TY - CHAP

T1 - Inverse Problems in a Bayesian Setting

AU - Matthies, Hermann G.

AU - Zander, Elmar

AU - Rosić, Bojana V.

AU - Litvinenko, Alexander

AU - Pajonk, Oliver

N1 - KAUST Repository Item: Exported on 2020-10-01

PY - 2016/2/13

Y1 - 2016/2/13

N2 - 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. We give a detailed account of this approach via conditional approximation, various approximations, and the construction of filters. 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 in form of a filter is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation 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 nonlinear Bayesian update in form of a filter on some examples.

AB - 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. We give a detailed account of this approach via conditional approximation, various approximations, and the construction of filters. 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 in form of a filter is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation 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 nonlinear Bayesian update in form of a filter on some examples.

UR - http://hdl.handle.net/10754/596466

UR - http://link.springer.com/chapter/10.1007%2F978-3-319-27996-1_10

UR - http://www.scopus.com/inward/record.url?scp=84964219513&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-27996-1_10

DO - 10.1007/978-3-319-27996-1_10

M3 - Chapter

SN - 978-3-319-27994-7

SP - 245

EP - 286

BT - Computational Methods for Solids and Fluids

PB - Springer Nature

ER -