We construct efficient algebraic multigrid preconditioners for singularly perturbed reaction-diffusion equations that arise from elliptic control with energy regularization. Here the standard L2(Ω)-norm regularization is replaced by the H−1(Ω)-norm leading to more focused controls z. In this case, the optimality system can be reduced to a single singularly perturbed diffusion-reaction equation known as differential filter in turbulence theory. We investigate the error between the finite element approximation uϱh to the state u and the desired state ud in terms of the mesh-size h and the regularization parameter ϱ. The choice ϱ = h2 ensures optimal convergence the rate of which only depends on the regularity of the target function ud . The resulting symmetric and positive definite system of finite element equations is solved by the conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) at MACH2 or balancing domain decomposition by constraints (BDDC) at RADON1. We numerically study robustness, efficiency, and performance of both preconditioners. These results can be found in our publication [6]. Figures 1 and 2 that are also taken from our paper [6] show the numerical results for a piecewise constant discontinuous target function ud which is one in the cube (1/4,3/4)3 that is inscribed in the computational domain Ω = (0,1)3 ⊂ R3 , and zero elsewhere.
Figure 1: Adaptive meshes at the 21th refinement step with 1, 895, 056 Dofs, uϱh in Ω and on the cutting plane x3 = 0.5 (up), control in Ω, near the interface, and on the line between [0,0.5,0.5] and [1,0.5,0.5] (down), ϱ = 10−6.
Figure 2: Number of AMG preconditioned CG iterations (left) and computational time in seconds (right) with respect to different ϱ ∈ {100, 10−1, ..., 10−6 }.
In [3,4,5,6,1], we consider different parabolic optimal control problems with different regularization terms including L2-, energy, and sparse regularizations. The reduced optimality system is then discretized by mean of space-time finite elements on unstructured simplicial space-time meshes. The system of finite element equations is then solved by AMG preconditioned GMRES methods. Different space-time methods for parabolic and hyperbolic partial differential equations can be found in the book [2] edited by U. Langer and O. Steinbach.
References
[1] U. Langer and A. Schafelner, Adaptive space-time finite element methods for parabolic optimal control problems, Journal of Numerical Mathematics, (2022), first published online, Nov. 3, 2021, https://doi.org/10.1515/jnma-2021-0059.
[2] U. Langer and O. Steinbach, editors, Space-Time Methods: Applications to Partial Differential Equations, Radon Series on Computational and Applied Mathematics, 25 (2019), Berlin, de Gruyter.
[3] U. Langer, O. Steinbach, F. Tröltzsch and H. Yang, Unstructured space-time finite element methods for optimal control of parabolic equation, SIAM Journal on Scientific Computing 43 (2021), no.2, A744–A771.
[4] U. Langer, O. Steinbach, F. Tröltzsch and H. Yang, Space-time finite element discretization of parabolic optimal control problems with energy regularization. SIAM Journal on Numerical Analysis 59 (2021), no.2, 675–695.
[5] U. Langer, O. Steinbach, F. Tröltzsch and H. Yang, Unstructured space-time finite element methods for optimal sparse control of parabolic equations, In ”Optimization and Control for PDEs”, ed by R. Herzog et al., Radon Series on Computational and Applied Mathematics, 29 (2022), 167–188, Berlin, de Gruyter.
[6] U. Langer, O. Steinbach, and H. Yang Robust discretization and solvers for elliptic optimal control problems with energy regularization Computational Methods in Applied Mathematics, 22 (2022), no.1, 97—111.