Fractional PDE Constrained Optimization: Box and Sparse Constrained Problems Fabio Durastante and Stefano Cipolla Abstract In this paper we address the numerical solution of two Fractional Par- tial Differential Equation constrained optimization problems: the two-dimensional semilinear Riesz Space Fractional Diffusion equation with box or sparse constraints. Both a theoretical and experimental analysis of the problems is carried out. The algorithmic framework is based on the L-BFGS-B method coupled with a Krylov subspace solver for the box constrained problem within an optimize-then- discretize approach and on the semismooth Newton–Krylov method for the sparse one. Suitable preconditioning strategies by approximate inverses and Generalized Locally Toeplitz sequences are taken into account. The numerical experiments are performed with benchmarked software/libraries enforcing the reproducibility of the results. Keywords Fractional differential equation · Constrained optimization · Preconditioner · Saddle matrix 1 Introduction Partial fractional differential equations model different phenomena not appropri- ately modeled by partial differential equations with ordinary derivatives: from the models of viscoplasticity and viscoelasticity to the modeling of diffusion processes in porous media and, indeed, many other problems exhibiting non-local properties; see [30] for a gallery of possible applications. Thus, the study of their controllability and the research of efficient algorithms for this task are becoming always more relevant. Already the discretize-then-optimize framework [2, 16] and the optimize- F. Durastante () Dipartimento di Informatica, Università di Pisa, Pisa (PI), Italy e-mail: fabio.durastante@di.unipi.it S. Cipolla Dipartimento di Matematica, Università di Padova, Padova (PD), Italy e-mail: cipolla@math.unipd.it © Springer Nature Switzerland AG 2018 M. Falcone et al. (eds.), Numerical Methods for Optimal Control Problems, Springer INdAM Series 29, https://doi.org/10.1007/978-3-030-01959-4_6 111