Reduced Order Optimal Control of the Convective FitzHugh-Nagumo Equation

In this paper, we compare three model order reduction methods: the proper orthogonal decomposition (POD), discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD) for the optimal control of the convective FitzHugh-Nagumo (FHN) equations. The convective FHN equations consists of the semi-linear activator and the linear inhibitor equations, modeling blood coagulation in moving excitable media. The semilinear activator equation leads to a non-convex optimal control problem (OCP). The most commonly used method in reduced optimal control is POD. We use DEIM and DMD to approximate efficiently the nonlinear terms in reduced order models. We compare the accuracy and computational times of three reduced-order optimal control solutions with the full order discontinuous Galerkin finite element solution of the convection dominated FHN equations with terminal controls. Numerical results show that POD is the most accurate whereas POD-DMD is the fastest

Challenges in Optimization with Complex PDE-Systems (hybrid meeting)

The workshop concentrated on various aspects of optimization problems with systems of nonlinear partial differential equations (PDEs) or variational inequalities (VIs) as constraints. In particular, discussions around several keynote presentations in the areas of optimal control of nonlinear or non-smooth systems, optimization problems with functional and discrete or switching variables leading to mixed integer nonlinear PDE constrained optimization, shape and topology optimization, feedback control and stabilization, multi-criteria problems and multiple optimization problems with equilibrium constraints as well as versions of these problems under uncertainty or stochastic influences, and the respectively associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, aspects of optimal control of data-driven PDE constraints (e.g. related to machine learning) were addressed

A Multigrid Method for the Efficient Numerical Solution of Optimization Problems Constrained by Partial Differential Equations

We study the minimization of a quadratic functional subject to constraints given by a linear or semilinear elliptic partial differential equation with distributed control. Further, pointwise inequality constraints on the control are accounted for. In the linear-quadratic case, the discretized optimality conditions yield a large, sparse, and indefinite system with saddle point structure. One main contribution of this thesis consists in devising a coupled multigrid solver which avoids full constraint elimination. To this end, we define a smoothing iteration incorporating elements from constraint preconditioning. A local mode analysis shows that for discrete optimality systems, we can expect smoothing rates close to those obtained with respect to the underlying constraint PDE. Our numerical experiments include problems with constraints where standard pointwise smoothing is known to fail for the underlying PDE. In particular, we consider anisotropic diffusion and convection-diffusion problems. The framework of our method allows to include line smoothers or ILU-factorizations, which are suitable for such problems. In all cases, numerical experiments show that convergence rates do not depend on the mesh size of the finest level and discrete optimality systems can be solved with a small multiple of the computational cost which is required to solve the underlying constraint PDE. Employing the full multigrid approach, the computational cost is proportional to the number of unknowns on the finest grid level. We discuss the role of the regularization parameter in the cost functional and show that the convergence rates are robust with respect to both the fine grid mesh size and the regularization parameter under a mild restriction on the next to coarsest mesh size. Incorporating spectral filtering for the reduced Hessian in the control smoothing step allows us to weaken the mesh size restriction. As a result, problems with near-vanishing regularization parameter can be treated efficiently with a negligible amount of additional computational work. For fine discretizations, robust convergence is obtained with rates which are independent of the regularization parameter, the coarsest mesh size, and the number of levels. In order to treat linear-quadratic problems with pointwise inequality constraints on the control, the multigrid approach is modified to solve subproblems generated by a primal-dual active set strategy (PDAS). Numerical experiments demonstrate the high efficiency of this approach due to mesh-independent convergence of both the outer PDAS method and the inner multigrid solver. The PDAS-multigrid method is incorporated in the sequential quadratic programming (SQP) framework. Inexact Newton techniques further enhance the computational efficiency. Globalization is implemented with a line search based on the augmented Lagrangian merit function. Numerical experiments highlight the efficiency of the resulting SQP-multigrid approach. In all cases, locally superlinear convergence of the SQP method is observed. In combination with the mesh-independent convergence rate of the inner solver, a solution method with optimal efficiency is obtained

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

Finite Element Methods for Fourth Order Variational Inequalities

In this work we study finite element methods for fourth order variational inequalities. We begin with two model problems that lead to fourth order obstacle problems and a brief survey of finite element methods for these problems. Then we review the fundamental results including Sobolev spaces, existence and uniqueness results of variational inequalities, regularity results for biharmonic problems and fourth order obstacle problems, and finite element methods for the biharmonic problem. In Chapter 2 we also include three types of enriching operators which are useful in the convergence analysis. In Chapter 3 we study finite element methods for the displacement obstacle problem of clamped Kirchhoff plates. A unified convergence analysis is provided for $C^1$ finite element methods, classical nonconforming finite element methods and $C^0$ interior penalty methods. The key ingredient in the error analysis is the introduction of the auxiliary obstacle problem. An optimal $O(h)$ error estimate in the energy norm is obtained for convex domains. We also address the approximations of the coincidence set and the free boundary. In Chapter 4 we study a Morley finite element method and a quadratic $C^0$ interior penalty method for the displacement obstacle problem of clamped Kirchhoff plates with general Dirichlet boundary conditions on general polygonal domains. We prove the magnitudes of the errors in the energy norm and the $L^{\infty}$ norm are $O(h^{\alpha})$, where $\alpha \u3e 1/2$ is determined by the interior angles of the polygonal domain. Numerical results are also presented to illustrate the performance of the methods and verify the theoretical results obtained in Chapter 3 and Chapter 4. In Chapter 5 we consider an elliptic optimal control problem with state constraints. By formulating the problem as a fourth order obstacle problem with the boundary condition of simply supported plates, we study a quadratic $C^0$ interior penalty method and derive the error estimates in the energy norm based on the framework we introduced in Chapter 3. The rate of convergence is derived for both quasi-uniform meshes and graded meshes. Numerical results presented in this chapter confirm our theoretical results

Convergence analysis of a Crank-Nicolson Galerkin method for an inverse source problem for parabolic equations with boundary observations

This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank-Nicolson Galerkin method is applied to the least squares functional with an quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach to zero with an appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and corresponding convergence rates are proved. Finally, several numerical experiments are presented to illustrate the theoretical findings.Comment: Inverse source problem, Tikhonov regularization, Crank-Nicolson Galerkin method, Source condition, Convergence rates, Ill-posedness, Parabolic proble

A mixed formulation for the direct approximation of L2L^2-weighted controls for the linear heat equation

This paper deals with the numerical computation of null controls for the linear heat equation. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a given positive time. In [Fernandez-Cara \& Münch, Strong convergence approximations of null controls for the 1D heat equation, 2013], a so-called primal method is described leading to a strongly convergent approximation of distributed control: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality conditions. In this work, we adapt the method to approximate the control of minimal square integrable-weighted norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner situation and is valid in any dimension