Some Alternating Direction Iteration Methods For Solving PDE-Constrained Optimization Problems

Numerically solving optimization problems subject to constraints given in terms of partial d-ifferential equations(PDEs)is one of the most important and challenging problems in applied mathematics.PDE-constrained optimization problems play an important role in industrial,med-ical,economical,and in other disciplines.The research on the traditional PDE-constrained optimization problems with L2-control cost in theories and numerical methods has obtained abundant research results.However,there are still not many researches on the PDE-constrained optimization problems with L1-control cost,which need to be further studied.Like the finite dimensional l1-regularization,it is well-known that L1-norm could lead to sparse optimal con-trol,i.e.the optimal control with small support.Such the sparse optimal control problem plays an important role for the placement of control devices.Motivated by the success of alternating direction algorithms in finite dimensional sparse optimization,we try to study the alternating di-rection algorithms for solving the PDE-constrained optimization problems with L1-control cost,and obtain the following research results:1.Elliptic PDE-constrained optimal control problems with L1-control cost(L1-EOCP)are considered.To discretize L1-EOCP,the piecewise linear finite element(FE)is considered.In order to overcome the difficulty that the discretized L1-norm does not have a decoupled form,a discretization scheme with decoupled form is proposed.An error estimate of order O(h)is obtained and confirmed by numerical experiments.Furthermore,to numerically solve the dis-cretized problem,an inexact heterogeneous ADMM(ihADMM)is proposed.Different from the classical ADMM,the ihADMM adopts two different weighted inner products to define the augmented Lagrangian function in two subproblems,respectively.Theoretical results on the global convergence as well as the iteration complexity results o(1/k)for ihADMM are given.In order to obtain a more accurate solution,a two-phase strategy is also presented,in which the primal-dual active set(PDAS)method is used as a postprocessor of ihADMM.2.Elliptic optimal control problems with L2-control cost(EOCP)are considered.To nu-merically solve EOCP,an inexact ADMM(iADMM)is firstly proposed on the continuous level with the aim of solving discretized problems to moderate accuracy.Then the standard piece-wise linear finite element is employed to discretize the related subproblems appearing in each iteration of iADMM algorithm.Such approach will give us the freedom to discretize two inner subproblems of iADMM algorithm by different discretized scheme,respectively.It is important to emphasise that the discretized version of iADMM algorithm is actually the ihADMM.3.Elliptic optimal control problems involving L1-control cost(L1-EOCP)is considered.To numerically discretize L1-EOCP,a common approach is employing a nodal quadrature formula to approximately discretize the L1-norm.It is inevitable that this technique will incur an addi-tional error.Different from the traditional approach,a duality-based approach and an inexact symmetric Gauss-Seidel based majorized accelerated block coordinate descent method(called sGS-imABCD)is proposed to solve L1-EOCP via its dual problem.Furthermore,based on the discretized dual problem,a new discretized scheme for the L1-norm is presented.Compare the new discretized scheme for L1-norm with the nodal quadrature formula,the advantages of the new discretized scheme can be demonstrated in terms of the approximation order.More im-portantly,finite element error estimates results for the primal problem with the new discretized scheme for the L1-norm are provided.4.A majorized accelerated block coordinate descent(mABCD)method in Hilbert space is analyzed to solve the sparse optimal control problem via its dual problem.The attractive O(1/k2)iteration complexity of ABCD method for the dual objective function values can be achieved.However the convergence for the sequence of the dual variables is not obvious.In order to accomplish this goal,a local second order growth condition for the dual objective function is presented.Then,the iteration complexity of O(1/k)for the iteration sequence of dual variables can be proved.Furthermore,making use of the relationship between the primal problem and dual problem,the iteration complexity results for the primal problem are proved.Based on the these convergence results,two types of mesh-independence for mABCD method are proved,which asserts that asymptotically the infinite dimensional ABCD method and finite dimensional discretizations have the same convergence property,and the number of iterations of mABCD method remains almost constant as the discretization is refined.