Almost all real-world applications involve a degree of uncertainty. This may be the result of noisy measurements, restrictions on observability, or simply unforeseen events. Since many models in both engineering and the natural sciences make use of partial differential equations (PDEs), it is natural to consider PDEs with random inputs. In this context, passing from modelling and simulation to optimization or control results in stochastic PDE-constrained optimization problems. This leads to a number of theoretical, algorithmic, and numerical challenges.

From a mathematical standpoint, the solution of the underlying PDE is a random field, which in turn makes the quantity of interest or the objective function an implicitly defined random variable. In order to minimize this distributed objective, one can use, e.g., stochastic order constraints, a distributionally robust approach, or risk measures. In this talk, we will make use of risk measures.

After motivating the approach via a model for the mitigation of an airborne pollutant, we build up an analytical framework and introduce some useful risk measures. This allows us to prove the existence of solutions and derive optimality conditions. We then present several approximation schemes for handling non-smooth risk measures in order to leverage existing numerical methods from PDE-constrained optimization. Finally, we discuss solutions techniques and illustrate our results with numerical examples.

# Past Computational Mathematics and Applications Seminar

During the last decade, the progress in the computational performance of commercial mixed-integer programming solvers have been significant. Part of this success is due to faster computers and better software engineering but a more significant part of it is due to the power of the cutting planes used in these solvers.

In the first part of this talk, we will discuss main components of a MIP solver and describe some classical families of valid inequalities (Gomory mixed integer cuts, mixed integer rounding cuts, split cuts, etc.) that are routinely used in these solvers. In the second part, we will discuss recent progress in cutting plane theory that has not yet made its way to commercial solvers. In particular, we will discuss cuts from lattice-free convex sets and answer a long standing question in the affirmative by deriving a finite cutting plane algorithm for mixed-integer programming.

In variational imaging and other inverse problem modeling, regularisation plays a major role. In recent years, high order regularizers such as the total generalised variation, the mean curvature and the Gaussian curvature are increasingly studied and applied, and many improved results over the widely-used total variation model are reported.

Here we first introduce the fractional order derivatives and the total fractional-order variation which provides an alternative regularizer and is not yet formally analysed. We demonstrate that existence and uniqueness properties of the new model can be analysed in a fractional BV space, and, equally, the new model performs as well as the high order regularizers (which do not yet have much theory).

In the usual framework, the algorithms of a fractional order model are not fast due to dense matrices involved. Moreover, written in a Bregman framework, the resulting Sylvester equation with Toeplitz coefficients can be solved efficiently by a preconditioned solver. Further ideas based on adaptive integration can also improve the computational efficiency in a dramatic way.

Numerical experiments will be given to illustrate the advantages of the new regulariser for both restoration and registration problems.

We will present a very general framework for unconstrained stochastic optimization which is based on standard trust region framework using random models. In particular this framework retains the desirable features such step acceptance criterion, trust region adjustment and ability to utilize of second order models. We make assumptions on the stochasticity that are different from the typical assumptions of stochastic and simulation-based optimization. In particular we assume that our models and function values satisfy some good quality conditions with some probability fixed, but can be arbitrarily bad otherwise. We will analyze the convergence and convergence rates of this general framework and discuss the requirement on the models and function values. We will will contrast our results with existing results from stochastic approximation literature. We will finish with examples of applications arising the area of machine learning.

Work with Jemima Tabeart, Sarah Dance, Nancy Nichols, Joanne Waller (University of Reading) and Stefano Migliorini, Fiona Smith (Met Office).

In environmental prediction variational data assimilation (DA) is a method for using observational data to estimate the current state of the system. The DA problem is usually solved as a very large nonlinear least squares problem, in which the fit to the measurements is balanced against the fit to a previous model forecast. These two terms are weighted by matrices describing the correlations of the errors in the forecast and in the observations. Until recently most operational weather and ocean forecasting systems assumed that the errors in the observations are uncorrelated. However, as we move to higher resolution observations then it is becoming more important to specify observation error correlations. In this work we look at the effect this has on the conditioning of the optimization problem. In the context of a linear system we develop bounds on the condition number of the problem in the presence of correlated observation errors. We show that the condition number is very dependent on the minimum eigenvalue of the observation error correlation matrix. We then present results using the Met Office data assimilation system, in which different methods for reconditioning the correlation matrix are tested. We investigate the effect of these different methods on the conditioning and the final solution of the problem.

Inverse problems are ubiquitous in many areas of Science and Engineering and, once discretised, they lead to ill-conditioned linear systems, often of huge dimensions: regularisation consists in replacing the original system by a nearby problem with better numerical properties, in order to find meaningful approximations of its solution. In this talk we will explore the regularisation properties of many iterative methods based on Krylov subspaces. After surveying some basic methods such as CGLS and GMRES, innovative approaches based on flexible variants of CGLS and GMRES will be presented, in order to efficiently enforce nonnegativity and sparsity into the solution.

We consider the Cauchy (or steepest descent) method with exact line search applied to a strongly convex function with Lipschitz continuous gradient. We establish the exact worst-case rate of convergence of this scheme, and show that this worst-case behavior is exhibited by a certain convex quadratic function. We also give worst-case complexity bound for a noisy variant of gradient descent method. Finally, we show that these results may be applied to study the worst-case performance of Newton's method for the minimization of self-concordant functions.

The proofs are computer-assisted, and rely on the resolution of semidefinite programming performance estimation problems as introduced in the paper [Y. Drori and M. Teboulle. Performance of first-order methods for smooth convex minimization: a novel approach. Mathematical Programming, 145(1-2):451-482, 2014].

Joint work with F. Glineur and A.B. Taylor.

Functions defined by evaluation programs involving smooth elementals and absolute values as well as max and min are piecewise smooth. For this class we present first and second order, necessary and sufficient conditions for the functions to be locally optimal, or convex, or at least possess a supporting hyperplane. The conditions generalize the classical KKT and SSC theory and are constructive; though in the case of convexity they may be combinatorial to verify. As a side product we find that, under the Mangasarin-Fromowitz-Kink-Qualification, the well established nonsmooth concept of subdifferential regularity is equivalent to first order convexity. All results are based on piecewise linearization and suggest corresponding optimization algorithms.