NA Seminar: "Fast and Reliable Solution of Fractional PDE Constrained Optimization Problems", Fabio Durastante, U Pisa
In this talk, based on [2,3], we consider the numerical solution of a PDE constrained optimization problem. For a given function, called the desired state, we try to compute numerically a source term for a semilinear Partial Fractional Differential Equation that is such that the solution of the latter approximates the desired state in a suitable norm. Particularly, we consider the cases in which the PDE is either the Fractional Advection-Dispersion Equation or the two-dimensional semilinear Riesz Space Fractional Diffusion Equation. We focus on extending the existing strategies for classic PDE constrained optimization problems to this new cases. Moreover, we consider also formulations in which the wanted source term has to be bounded or sparse, i.e., cases in which we are adding other constraints to the problem. We present both a theoretical and experimental analysis of the framework. From the algorithmic point of view, we couple an optimization method, namely the L-BFGS/L-BFGS-B and Semismooth Newton methods, and a Krylov subspace solver. To speed up the computation, we take into account a suitable preconditioning strategy by approximate inverse factorizations  for the Krylov solver. We perform some numerical experiments with benchmarked software/libraries thus enforcing the reproducibility of the results.
This is a joint work with S. Cipolla (Università di Padova, cipolla [at] math [dot] unipd [dot] it)
 Bertaccini, D., Durastante, F. 2017. Solving mixed classical and fractional partial differential equations using short--memory principle and approximate inverses. Numer. Algorithms, 74(4), 1061-1082.
 Cipolla, S. and Durastante, F. 2018 Fractional PDE constrained optimization: An optimize-then-discretize approach with L-BFGS and approximate inverse preconditioning. Appl. Numer. Math. 123, 43-57.
 Durastante, F. and Cipolla, S. 2018. Fractional PDE constrained optimization: box and sparse constrained problems. Falcone M. R. Ferretti, Grune L; McEneaney, W (Ed.): Numerical methods for optimal control problems, INdAM Springer, 2017, (In Press).