Erkki Somersalo - Publications

Publications

The complete list of my publications can be found in my Google Scholar profile

https://scholar.google.com/citations?user=IrKBbAoAAAAJ&hl=en

Here are some of my most recent publications

Calvetti D, Cosmo A, Perotto S and Somersalo E (2020) Bayesian Mesh Adaptation for Estimating Distributed Parameters to appear in SISC

Abstract: The problem of estimating numerically a distributed parameter from indirect measurements arises in many applications, and in that context the choice of the discretization plays an important role. In fact, to guarantee a certain level of accuracy of the forward model that maps the unknown to the observations may require a fine discretization, adding to the complexity of the problem and to the computational cost. On the other hand, reducing the complexity of the problem by adopting a coarser discretization may increase the modeling error and can be very detrimental for ill-posed inverse problems. To balance accuracy and complexity, we propose an adaptive algorithm for adjusting the discretization level automatically and dynamically while estimating the unknown distributed parameter by an iterative scheme. In the Bayesian paradigm, all unknowns, including the metric that defines the discretization, are modeled as random variables. Our approach couples the discretization with a Bayesian hierarchical hyperparameter that is estimated simultaneously with the unknown parameter of primary interest. The viability of the proposed algorithm, the Bayesian Mesh Adaptation (BMA) is assessed on two test cases, a fan-beam X-ray tomography and an inverse source problem for a Darcy flow model.

Calvetti D, Pragliola M and Somersalo E (2020) Sparsity promoting hybrid solvers for hierarchical Bayesian inverse problems to appear in SISC

Abstract: The recovery of sparse generative models from few noisy measurements is an important and challenging problem. Many deterministic algorithms rely on some form of $\ell_1$-$\ell_2$ minimization to combine the computational convenience of the $\ell_2$ penalty and the sparsity promotion of the $\ell_1$. It was recently shown within the Bayesian framework that sparsity promotion and computational efficiency can be attained with hierarchical models with conditionally Gaussian priors and gamma hyperpriors. The related Gibbs energy function is a convex functional and its minimizer, which is the MAP estimate of the posterior, can be computed efficiently with the globally convergent Iterated Alternating Sequential (IAS) algorithm. Generalization of the hyperpriors for these sparsity promoting hierarchical models to generalized gamma family yield either globally convex Gibbs energy functionals, or can exhibit local convexity for some choices for the hyperparameters. The main problem in computing the MAP solution for greedy hyperpriors that strongly promote sparsity is the presence of local minima. To overcome the premature stopping at a spurious local minimizer, we propose two hybrid algorithms that first exploit the global convergence associated with gamma hyperpriors to arrive in a neighborhood of the unique minimizer, then adopt a generalized gamma hyperprior that promote sparsity more strongly. The performance of the two algorithms is illustrated with computed examples.