Sparse discretization of sparse control problems

We consider optimal control problems that inherit a sparsity structure, especially we look at problems governed by measure controls. Our goal is to achieve maximal sparsity on the discrete level. We use variational discretization of the control problems utilizing a Petrov‐Galerkin approximation of the state, which induces controls that are composed of Dirac measures. In the parabolic case this allows us to achieve sparsity on the discrete level in space and time. Numerical experiments show the differences of this approach to a full discretization approach.

We consider optimal control problems that inherit a sparsity structure, especially we look at problems governed by measure controls. Our goal is to achieve maximal sparsity on the discrete level. We use variational discretization of the control problems utilizing a Petrov-Galerkin approximation of the state, which induces controls that are composed of Dirac measures. In the parabolic case this allows us to achieve sparsity on the discrete level in space and time. Numerical experiments show the differences of this approach to a full discretization approach.
where the state y ∈ L q (Q) solves a heat equation with right hand side u, initial data u 0 and zero boundary values in a very weak sense, i.e.
Considering it the generic case thatw is not constant on sets of measure greater than zero, this implies that the support sets of the controls are of measure zero. This motivates us to suggest a discretization strategy, which preserves this sparsity structure on the discrete level in space and time.

Variational discretization
We want to achieve the desired maximal discrete sparsity, i.e. Dirac-measures in space-time, by choosing the Petrov-Galerkinansatz and -test spaces that will induce this structure in combination with the variational discretization concept introduced in [5]. This concept, via the discretization of the test space and the optimality conditions, induces an implicit discretization of the controls (u 0 , u) ∈ M(Ω c ) × M(Q c ). This is how we control the discrete structure of the controls.
In the following we will indicate discretization in space-time by index σ := (τ, h). We define the discrete state space consisting of piecewise linear and continuous finite elements in space and piecewise constant functions with respect to time and the discrete test space consisting of continuous and piecewise linear functions in space and time. This setting yields a Crank-Nicholson scheme with smoothing step (see [3] and [4]). Since the space-time discrete adjoint statew σ is piecewise linear in space and time, we deduce from the sparsity pattern in the generic case that the support ofū is a subset of the spacetime grid points and the support ofū 0 is a subset of the space grid points. Consequently the induced discrete control spaces consists of dirac measures in space and space-time concentrated in grid points, respectively. Similar to the results in [1] we can prove that the projections of all continuous solutions (ū 0 ,ū) onto the induced discrete control spaces coincide with the unique discrete solution. Therefore it is meaningful to work with this discrete control space in numerical examples.
We will compare our approach to the discontinuous Galerkin discretization from [2,Chapter 4.1.], where the discrete state and test space are identical to the discrete state space in our approach and an implicit Euler time stepping scheme is used to constitute the discrete state equation.

Computational results
We numerically solve the variationally discretized and fully discretized problems, both by a semi smooth Newton's method, using the respective optimality systems. To simplify, we fix u 0 = 0 and assume Q c = Q. We choose Ω = (0, 1) and T = 1.5 and set q = 4 3 . To visualize the occuring differences, we use a coarse equidistant 10 × 15 space-time grid. We generate a numerical example by calculating the associated state y(u) for u = δ 0.5,0.5 and then use the interpolation of y(u) as the desired state y d in the problem (P ). Consequently this example problem is a source identification that inherits sparsity. If the parameter α equals zero, the only term remaining in the target functional is 1 q y σ − y d q L q (Q) and consequently the discrete controls u σ,VD and u σ,DG can be calculated by the inverse of the discrete solution operator of the heat equation. Here the indices "VD" and "DG" indicate the two discretization approaches -variational discretization and discontinuous Galerkin -respectively. Due to the different discretization approaches these differ, which can be observed in figure 1. As a consequence of the discretization error in y d , we are not able to reproduce u in either case. Now we choose α = 0.65. Note that the chosen u is not the true solution for problem (P ) with α = 0.65, but rather the true source in the heat equation. Hence we do not expect to find u but rather the location of u, when we solve for the optimal controlsū σ,VD andū σ,DG with associated statesȳ σ,VD andȳ σ,DG , see figure 2. Since u was located on the grid, we are able to find the exact location (0.5, 0.5) in the variational discrete case. The pertubations in the stateȳ σ,VD are caused by the ratio of time and space gridsizes in the Crank-Nicholson scheme. For the discontinuous Galerkin scheme, we observe that due to choice of the control space the discrete control u σ,DG is piecewise constant on the time interval [0.5, 0.6). As a consequence of the implicit Euler time stepping scheme the discrete stateȳ σ,DG is smoother.