# Direct and Inverse Problems for PDEs with Random Coefficients - Abstract

**Manzoni, Andrea**

*A reduced-order framework for the efficient solution of PDE-constrained inverse UQ problems*

joint work with S. Pagani In several contexts, PDE models depend on unknown or uncertain parameters (or fields), which need to be estimated from indirect observations/measurements of suitable quantities of interest. Parameter identification and inverse uncertainty quantification (UQ) problems in a Bayesian framework hinge upon sampling a posterior distribution, which results from the combination of prior knowledge on the parameters and the outcome of the forward computational model. Sampling techniques, such as Monte Carlo Markov Chain (MCMC) or Kalman Filters, are thus computationally demanding (often even not affordable), since they may require $O(10^5-10^6)$ queries to the forward PDE model. Indeed, each query entails the solution of a huge problem when relying on high-fidelity techniques such as the finite element method. In the case of time-dependent nonlinear PDEs, solving inverse UQ problems is even more challenging citeMPT. We speedup the solution of the inverse problem by replacing the high-fidelity approximation of the forward problem with a (certified, inexpensive) reduced basis (RB) approximation citemanz, taking advantage of the discrete empirical interpolation method (DEIM) to deal with nonaffine and nonlinear terms, as proposed in citenegri. In this way, each forward query is inexpensive and the overall computational cost entailed by the Bayesian inversion can be greatly reduced. Moreover, we apply some reduced error models citeMPT to control the propagation of *reduction errors* along the inversion process. The proposed approach is rather general: as a proof of concept, we apply it to a problem dealing with the identification of an ischemic region over the cardiac tissue. beginthebibliography1 bibitemMPT A. Manzoni, S. Pagani, and T. Lassila. Accurate solution of Bayesian inverse uncertainty quantification problems using model and error reduction methods, em submitted, 2014. MATHICSE Report 47-2014 bibitemnegri F. Negri, A. Manzoni, and D.Amsallem. Efficient model reduction of parametrized systems by matrix discrete empirical interpolation, em submitted, 2015. MATHICSE Report 2-2015. bibitemmanz A. Quarteroni, A. Manzoni, and F. Negri. Reduced Basis Methods for Partial Differential Equations. An Introduction. Springer, Unitext, vol. 92, 2015. endthebibliography