The talk will introduce the concepts of multilevel optimization and motivate them in the context of problems arising from the discretization of infinite dimensional applications. It will be shown how optimization methods can accomodate a number of useful (and classical) ideas from the multigrid

community, and thereby produce substantial efficiency improvements compared to existing large-scale minimization techniques. Two different classes of multilevel methods will be discussed: trust-region and linesearch algorithms.

The first class will be described in the context of a multilevel generalization of the well-known trust-region-Newton method. The second will focus on limited-memory quasi-Newton algorithms. Preliminary numerical results will be presented which indicate that both types of multilevel algorithms may be practically very advantageous.

# Past Computational Mathematics and Applications Seminar

In numerical atmosphere models, values of relevant physical parameters are often uncertain by more than 100% and weather forecast skill is significantly reduced after a couple of days. Still, numerical operations are typically calculated in double precision with 15 significant decimal digits. If we reduce numerical precision, we can reduce power consumption and increase computational performance significantly. If savings are reinvested to build larger supercomputers, this would allow an increase in resolution in weather and climate models and might lead to better predictions of future weather and climate.

I will discuss approaches to reduce numerical precision beyond single precision in high performance computing and in particular in weather and climate modelling. I will present results that show that precision can be reduced significantly in atmosphere models and that potential savings can be huge. I will also discuss how rounding errors will impact model dynamics and interact with model uncertainty and predictability.

We propose the use of a constraint preconditioner for the iterative solution of the linear system arising from the finite element discretization of the coupled Stokes-Darcy system. The Stokes-Darcy system is a set of coupled PDEs that can be used to model a freely flowing fluid over porous media flow. The fully coupled system matrix is large, sparse, non-symmetric, and of saddle point form. We provide for exact versions of the constraint preconditioner spectral and field-of-values bounds that are independent of the underlying mesh width. We present several numerical experiments, using the deal.II finite element library, that illustrate our results in both two and three dimensions. We compare exact and inexact versions of the constraint preconditioner against standard block diagonal and block lower triangular preconditioners to illustrate its favorable properties.

We develop a novel, fundamental and surprisingly simple randomized iterative method for solving consistent linear systems. Our method has six different but equivalent interpretations: sketch-and-project, constrain-and-approximate, random intersect, random linear solve, random update and random fixed point. By varying its two parameters—a positive definite matrix (defining geometry), and a random matrix (sampled in an i.i.d. fashion in each iteration)—we recover a comprehensive array of well known algorithms as special cases, including the randomized Kaczmarz method, randomized Newton method, randomized coordinate descent method and random Gaussian pursuit. We naturally also obtain variants of all these methods using blocks and importance sampling. However, our method allows for a much wider selection of these two parameters, which leads to a number of new specific methods. We prove exponential convergence of the expected norm of the error in a single theorem, from which existing complexity results for known variants can be obtained. However, we also give an exact formula for the evolution of the expected iterates, which allows us to give lower bounds on the convergence rate.

When formulated appropriately, the broad families of sequential quadratic programming, augmented Lagrangian and interior-point methods all require the solution of symmetric saddle-point linear systems. When regularization is employed, the systems become symmetric and quasi definite. The latter are

indefinite but their rich structure and strong relationships with definite systems enable specialized linear algebra, and make them prime candidates for matrix-free implementations of optimization methods. In this talk, I explore various formulations of the step equations in optimization and corresponding

iterative methods that exploit their structure.

Security Constrained Optimal Power Flow is an increasingly important problem for power systems operation both in its own right and as a subproblem for more complex problems such as transmission switching or

unit commitment.

The structure of the problem resembles stochastic programming problems in that one aims to find a cost optimal operation schedule that is feasible for all possible equipment outage scenarios

(contingencies). Due to the presence of power flow constraints (in their "DC" or "AC" version), the resulting problem is a large scale linear or nonlinear programming problem.

However it is known that only a small subset of the contingencies is active at the solution. We show how Interior Point methods can exploit this structure both by simplifying the linear algebra operations as

well as generating necessary contingencies on the fly and integrating them into the algorithm using IPM warmstarting techniques. The final problem solved by this scheme is significantly smaller than the full

contingency constrained problem, resulting in substantial speed gains.

Numerical and theoretical results of our algorithm will be presented.

Can we extend the FEM to general polytopic, i.e. polygonal and polyhedral, meshes while retaining

the ease of implementation and computational cost comparable to that of standard FEM? Within this talk, I present two approaches that achieve just that (and much more): the Virtual Element Method (VEM) and an hp-version discontinuous Galerkin (dG) method.

The Virtual Element spaces are like the usual (polynomial) finite element spaces with the addition of suitable non-polynomial functions. This is far from being a novel idea. The novelty of the VEM approach is that it avoids expensive evaluations of the non-polynomial "virtual" functions by basing all

computations solely on the method's carefully chosen degrees of freedom. This way we can easily deal

with complicated element geometries and/or higher continuity requirements (like C1, C2, etc.), while

maintaining the computational complexity comparable to that of standard finite element computations.

As you might expect, the choice and number of the degrees of freedom depends on such continuity

requirements. If mesh flexibility is the goal, while one is ready to give up on global regularity, other approaches can be considered. For instance, dG approaches are naturally suited to deal with polytopic meshes. Here I present an extension of the classical Interior Penalty dG method which achieves optimal rates of convergence on polytopic meshes even under elemental edge/face degeneration.

The next step is to exploit mesh flexibility in the efficient resolution of problems characterised by

complicated geometries and solution features, for instance within the framework of automatic FEM

adaptivity. I shall finally introduce ongoing work in this direction.

In this seminar I will present a semi-langrangian discretisation of the Monge-Ampère operator, which is of interest in optimal transport

and differential geometry as well as in related fields of application.

I will discuss the proof of convergence to viscosity solutions. To address the challenge of uniqueness and convexity we draw upon the classical relationship with Hamilton-Jacobi-Bellman equations, which we extend to the viscosity setting. I will explain that the monotonicity of semi-langrangian schemes implies that they possess large stencils, which in turn requires careful treatment of the boundary conditions.

The contents of the seminar is based on current work with X Feng from the University of Tennessee.