For the Helmholtz equation with a Gaussian random field coefficient, we approximate and characterize spectrally the source-to-measurement map. To this end, we first analyze the case with a deterministic coefficient, and here discover and quantify a ’spectral leakage’ effect. We compare the theoretically predicted forward operator spectrum with a Finite Element Method computation. Our results are applicable in the analysis of the robustness of solution of inverse source problems in the presence of deterministic and random media.

Photonic nanojets (PNJs) have promising applications as optical probes in super-resolution optical microscopy, Raman microscopy, as well as fluorescence microscopy. In this work, we consider optimal design of PNJs using a heterogeneous lens refractive index with a fixed lens geometry and uniform plane wave illumination. In particular, we consider the presence of manufacturing error of heterogeneous lens, and propose a computational framework of Optimization Under Uncertainty (OUU) for robust optimal design of PNJ. We formulate a risk-averse stochastic optimization problem with the objective to minimize both the mean and the variance of a target function, which is constrained by the Helmholtz equation that governs the 2D transverse electric (2D TE) electromagnetic field in a neighborhood of the lens. The design variable is taken as a spatially-varying field variable, where we use a finite element method for its discretization, impose a total variation penalty to promote its sparsity, and employ an adjoint-based BFGS method to solve the resulting high-dimensional optimization problem. We demonstrate that our proposed OUU computational framework can achieve more robust optimal design than a deterministic optimization scheme to significantly mitigate the impact of manufacturing uncertainty.

In Bayesian Inverse Problems the aim is to recover the posterior distribution for the quantity of interest. This distribution is given in terms of the prior distribution modeling a priori knowledge and the likelihood distribution modeling the noise. In many applications, one single estimator, e.g., the posterior mean, is desired and reported, however it is crucial for the fundamental understanding that this estimator is consistent, meaning that the estimator converges in probability to the ground truth when the noise level tends to zero.

In this talk we will explore the fundamental questions and see, how consistency indeed is possible in the case of PDE driven problems such as Photo-Acoustic Tomography with parametrized inclusions.

We study some uncertainty quantification problems in nonlinear inverse coefficient problems for PDEs. We are interested in characterizing the impact of unknown parameters in the PDE models on the reconstructed coefficients. We argue that, unlike the situation in forward problems, uncertainty propagation in inverse problems is influenced by both the forward model and the inversion method used in the reconstructions. For ill-conditioned problems, errors in reconstructions can sometimes dominate the uncertainty caused by the unknown parameters in the model. Based on such observations, we will propose methods that quantify uncertainties more accurately than a generic method by compensating for the errors due to the reconstruction algorithms.