Error Estimates for Sparse Optimal Control Problems by Piecewise Linear Finite Element Approximation (Journal of Computational Mathematics, 2021)
Optimization problems with $L^1$-control cost functional subject to an elliptic partialdifferential equation (PDE) are considered. However, different from the finite dimensional $l^1$-regularization optimization, the resulting discretized $L^1$-norm does not have a decoupled form when the standard piecewise linear finite element is employed to discretize thecontinuous problem. A common approach to overcome this difficulty is employing a nodalquadrature formula to approximately discretize the $L^1$-norm. In this paper, a new discretized scheme for the $L^1$-norm is presented. Compared to the new discretized scheme for$L^1$-norm with the nodal quadrature formula, the advantages of our new discretized schemecan be demonstrated in terms of the order of approximation. Moreover, finite elementerror estimates results for the primal problem with the new discretized scheme for the$L^1$-norm are provided, which confirms that this approximation scheme will not change theorder of error estimates. To solve the new discretized problem, a symmetric Gauss-Seidelbased majorized accelerated block coordinate descent (sGS-mABCD) method is introducedto solve it via its dual. The proposed sGS-mABCD algorithm is illustrated at two numerical examples. Numerical results not only confirm the finite element error estimates, butalso show that our proposed algorithm is efficient.
Keywords: Finite element method; ABCD method; Approximate discretization; Error estimates.