A Space-Time Variational Method for Optimal Control Problems

We consider a space-time variational formulation of a PDE-constrained optimal control problem with box constraints on the control and a parabolic PDE with Robin boundary conditions. In this setting, the optimal control problem reduces to an optimization problem for which we derive necessary and sufficient optimality conditions. Next, we introduce a space-time (tensorproduct) discretization using finite elements in space and piecewise linear functions in time. This setting is known to be equivalent to a Crank-Nicolson time stepping scheme for parabolic problems. The optimization problem is solved by a projected gradient method. We show numerical comparisons for problems in 1d, 2d and 3d in space. It is shown that the classical semi-discrete primal-dual setting is more efficient for small problem sizes and moderate accuracy. However, the space-time discretization shows good stability properties and even outperforms the classical approach as the dimension in space and/or the desired accuracy increases.Comment: 20 page

Error estimates of mixed finite elements combined with Crank-Nicolson scheme for parabolic control problems

In this paper, a mixed finite element method combined with Crank-Nicolson scheme approximation of parabolic optimal control problems with control constraint is investigated. For the state and co-state, the order Raviart-Thomas mixed finite element spaces and Crank-Nicolson scheme are used for space and time discretization, respectively. The variational discretization technique is used for the control variable. We derive optimal priori error estimates for the control, state and co-state. Some numerical examples are presented to demonstrate the theoretical results

Error estimates for the discretization of the velocity tracking problem

In this paper we are continuing our work (Casas and Chrysafinos, SIAM J Numer Anal 50(5):2281–2306, 2012), concerning a priori error estimates for the velocity tracking of two-dimensional evolutionary Navier–Stokes flows. The controls are of distributed type, and subject to point-wise control constraints. The discretization scheme of the state and adjoint equations is based on a discontinuous time-stepping scheme (in time) combined with conforming finite elements (in space) for the velocity and pressure. Provided that the time and space discretization parameters, t and h respectively, satisfy t = Ch2, error estimates of order O(h2) and O(h 3/2 – 2/p ) with p > 3 depending on the regularity of the target and the initial velocity, are proved for the difference between the locally optimal controls and their discrete approximations, when the controls are discretized by the variational discretization approach and by using piecewise-linear functions in space respectively. Both results are based on new duality arguments for the evolutionary Navier–Stokes equations

NEW COMPUTATIONAL METHODS FOR OPTIMAL CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS

