Past Numerical Analysis Group Internal Seminar

12 June 2018
14:30
Adilet Otemissov
Abstract


(Joint work with Coralia Cartis) The problem of finding the most extreme value of a function, also known as global optimization, is a challenging task. The difficulty is associated with the exponential increase in the computational time for a linear increase in the dimension. This is known as the ``curse of dimensionality''. In this talk, we demonstrate that such challenges can be overcome for functions with low effective dimensionality --- functions which are constant along certain linear subspaces. Such functions can often be found in applications, for example, in hyper-parameter optimization for neural networks, heuristic algorithms for combinatorial optimization problems and complex engineering simulations.
We propose the use of random subspace embeddings within a(ny) global minimisation algorithm, extending the approach in Wang et al. (2013). We introduce a new framework, called REGO (Random Embeddings for GO), which transforms the high-dimensional optimization problem into a low-dimensional one. In REGO, a new low-dimensional problem is formulated with bound constraints in the reduced space and solved with any GO solver. Using random matrix theory, we provide probabilistic bounds for the success of REGO, which indicate that this is dependent upon the dimension of the embedded subspace and the intrinsic dimension of the function, but independent of the ambient dimension. Numerical results demonstrate that high success rates can be achieved with only one embedding and that rates are for the most part invariant with respect to the ambient dimension of the problem.
 

  • Numerical Analysis Group Internal Seminar
5 June 2018
14:00
Abstract

In this talk, first we  address the convergence issues of a standard finite volume element method (FVEM) applied to simple elliptic problems. Then, we discuss discontinuous finite volume element methods (DFVEM) for elliptic problems  with emphasis on  computational and theoretical  advantages over the standard FVEM. Further, we present a natural extension of DFVEM employed for the elliptic problem to the Stokes problems. We also discuss suitability of these methods for the approximation of incompressible miscible displacement problems.
 

  • Numerical Analysis Group Internal Seminar
29 May 2018
14:30
Jonathan Grant-Peters
Abstract

The vast majority of the world's electricity is generated by converting thermal energy into electric energy by use of a steam turbine. Siemens are one of the worlds leading manufacturers of such
turbines, and aim to design theirs to be as efficient as possible. Using an internally built software, Siemens can estimate the efficiency which would result from a turbine design. In this presentation, we present the approaches that have been taken to improve turbine design using mathematical optimisation software. In particular, we focus on the failings of the approach currently taken, the obstacles in place which make solving this problem difficult, and the approach we intend to take to find a locally optimal solution.

  • Numerical Analysis Group Internal Seminar
29 May 2018
14:00
Chris Farmer
Abstract

This talk will review the main Tikhonov and Bayesian smoothing formulations of inverse problems for dynamical systems with partially observed variables and parameters. The main contenders: strong-constraint, weak-constraint and penalty function formulations will be described. The relationship between these formulations and associated optimisation problems will be revealed.  To close we will indicate techniques for maintaining sparsity and for quantifying uncertainty.

  • Numerical Analysis Group Internal Seminar
22 May 2018
14:30
Dr Dante Kalise
Abstract

In this talk we address the numerical approximation of Mean Field Games with local couplings. For finite difference discretizations of the Mean Field Game system, we follow a variational approach, proving that the schemes can be obtained as the optimality system of suitably defined optimization problems. In order to prove the existence of solutions of the scheme with a variational argument, the monotonicity of the coupling term is not used, which allow us to recover general existence results. Next, assuming next that the coupling term is monotone, the variational problem is cast as a convex optimization problem for which we study and compare several proximal type methods. These algorithms have several interesting features, such as global convergence and stability with respect to the viscosity parameter. We conclude by presenting numerical experiments assessing the performance of the proposed methods. In collaboration with L. Briceno-Arias (Valparaiso, CL) and F. J. Silva (Limoges, FR).

  • Numerical Analysis Group Internal Seminar
22 May 2018
14:00
Abstract

Semidefinite convex optimization problems often have low-rank solutions that can be represented with O(p)-storage. However, semidefinite programming methods require us to store the matrix decision variable with size O(p^2), which prevents the application of virtually all convex methods at large scale.

Indeed, storage, not arithmetic computation, is now the obstacle that prevents us from solving large- scale optimization problems. A grand challenge in contemporary optimization is therefore to design storage-optimal algorithms that provably and reliably solve large-scale optimization problems in key scientific and engineering applications. An algorithm is called storage optimal if its working storage is within a constant factor of the memory required to specify a generic problem instance and its solution.

