EFFICIENT DUALITY-BASED NUMERICAL METHODS FOR SPARSE PARABOLIC OPTIMAL CONTROL PROBLEMS

Abstract

In this thesis, I design efficient algorithms for sparse optimal parabolic control problems(SOPCPs). Firstly, I provide a new discretization based on the dual problem of the SOPCPs and give an error estimate for the new model. Later I utilize the symmetric Gauss-Seidel(SGS) based inexact majorized accelerated block coordinate descent(imABCD) method to solve it and exploit the convergence result.

Secondly, I apply the Semismooth Newton Augmented Lagrangian(SSNAL) method to solve cases when regularization parameter alpha is very small or is zero. I prove the convergence results and the uniformly mesh-independence property. Furthermore, I derive the robustness property to alpha. For the case alpha is zero, SSNAL still deals with it very efficiently.

Numerical results show that sGS-imABCD method performs well for cases alpha not very small, and SSNAL method performs best for all cases, comparing to inexact semi-proximal alternative direction method(isPADMM), and the globalized semismooth Newton method(SSN).