Prof. Dr. Christoph Schwab (ETH Zürich) High-Dimensional Numerical Integration
in Bayesian Estimation for Parametric Operator Equations
Thursday, 20.11.2014 16:30 im Raum M5
We consider numerical analysis of Bayesian estimation for uncertain
systems governed by partial or ordinary differential equations, with
uncertain parameters in high-dimensional parameter spaces, subject to
given measured data which is assumed to be subject to additive
Gaussian observation noise.
Bayesian estimation, conditional to given data, then takes the form of
a mathematical expectation of system responses over all possible
realizations of the uncertain input, conditional on the measurement
For uncertainties in an infinite-dimensional function space which
admits an unconditional basis, this expectation is written as
infinite-dimensional iterated integral w.r. to the prior measure.
Regularity of the uncertain input translates into a sparsity result of
the Bayesian posterior density.
Based on this sparsity, we survey several dimension-adaptive quadrature approaches that allow
the approximate evaluation of the Bayesian estimates with convergence
rates that depend only on the sparsity, and that are independent of
the dimension of the parametric domain.
Numerical examples from PDEs with random inputs, shape-uncertainty,
and large nonlinear systems of parametric ODEs arising in biological
systems engineering illustrate applications of the theory.
Work supported by ERC under AdG247277 and by SNF.