# Past Computational Mathematics and Applications Seminar

Equations of quantum mechanics in the semiclassical regime present an enduring challenge for numerical analysts, because their solution is highly oscillatory and evolves on two scales. Standard computational approaches to the semiclassical Schrödinger equation do not allow for long time integration as required, for example, in quantum control of atoms by short laser bursts. This has motivated our approach of asymptotic splittings. Combining techniques from Lie-algebra theory and numerical algebra, we present a new computational paradigm of symmetric Zassenhaus splittings, which lends itself to a very precise discretisation in long time intervals, at very little cost. We will illustrate our talk by examples of quantum phenomena – quantum tunnelling and quantum scattering – and their computation and, time allowing, discuss an extension of this methodology to time-dependent semiclassical systems using Magnus expansions

This talk focuses on the direct search method, arguably one of the simplest optimization algorithms. The algorithm minimizes an objective function by iteratively evaluating it along a number of (polling) directions, which are typically taken from so-called positive spanning sets. It does not use derivatives.

We first introduce the worst case complexity theory of direct search, and discuss how to choose the positive spanning set to minimize the complexity bound. The discussion leads us to a long-standing open

problem in Discrete Geometry. A recent result on this problem enables us to establish the optimal order for the worst case complexity of direct search.

We then show how to achieve even lower complexity bound by using random polling directions. It turns out that polling along two random directions at each iteration is sufficient to guarantee the convergence

of direct search for any dimension, and the resultant algorithm enjoys lower complexity both in theory and in practice.

The last part of the talk is devoted to direct search based on inaccurate function values. We address three questions:

i) what kind of solution can we obtain by direct search if the function values are inaccurate?

ii) what is the worst case complexity to attain such a solution? iii) given

the inaccuracy in the function values, when to stop the algorithm in order

to guarantee the quality of the solution and also avoid “over-optimization”?

This talk is based on joint works with F. Delbos, M. Dodangeh, S. Gratton, B. Pauwels, C. W. Royer, and L. N. Vicente.

I will review some recent work on the problem of reliable automatic detection of blow-up behaviour for nonlinear parabolic PDEs. The adaptive algorithms developed are based on rigorous conditional a posteriori error bounds. The use of space-time adaptivity is crucial in making the problem computationally tractable. The results presented are applicable to quite general spatial operators, rendering the approach potentially useful in informing respective PDE theory. The new adaptive algorithm is shown to accurately estimate the blow-up time of a number of problems, including ones exhibiting regional blow-up.

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.

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.