1)Domain Decomposition (1)View MoreType
Technical Report (3)

Year (Issue Date)2017 (1)2014 (1)2013 (1)Item AvailabilityOpen Access (3)

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

Litvinenko, Alexander (2017-08-09) [Technical Report]

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.

Computation of the Response Surface in the Tensor Train data format

Dolgov, Sergey; Khoromskij, Boris N.; Litvinenko, Alexander; Matthies, Hermann G. (2014-06-11) [Technical Report]

We apply the Tensor Train (TT) approximation to construct the Polynomial Chaos Expansion (PCE) of a random field, and solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization. We compare two strategies of the polynomial chaos expansion: sparse and full polynomial (multi-index) sets. In the full set, the polynomial orders are chosen independently in each variable, which provides higher flexibility and accuracy. However, the total amount of degrees of freedom grows exponentially with the number of stochastic coordinates. To cope with this curse of dimensionality, the data is kept compressed in the TT decomposition, a recurrent low-rank factorization. PCE computations on sparse grids sets are extensively studied, but the TT representation for PCE is a novel approach that is investigated in this paper. We outline how to deduce the PCE from the covariance matrix, assemble the Galerkin operator, and evaluate some post-processing (mean, variance, Sobol indices), staying within the low-rank framework. The most demanding are two stages. First, we interpolate PCE coefficients in the TT format using a few number of samples, which is performed via the block cross approximation method. Second, we solve the discretized equation (large linear system) via the alternating minimal energy algorithm. In the numerical experiments we demonstrate that the full expansion set encapsulated in the TT format is indeed preferable in cases when high accuracy and high polynomial orders are required.

Inverse problems and uncertainty quantification

Litvinenko, Alexander; Matthies, Hermann G. (2013-12-18) [Technical Report]

In a Bayesian setting, inverse problems and uncertainty quantification (UQ)— the propagation of uncertainty through a computational (forward) model—are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. This is especially the case as together with a functional or spectral approach for the forward UQ there is no need for time- consuming and slowly convergent Monte Carlo sampling. The developed sampling- free non-linear Bayesian update is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisa- tion to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of different observation or measurement operators on the update.

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