Partial differential equations are the chief means of providing mathematical models in science, engineering and other fields. Optimal control of partial differential equations (PDEs) has tremendous applications in engineering and science, such as shape optimization, image processing, fluid dynamics, and chemical processes. In this thesis, we develop and analyze several efficient numerical methods for the optimal control problems governed by elliptic PDE, parabolic PDE, and wave PDE, respectively. The thesis consists of six chapters. In Chapter 1, we briefly introduce a few motivating applications and summarize some theoretical and computational foundations of our following developed approaches. In Chapter 2, we establish a new multigrid algorithm to accelerate the semi-smooth Newton method that is applied to the first-order necessary optimality system arising from semi-linear control-constrained elliptic optimal control problems. Under suitable assumptions, the discretized Jacobian matrix is proved to have a uniformly bounded inverse with respect to mesh size. Different from current available approaches, a new strategy that leads to a robust multigrid solver is employed to define the coarse grid operator. Numerical simulations are provided to illustrate the efficiency of the proposed method, which shows to be computationally more efficient than the popular full approximation storage (FAS) multigrid method. In particular, our proposed approach achieves a mesh-independent convergence and its performance is highly robust with respect to the regularization parameter. In Chaper 3, we present a new second-order leapfrog finite difference scheme in time for solving the first-order necessary optimality system of the linear parabolic optimal control problems. The new leapfrog scheme is shown to be unconditionally stable and it provides a second-order accuracy, while the classical leapfrog scheme usually is well-known to be unstable. A mathematical proof for the convergence of the proposed scheme is provided under a suitable norm. Moreover, the proposed leapfrog scheme gives a favorable structure that leads to an effective implementation of a fast solver under the multigrid framework. Numerical examples show that the proposed scheme significantly outperforms the widely used second-order backward time differentiation approach, and the resultant fast solver demonstrates a mesh-independent convergence as well as a linear time complexity. In Chapter 4, we develop a new semi-smooth Newton multigrid algorithm for solving the discretized first-order necessary optimality system that characterizes the optimal solution of semi-linear parabolic PDE optimal control problems with control constraints. A new leapfrog discretization scheme in time associated with the standard five-point stencil in space is established to achieve a second-order accuracy. The convergence (or unconditional stability) of the proposed scheme is proved when time-periodic solutions are considered. Moreover, the derived well-structured discretized Jacobian matrices greatly facilitate the development of an effective smoother in our multigrid algorithm. Numerical simulations are provided to illustrate the effectiveness of the proposed method, which validates the second-order accuracy in solution approximations as well as the optimal linear complexity of computing time. In Chapter 5, we offer a new implicit finite difference scheme in time for solving the first-order necessary optimality system arising in optimal control of wave equations. With a five-point central finite difference scheme in space, the full discretization is proved to be unconditionally convergent with a second-order accuracy, which is not restricted by the classical Courant-Friedrichs-Lewy (CFL) stability condition on the spatial and temporal step sizes. Moreover, based on its advantageous developed structure, an efficient preconditioned Krylov subspace method is provided and analyzed for solving the discretized sparse linear system. Numerical examples are presented to confirm our theoretical conclusions and demonstrate the promising performance of proposed preconditioned iterative solver. Finally, brief summaries and future research perspectives are given in Chapter 6

B-Spline Based Methods: From Monotone Multigrid Schemes for American Options to Uncertain Volatility Models

In the first part of this thesis, we consider B-spline based methods for pricing American options in the Black-Scholes and Heston model. The difference between these two models is the assumption on the volatility of the underlying asset. While in the Black-Scholes model the volatility is assumed to be constant, the Heston model includes a stochastic volatility variable. The underlying problems are formulated as parabolic variational inequalities. Recall that, in finance, to determine optimal risk strategies, one is not only interested in the solution of the variational inequality, i.e., the option price, but also in its partial derivatives up to order two, the so-called Greeks. A special feature for these option price problems is that initial conditions are typically given as piecewise linear continuous functions. Consequently, we have derived a spatial discretization based on cubic B-splines with coinciding knots at the points where the initial condition is not differentiable. Together with an implicit time stepping scheme, this enables us to achieve an accurate pointwise approximation of the partial derivatives up to order two. For the efficient numerical solution of the discrete variational inequality, we propose a monotone multigrid method for (tensor product) B-splines with possible internal coinciding knots. Corresponding numerical results show that the monotone multigrid method is robust with respect to the refinement level and mesh size. In the second part of this thesis, we consider the pricing of a European option in the uncertain volatility model. In this model the volatility of the underlying asset is a priori unknown and is assumed to lie within a range of extreme values. Mathematically, this problem can be formulated as a one dimensional parabolic Hamilton-Jacobi-Bellman equation and is also called Black-Scholes-Barenblatt equation. In the resulting non-linear equation, the diffusion coefficient is given by a volatility function which depends pointwise on the second derivative. This kind of non-linear partial differential equation does not admit a weak H^1-formulation. This is due to the fact that the non-linearity depends pointwise on the second derivative of the solution and, thus, no integration by parts is possible to pass the partial derivative onto a test function. But in the discrete setting this pointwise second derivative can be approximated in H^1 by L^1-normalized B-splines. It turns out that the approximation of the volatility function leads to discontinuities in the partial derivatives. In order to improve the approximation of the solution and its partial derivatives for cubic B-splines, we develop a Newton like algorithm within a knot insertion step. Corresponding numerical results show that the convergence of the solution and its partial derivatives are nearly optimal in the L^2-norm, when the location of volatility change is approximated with desired accuracy