So far, convex methods have completely failed to satisfy storage optimality. As a result, the literature has largely focused on storage optimal non-convex methods to obtain numerical solutions. Unfortunately, these algorithms have been shown to be provably correct only under unverifiable and unrealistic statistical assumptions on the problem template. They can also sacrifice the key benefits of convexity, as they do not use key convex geometric properties in their cost functions.

To this end, my talk introduces a new convex optimization algebra to obtain numerical solutions to semidefinite programs with a low-rank matrix streaming model. This streaming model provides us an opportunity to integrate sketching as a new tool for developing storage optimal convex optimization methods that go beyond semidefinite programming to more general convex templates. The resulting algorithms are expected to achieve unparalleled results for scalable matrix optimization problems in signal processing, machine learning, and computer science.

  • Numerical Analysis Group Internal Seminar
15 May 2018
14:30
Pranav Singh
Abstract

The Schrödinger equation with a time-dependent potential occurs in a wide range of applications in theoretical chemistry, quantum physics and quantum computing. In this talk I will discuss a variety of Magnus expansion based schemes that have been found to be highly effective for numerically solving these equations since the pioneering work of Tal Ezer and Kosloff in the early 90s. Recent developments in the field focus on approximation of the exponential of the Magnus expansion via exponential splittings including some asymptotic splittings and commutator-free splittings that are designed specifically for this task.

I will also present a very recently developed methodology for the case of laser-matter interaction. This methodology allows us to extend any fourth-order scheme for Schrödinger equation with time-independent potential to a fourth-order method for Schrödinger equation with laser potential with little to no additional cost. These fourth-order methods improve upon many leading schemes of order six due to their low costs and small error constants.

 

  • Numerical Analysis Group Internal Seminar
15 May 2018
14:00
Radu Cimpeanu
Abstract

Many problems that involve the propagation of time-harmonic waves are naturally posed in unbounded domains. For instance, a common problem in the are a of acoustic scattering is the determination of the sound field that is generated when an incoming time-harmonic wave (which is assumed to arrive ``from infinity'') impinges onto a solid body (the scatterer). The boundary
conditions to be applied on the surface of the scatterer (most often of Dirichlet, Neumann or Robin type) tend to be easy to enforce in most numerical solution schemes. Conversely, the imposition of a suitable decay condition (typically a variant of the Sommerfeld radiation condition), which is required to ensure the well-posedness of the solution, is considerably more involved. As a result, many numerical schemes generate spurious reflections from the outer boundary of the finite computational domain.


Perfectly matched layers (PMLs) are in this context a versatile alternative to the usage of classical approaches such as employing absorbing boundary conditions or Dirichlet-to-Neumann mappings, but unfortunately most PML formulations contain adjustable parameters which have to be optimised to give the best possible performance for a particular problem. In this talk I will present a parameter-free PML formulation for the case of the two-dimensional Helmholtz equation. The performance of the proposed method is demonstrated via extensive numerical experiments, involving domains with smooth and polygonal boundaries, different solution types (smooth and singular, planar and non-planar waves), and a wide range of wavenumbers (R. Cimpeanu, A. Martinsson and M.Heil, J. Comp. Phys., 296, 329-347 (2015)). Possible extensions and generalisations will also be touched upon.

  • Numerical Analysis Group Internal Seminar
8 May 2018
14:30
Afaf Bouharguane
Abstract

We consider numerical methods for solving  time dependent partial differential equations with convection-diffusion terms and anti-diffusive fractional operator of order $\alpha \in (1,2)$. These equations are motivated by two distinct applications: a dune morphodynamics model and a signal filtering method. 
We propose numerical schemes based on local discontinuous Galerkin methods to approximate the solutions of these equations. Numerical stability and convergence of these schemes are investigated. 
Finally numerical experiments are given to illustrate qualitative behaviors of solutions for both applications and to confirme the convergence results. 

  • Numerical Analysis Group Internal Seminar
8 May 2018
14:00
Nour Seloula
Abstract

We introduce and analyze a discontinuous Galerkin method for the Oseen equations in two dimension spaces. The boundary conditions are mixed and they are assumed to be of three different types:
the vorticity  and the normal component of the velocity are given on a first part of the boundary, the pressure and the tangential component of the velocity are given on a second part of the boundary and the Dirichlet condition is given on the remainder part . We establish a priori error estimates in the energy norm for the velocity and in the L2 norm for the pressure. An a posteriori error estimate is also carried out yielding optimal convergence rate. The analysis is based on rewriting the method in a non-consistent manner using lifting operators in the spirit of Arnold, Brezzi, Cockburn and Marini.

  • Numerical Analysis Group Internal Seminar

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