## Abstracts of Student Conference, 1st December 2014

## Doaa Elsakout

### Explore the Feasibility of Multilevel Monte Carlo (MLMC) in Reservoir Simulation

**Abstract:**

The comparison between the performance of Multilevel Monte Carlo (MLMC) and Standard Monte Carlo (SMC) is the aim of the talk. We are going to look at Advection, Buckley-Leveret and Pressure equations, and see the differences between these two techniques in different scenarios.

MLMC is a variance reduction technique which accelerates the convergence for SMC because it suffers from low convergence. SMC technique is well known for quantify the uncertainty in porous media flow problem, easy to implement but unfortunately time consuming.

The idea for MLMC technique is telescoping sum: which is representing the quantity of interest (like mean) on the finest grid in terms of the same quantity on the coarse grid and ‘correction terms’. The main methodology is twofold: Firstly, numerous SMC realizations are run on the coarse grid, which is cheap but is not accurate and a fraction is run on the correction terms. Secondly, each term in the correction terms is determined from the difference between the two grids should have the same stochastic behaviour.

## Pavel Markov

### Multistage Upscaling Of Multiphase Flow in Porous Media

**Abstract:**

The presence of various scales in natural reservoirs demands from us using of different approaches and mathematical tools for modelling. In this presentation will be shown three transitions from different scales:

The transition from a continuous space of differential equations to a discrete space of difference models with help of Lie groups theory.

The transition from pore and core scales to the scale of a grid with using of porosimetry data analysis and calculations of relative permeability on the basis of percolation theory.

The transition from a fine grid to a coarse grid with using of relative permeability upscaling which is based on the assignment of regions for flow residuals.

## Gwendolyn E. Barnes

**The structure of quantum space-time**

**Abstract:**

Light chooses to follow the shortest path between two points but it is observed to bend in the vicinity of massive bodies in space in a process called gravitational lensing. This suggests that spacetime itself is curved by massive objects. Does this same effect happen on the quantum level? What is the nature of quantum space-time?

Gravitational lensing: http://www.cfhtlens.org/public/what-gravitational-lensing

## Daniel O. Schulte

**A MATLAB Toolbox for Optimization of Deep Borehole Heat Exchanger Arrays**

**Abstract:**

Due to their slow thermal response arrays of borehole heat exchangers (BHE) represent suitable storage systems for seasonally fluctuating sources like solar energy. Excess heat is fed in during summer and extracted in winter. Certain requirements have to be met by such a system: the stored heat must remain in place and the working fluid must maintain an extraction temperature sufficiently high for the specific heating purpose at all times. Since drilling is the most critical cost factor, the required number of BHEs and their respective distance and length need to be optimized. For this purpose a MATLAB toolbox code was developed. It deploys a tetrahedron mesh-based finite element code, a thermal resistance and capacity model for BHE interaction and mathematical optimization techniques. It can effectively simulate and optimize a BHE heat storage system.

## Kimon Fountoulakis

### Robust Block Coordinate Descent

**Abstract:**

We are concerned with the solution of the following optimization problem

minimize f(x) + Ψ(x),

where x ∈ ℝ^{N}, f(x) is a smooth convex function and Ψ(x) is a (possibly) nonsmooth, block separable,

extended real valued convex function. Problems of this form arise in many important scientific fields,

and applications include support vector machine, regression and signal processing to mention a few.

In this talk, we will present a robust and inexpensive higher-order method that can handle large-

scale ill-conditioned problems. The proposed approach is a novel randomized block coordinate descent

method for the minimization of a convex composite objective function. The method uses (approxi-

mate) partial second-order (curvature) information, so that the algorithm performance is more robust

when applied to highly nonseparable or ill-conditioned problems. We call the method Robust Co-

ordinate Descent (RCD). At each iteration of RCD, a block of coordinates is sampled randomly, a

quadratic model is formed for that block and the model is minimized approximately/inexactly to

determine the search direction. An inexpensive line search is then employed to ensure a monotonic

decrease in the objective function and acceptance of large step sizes.

We prove global convergence of the RCD algorithm, and we also discuss several results on the

local convergence of RCD for strongly convex functions. Finally, we present numerical results on

large-scale problems to demonstrate the practical performance of the method.