Objective acceleration for unconstrained optimization

Acceleration schemes can dramatically improve existing optimization procedures. In most of the work on these schemes, such as nonlinear Generalized Minimal Residual (N-GMRES), acceleration is based on minimizing the $\ell_2$ norm of some target on subspaces of $\mathbb{R}^n$. There are many numerical examples that show how accelerating general purpose and domain-specific optimizers with N-GMRES results in large improvements. We propose a natural modification to N-GMRES, which significantly improves the performance in a testing environment originally used to advocate N-GMRES. Our proposed approach, which we refer to as O-ACCEL (Objective Acceleration), is novel in that it minimizes an approximation to the \emph{objective function} on subspaces of $\mathbb{R}^n$. We prove that O-ACCEL reduces to the Full Orthogonalization Method for linear systems when the objective is quadratic, which differentiates our proposed approach from existing acceleration methods. Comparisons with L-BFGS and N-CG indicate the competitiveness of O-ACCEL. As it can be combined with domain-specific optimizers, it may also be beneficial in areas where L-BFGS or N-CG are not suitable.Comment: 18 pages, 6 figures, 5 table

Increasing the Creativity of ENGINO Toy Sets and Generating Automatic Building Instructions

During the First Study Group with Industry which was held in Limassol, Cyprus, the ENGINO⃝R TOY SYSTEM introduced two challenging problems. The first is to get bounds on the number of possible models/toys which can be constructed using a given package of building blocks. And the second is to generate automatically the assembly instructions for a given toy. In this report we summarize our insights and provide preliminary results for the two challenges

Segmentation and Scene Content in Moving Images

The problem of scene content in moving images was brought by Aralia. The goal in this study group was to consider two problems. The first was image segmentation and the second is the context of the scene. These problems were explored in different areas, namely the Bayesian approach to image segmentation, shadow detection, shape recognition and background separation

Mathematical modelling of the Germasogeia aquifer

Two challenges related to improving the management of the Germasogeia aquifer were presented to the Study Group by the Cyprus Water Development Department (WDD), the public organisation responsible for managing the wa- ter resources in Cyprus. The rst challenge was how to optimally recharge the aquifer in order to compensate for the extraction of drinking and irrigation water whilst preventing sea water intrusion. In order to address this challenge we developed model for the water in the aquifer. Note that by exploiting the long, thin nature of the aquifer we only develop two-dimensional models in this work. We rst develop a simple model based on Darcy ows for porous media which gives the water table height for given dam seepage rate, recharge and extraction rates; we neglect seawater intrusion. We then use the steady version of this model to develop an optimized recharge strategy with which we can identify minimal recharge required for a desired extracted water volume such that the minimum prescribed water table is respected. We explore 4 di erent scenarios and we nd that in certain cases there can be a considerable reduction in the amount of recharged water compared to the current empirical strategy the Water Development Department is employing, where water is recharged and extracted in equal proportions. To incorporate the e ects of seawater intrusion, which can be very damaging to the water quality, we next develop transient two- dimensional models of saturated-unsaturated groundwater ow and solve them numerically using the open source software SUTRASuite and the commercial package ANSYS FLUENT; the position of the water table and the seawater- freshwater interface are determined for various extraction/recharge strategies. Data from the WDD are used in some of the simulations. The second important challenge we were asked to look at was to predict the transport of pollutants in the aquifer in the case of an accidental leakage. An advection-difusion equation for the contaminant concentration is introduced and simulations are under- taken using the commercial package COMSOL. The concentration pro les of the contaminant are studied and we nd that the e ect of contamination varies depending on where the contamination site is; the closer the contamination site is to the dam, the larger the extent of contamination will be

Algorithms for decision making

We investigate algorithms for different steps in the decision making process, focusing on systems where we are uncertain about the outcomes but can quantify how probable they are using random variables. Any decision one makes in such a situation leads to a distribution of outcomes and requires a way to evaluate a decision. The standard approach is to marginalise the distribution of outcomes into a single number that tries in some way to summarise the value of each decision. After selecting a marginalisation approach, mathematicians and decision makers focus their analysis on the marginalised value but ignore the distribution. We argue that we should also be investigating the implications of the chosen mathematical approach for the whole distribution of outcomes. We illustrate the effect different mathematical formulations have on the distribution with one-stage and sequential decision problems. We show that different ways to marginalise the distributions can result in very similar decisions but each way has a different complexity and computational cost. It is often computationally intractable to approximate optimal decisions to high precision and much research goes into developing algorithms that are suboptimal in the marginalised sense, but work within the computational budget available. If the performance of these algorithms is evaluated they are mainly judged based on the marginalised values, however, comparing the performance using the full distribution provides interesting information: We provide numerical examples from dynamic pricing applications where the suboptimal algorithm results in higher profit than the optimal algorithm in more than half of the realisations, which is paid for with a more significant underperformance in the remaining realisations. All the problems discussed in this thesis lead to continuous optimisation problems. We develop a new algorithm that can be used on top of existing optimisation algorithms to reduce the cost of approximating solutions. The algorithm is tested on a range of optimisation problems and is shown to be competitive with existing methods.</p

Model Predictive Control for Scheduling and Routing in a Solid Waste Management System

Solid waste collection and hauling account for the greater part of the total cost in modern solid waste management systems. In a recent initiative, 3,300 Swedish recycling containers have been fitted with level sensors and wireless communication equipment thereby giving waste collection operators access to real-time information on the status of each container. In a previous study (Johansson, 2006), analytical modeling and discrete-event simulation have been used to evaluate different scheduling and routing policies utilizing the real-time data, and it has been shown that dynamic scheduling and routing policies exist that have lower operating costs, shorter collection and hauling distances, and reduced labor hours compared to the static policy with fixed routes and pre-determined pick-up frequencies employed by many waste collection operators today. This study aims at further refining the scheduling and routing policies by employing a model predictive control (MPC) framework on the system. In brief, the MPC controller should minimize an objective cost function consisting of fixed and variable collection and hauling costs for a fixed future horizon by calculating a sequence of tactical scheduling and routing decisions that satisfies system constraints using a receding horizon strategy

Firedrake: An Automated Finite Element System

Version of Firedrake in the Firedrake paper. This release is specifically created to document the version of Firedrake used in a particular set of experiments. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to http://www.firedrakeproject.org/publications.html</span

firedrake: an automated finite element system

Version of Firedrake in the Firedrake paper. This release is specifically created to document the version of Firedrake used in a particular set of experiments. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to http://www.firedrakeproject.org/publications.htm