In the second part of the talk, we will discuss inverse problems form PDEs which vary on a very fine scale $\varepsilon$ using Bayesian techniques. We propose a new strategy based on a homogenized forward model. Using G-convergence, we show that true posterior converges to the “homogenized posterior” in the Hellinger metric as $\varepsilon\rightarrow 0$. We then propose a numerical technique based on numerical homogenization and reduced basis techniques for a cheap evaluation of the forward model in a Markov Chain Monte Carlo procedure. We also comment on a procedure to account for the modeling error at fixed $\varepsilon$.

References:

A. Abdulle, G. Garegnani, Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration,

Preprint

(2017).
A. Abdulle, G. Garegnani, Probabilistic geometric integration of Hamiltonian systems based on random time steps,

Preprint

(2018).
A. Abdulle, A. Di Blasio, Numerical homogenization and model order reduction for multiscale inverse problems, submitted to publication,

Preprint

(2017).
A. Abdulle, A. Di Blasio, A Bayesian numerical homogenization method for elliptic multiscale inverse problems,

Preprint

(2018).So-called functional error estimators (Repin, 2000) provide a valuable tool for reliably estimating the discretization error for a sum of two convex functions. We apply this concept to Tikhonov regularization for the solution of inverse problems for partial differential equations, not only for quadratic Hilbert space regularization terms but also for nonsmooth Banach space penalties. Examples include the measure-space norm (i.e., sparsity regularization) or the indicator function of an $L^\infty$ ball (i.e., Ivanov regularization). The error estimators can be written in terms of residuals in the optimality system that can then be estimated by conventional techniques, thus leading to explicit estimators. This is illustrated by means of an elliptic inverse source problem with the above-mentioned penalties, and numerical results are provided for the case of sparsity regularization.

For the optical modeling, we need to solve Maxwell's equations across a wide range of wavelengths throughout the visible solar spectrum in order to predict the generation rate of free electrons. We use a Fourier based technique with a special solver, called the Rigorous Coupled Wave Analysis (RCWA) method. This method is fast and does not require remeshing between solar cells with different geometrical structures. The output from RCWA is used as a source term in the electrical model, driving the flow of electrons and holes in semiconductor elements of the device. This electronic behaviour is modeled using the classical drift-diffusion equations, which are approximated using the Hybridizable Discontinuous Galerkin (HDG) method. This formulation naturally allows upwinding and the handling of heterojunctions. I shall describe each component of the model, giving numerical results, and show how we can achieve a successful optimal design process.

References:

BS18: A. Buhr and K. Smetana. Randomized local model order reduction. Technical report, arXiv:1706.09179, preprint (submitted), 2017.

BEOR17: A. Buhr, C. Engwer, M. Ohlberger, and S. Rave. ArbiLoMod, a Simulation Technique Designed for Arbitrary Local Modifications. SIAM J. Sci. Comput., 39(4):A1435-A1465, 2017.

MRS2016: R. Milk, S. Rave, F. Schindler. pyMOR- generic algorithms and interfaces for model order reduction. SIAM J. Sci. Comput. 38(5):S194–S216, 2016.

OSS18: M. Ohlberger, M. Schaefer, and F. Schindler. Localized model reduction in pde constrained optimization. Accepted for publication in: Procceedings of the DFG-AIMS Workshop on Shape optimization, homogenization and control, AIMS Sénégal, Mbour, Sénégal. 2017.

OS15: M. Ohlberger, F. Schindler. Error control for the localized reduced basis multi-scale method with adaptive on-line enrichment. SIAM J. Sci. Comput. 37(6):A2865–A2895, 2015.

The a priori and a posteriori analysis for such types of problems typically require some regularity of the simplified problems. In particular, for the a posteriori error analysis the (estimated) value of the regularity constant (of the simplified problem) with respect to a $W^{1,p}$ norm for some $p>2$ is required.

In our talk, we will present model problems where the a posteriori analysis leads to such regularity problems and explain theoretical tools for their estimation.

Collaboration with: Andrea Bertozzi (UCLA), Michael Luo (UCLA), Kostas Zygalakis (Edinburgh), Matt Dunlop (Caltech), Dejan Slepcev (CMU), Matt Thorpe (Cambridge), Victor Chen (Caltech), Omiros Papaspiliopoulos (UPF-Barcelona)

https://arxiv.org/abs/1703.08816

https://arxiv.org/abs/1703.08816

The simulation of such problems is quite challenging due to the general wave nature of the problem and the additional fine-scale oscillations from the material inhomogeneities. At the example of the time-harmonic Maxwell's equations, we show that a (fine-scale) corrector is needed for a good approximation in $L^2$. We discuss how this corrector can be efficiently computed as the solution of local cell problems. We present corresponding numerical multiscale methods and their a priori error analysis.

Numerical examples conform the theoretical results and illustrate some of the interesting physical phenomena. For instance, a high contrast between two composites in a periodic microstructure can lead to frequency band gaps caused by (Mie) resonances.

This talk is based on a joint work with Philippe Chartier, Mohammed Lemou, and Florian Méhats (Rennes).

Preprints available at http://www.unige.ch/~vilmart.