# ATMW New Directions in PDE Constrained Optimisation (2018)- Speakers and Syllabus

Partial differential equations (PDE) are found in models throughout all branches of engineering and the natural sciences. Passing from modelling and simulation to the optimization or control of physical systems naturally leads to PDE-constrained optimization problems. For example, we can model the flow of air over a given aircraft wing using the Euler or Navier-Stokes equations. However, if we wish to find a new wing design with less drag and sufficient lift, then these equations along with additional bounds on certain physical properties could arise as constraints in an optimization problem. The analysis, development of efficient algorithms, and numerical solution of such problems requires techniques that go beyond the usual results found in finite dimensional non-linear optimization. The five day winter school will comprise several tutorials and a few selected invited talks on both the basic theory and numerical methods for PDE-constrained optimization. Additional talks on cutting edge research topics will also be presented. The tutorials are targeted toward graduate students, postdocs and researchers interested in entering the field. The tutorials will provide an overview of PDE constrained optimization, with special emphasis on variational inequalities, uncertainty, model order reduction, shape optimization, and software implementation. The invited presentations will complement the tutorials.

Workshop Venue:-Ramanujan Hall, Department of Mathematics,IIT Bombay on 12th and 13th March and Room no 14, VMCC Hall on 14th,15th, 16th,March

Syllabus to be covered in terms of modules of 6 lectures each:

 Speaker Detailed Syllabus Harbir Antil(George Mason University, Virginia, USA) Title: Brief Introduction to PDE-constrained Optimization 1) Basics: Problem statement, examples of elliptic/parabolic optimal control problems (OCP) Notation Steps to solve an OCP: existence/uniqueness of solution; first order necessary and second order sufficient optimality conditions; Numerical approximation and analysis of the discretized problem; Implementation of the discretized problem - Examples 2) Uniqueness and regularity of weak solution to elliptic BVPs. 3) Linear elliptic control problems: Reduced functional Existence Differentiation in Banach spaces: Gateaux, Frechet First order necessary optimality conditions Formal Lagrange Method  4) Semilinear elliptic control problems Existence of solution to state equation and the control problem Nemytskii operator Differentiability of control to state map First order necessary conditions  5) Finite element discretization and provide a finite element solver Sören Bartels(University of Freiburg, Germany) Title: Finite element approximation of the ROF model problemAbstract: The ROF model problem is a convex, nondifferentiable minimization problem that arises in image processing and serves as a model problem for nonsmooth PDEs. It defines a regularization of a noisy image via a simple optimization procedure involving the total variation and was introduced by Rudin, Osher, and Fatemi in 1992. Its discretization with finite element or finite difference methods leads to reduced convergence rates and requires suitable iterative methods for their practical solution. We present an error estimate that avoids unrealistic regularity assumptions and discuss the stability of a semi-implicit gradient flow for a suitable regularization. A corresponding Matlab code will be presented and explained in the tutorial. 1) P1-FEM and application to the ROF model problem2) Iterative minimization via a regularized gradient flow3) Discussion of a Matlab implementation Michael Hintermüller (WIAS and Humboldt University, Berlin) Title: Semismooth Newton methods in PDE constrained optimization. 1. Introduction A wide array of mathematical models in the engineering and natural sciences, including mathematical imaging, biomedical sciences, mathematical finance and economics, are subject to non-smooth structures. This non-smoothness can be a direct result of the model itself, e.g., a non-smooth energy or objective functional or due to its formulation as a (quasi)-variational inequality. On the other hand, the non-smoothness may be implicit and only appear as a result of the presence of inequality constraints, e.g., control or state constraints, when considering second-order solution methods. 2. Non-Smooth Operator Equations and Generalized Newton Methods Following a discussion of several modern applications taken from the fields mentioned above, a non-smooth operator equation (re) formulation will be presented, which is amenable to a generalized Newton framework in function space. The necessary notions of generalized differentiation and semi-smoothness of operators in function space, which are pertinent to further discussion, will be presented in detail. Building upon these concepts, a general locally superlinear convergent numerical solution algorithm associated with a semismooth Newton iteration will be derived. 3. Mesh Independence For the discretization of infinite dimensional PDE-constrainedoptimization and optimal control problems, it is essential that the associated numerical solution algorithm exhibits mesh independent convergence behaviour upon adaptive refinements of the underlying mesh. It will be demonstrated for a large class of pertinent problems that semismooth Newton methods are mesh-independent.  4. Applications In the final part of the course, the efficiency and wide applicability of the semismooth Newton framework will be highlighted by considering constrained optimal control problems for fluid flow, contact problems with or without adhesive forces, phase-separation phenomena, which rely on non-smooth homogeneous free energy densities and image restoration problems in mathematical imaging. Thomas M. Surowiec (Philipps University of Marburg, Germany) Title: PDE constrained optimization with uncertainty.  1. Introduction The control and optimization of PDE under uncertainty is a rapidly growing field that combines aspects of mathematical optimization, uncertainty quantification, non­linear functional analysis, and techniques of risk management. The first part of the course will comprise several examples, which motivate the field and introduce the general thought process.  2. Existence and Optimality Conditions  The second stage of the course will be reserved for a discussion of the proper function space setting and the necessary conditions used to derive existence results and optimality conditions. Some classical topics from convex analysis, the calculus of variations, and differentiability properties of Nemytskii operators will be presented and discussed as needed. A primer on risk measures taken from the management sciences will be given as well.  3. Approximation and Discretization  The third portion of the course will be divided into a discussion on smoothing, regularization, and discretization techniques. The need for smoothing techniques is motivated by numerical examples. The asymptotic behavior is investigated from a qualitative and quantitative standpoint. Finally, several means of discretizing the random variables and fields will be discussed. 4.4.   Numerical  Examples  The  shortest  and  final  section  of  the  course  will  be  dedicated  to  the demonstration and interpretation of the results in  1.­3. in  a  computational setting.

Names of the tutors / course associate with their affiliation and status

1. Harbir Antil (George Mason University, Virginia, USA)
2. Sören Bartels (University of Freiburg, Germany)
3. Michael Hintermüller (WIAS and Humboldt University, Berlin)
4. Thomas M. Surowiec (Philipps University of Marburg, Germany)

Tentative time-table:

 Day Date Lecture 1 9.30to11.00 Tea 11.00to11.30 Lecture 2 11.30to1.00 Lunch 1.00to2.00 Tutorial session (lab session) 2.00to3.30 Tea 3.30to4.00 Invited talks 4.00to5.00 Tea&Snacks 5.00Onwards name of the speaker name of the speaker name of the speaker Name of the speaker / tutor Mon 12.03.2018 H.Antil H. Antil H. Antil + T.Surowiec (Software implementation + tutorial) C. Carstensen (Axioms of adaptivity for optimal convergence rates) Tues 13.03.2018 S. Bartels S.Bartels S. Bartels (Software implementation + tutorial) C. Carstensen(Separate marking for adaptive least squares finite element schemes ) Wed 14.03.2018 M. Hintermuller M. Hintermueller S. Bartels (Tutorial) 3:45 PM to 06:00 PMShort research talks by participants Thu 15.03.2018 T.Surowiec T.Surowiec T. Surowiec             (Optimal Control of Electrowetting on Dielectric) M. Vanninathan Optimal Design Problems and Homogenization Fri 16.03.2018 S. Bartels(Numerical solution of a nonlinear eigenvalue problem arising in optimal insulation) H. Antil (Fractional PDEs) H. Antil D. Mitra (Stabilizability of the compressible Navier-Stokes system)