Modelling Week Projects

8) Where is the boat?

posted Jul 4, 2011, 9:23 AM by Ecmi Milano   [ updated Jul 7, 2011, 11:51 PM ]

Instructor: Ana Leonor Silvestre

Suppose that, during a trip, a small boat has an engine failure and stops at sea, with no means to communicate and ask for help. After some time, the disappearance of the boat is reported to the authorities, who decide to start the search and rescue activities using only maritime resources. The sea is calm and it is possible to predict that the boat must have to be adrift in a certain region, which, however, can be very wide. The aim of this project is to show that it is possible to have a good prevision of the location of the boat from appropriate informations on the state of the water in the surface, far way from the boat. More precisely, using a set of measurements of velocity and pressure of the water and an algorithm described in the course, the objective is to find the location of the lost boat.

Preferred mathematical background: Partial Differential Equations and Numerical Analysis

13) Another slice of the whole

posted Jul 1, 2011, 3:29 AM by Ecmi Milano   [ updated Jul 7, 2011, 11:52 PM ]

Instructor: Giovanni Naldi

Project description

Computed tomography images are becoming an invaluable mean for abdominal organ investigation. In the field of medical image processing, some of the current interests are the automatic diagnosis of pathologies, and the 3D volume rendering of these abdominal organs. Their automatic segmentation is the first and fundamental step in all these studies, but it is still an open problem. Also in the Radiotherapy planning for the treatment of cancer the geometrical information about the disposition of the structures of interest in a patient could increase the efficacy and risk assessment of the process.

The Problem

We would like to reconstruct 3D organ from CT slices. As a basic step we want to develop an automatic method in order to compute the 2D contour of the organ in each plane. As a working data we consider CT data of bladder adapted from Medical Physics Unit, Fondazione IRCCS Istituto Nazionale Tumori (Milano, Italy). The stack of all contours is the boundary of the 3D volume. The problem involves the following steps: reduce the original image, denoising of the image in order to have a first almost piecewise image, detecting the contour of the organ in the slice, collecting all the contours in order to reconstruct the organ volume. For this project is expected to know basic facts about image processing, MATLAB programming, basic numerical algorithms for partial differential equations.

Preferred mathematical background: MATLAB programming, basic numerical algorithms for partial differential equations.

12) Computational homogenization in heat conduction problems with application to heterogeneous materials

posted Jul 1, 2011, 3:28 AM by Ecmi Milano   [ updated Jul 7, 2011, 11:54 PM ]

Instructor: Paola Causin

Several material commonly adopted in engineering applications – for example, refractories used in thermal coatings or microelectronics - are strongly heterogeneous and display a multi-phase porous microstructure. An accurate prediction of the behavior of such material under different conditions requires a comprehensive understanding of the distribution of the relevant physical fields –for example the temperature when considering heat conduction problems-both at the macroscopic and microscopic level. A possible approach for such an analysis is based on the principle of scale separation, according to which one can take into account the local microstructural features and transfer this information to the macroscopic level in a physically consistent manner. In this project, we aim at investigating a computational methodology of homogenization as a tool for the transfer of the information between the micro and macro scale. The application will be the simulation of evolving thermal fields within materials of complex microstructure, with properties possibly depending on the temperature itself.

Preferred mathematical background: one basic course on numerical analysis and physics. Knowledge of the finite element method (at least from a theoretical point of view) is preferable. Matlab programming.

11) Modelling of local mobility fluctuations in nanoscale electronic devices

posted Jul 1, 2011, 3:27 AM by Ecmi Milano   [ updated Jul 7, 2011, 11:54 PM ]

Instructor: Alessandra Micheletti

Elementary electronic devices (such as MOS transistors) have been continuously shrinked in size since many years. State-of-the-art transistors have typical dimensions of a few tens of nm, therefore only a few dopant atoms are present in the active region. Since the dopants number and position are random variables, the electrical characteristics differ stochastically from one device instance to another.
In particular electron transport depends on the coulombian interaction with such dopant atoms. Since this interaction decrease strongly with the electron-dopant distance, the resulting local mobility (ratio between velocity and electric field) varies across the device depending on the local dopant arrangement.
The problem we want to solve is to find a suitable analytical model (to be used in a Drift-Diffusion framework) describing the local mobility as a function of the discrete dopants number and position.
Given a statistical set of doping arrangements and of the corresponding local mobility data (provided by Montecarlo simulations, assumed as a reference) first of all the most appropriate functional form has to be found, then its parameters have to be optimized to provide the best overall agreement with the mobility data.

Preferred mathematical background: Multivariate statistics, mathematical physics for semiconductors, Montecarlo simulation

10) The Homogeneous Areas Problem (HAP)

posted Jul 1, 2011, 3:25 AM by Ecmi Milano   [ updated Jul 8, 2011, 1:32 AM ]

Instructor: Marco Trubian

In Italy, provinces play a coordination role: they support all local activities involving more than one town. Examples of such common activities are the management of some parks, the maintenance of some roads, the management of incentives to locally relevant economic sectors, the organization of cultural happenings, and so on. Experts on the technical and legislative issues related to these specific fields work for the different departments of the provincial administrations. For each given activity, the employees of town administrations must refer to these experts and interact with the employees of the other towns involved.
To provide a simple and effective structure from the point of view of the towns, the province administration offers a sort of "help desk", or friendly interface, under the form of a team of employees. This team supports the town employees on any matter of interest, addresses them to the experts in the right departments, keeps track of their needs, organizes meetings, and so on. This is also a way to improve cooperation by means of personal relationships between the employees of the towns and those of the province. In small provinces, a single team is sufficient. When the number of towns and activities grows larger, however, it is no longer possible to guarantee that all employees of the team have the necessary expertise on all the activities involving the province. Larger provinces, therefore, are partitioned into areas so that the team supporting each area can specialize on the corresponding activities. A desirable partitioning should group into the same area towns involved in the same activities and should disperse in different areas those involved in different activities. In short, it should yield "homogeneous" areas. Moreover, each area should correspond to a connected subset of towns, both because this makes it simpler to organize meetings for the town employees of the area and because disconnected areas are considered historically and politically "unacceptable".
The size of the teams cannot be too large, because the members of each team should be interchangeable, and it cannot be too small, to avoid that vacations and illnesses might deprive a whole area of all support. Two to four officers are considered a reasonable size. Since the total workload cannot exceed the available number of work hours, the size of the teams and the total workload imply a small range of reasonable values for the number of teams. Moreover, for equity reasons, the workload should also be distributed fairly among the teams.
The workload for each activity derives from two groups of elementary tasks: the former is devoted to the activity itself (keeping up-to-date with the legislation, organizing meetings, sending standard letters, etc\ldots), the latter is repeated for each town singularly (paying visits, answering questions, etc\ldots). An estimate of the workload can be made taking into account both groups of tasks.
Due to the connection and capacity constraints, as well as the heterogeneous nature of the activities, it might be impossible to keep all towns involved in the same activity in a single area, so that two or more teams might be forced to replicate the first group of elementary tasks. This is a source of redundancy and inefficiency, which should be minimized.
Therefore, the optimization problems is the following: we are looking for a partition of the towns into connected areas with a balanced workload, such that the amount of replicated tasks is as little as possible.
The problem can be fromulated as an integer linear programming (ILP) optimization problem. The spreadsheets with the data of two provinces, of about 50 and 130 towns, are available.

Preferred mathematical background: optimization, integer linear programming.

9) Spider web vibrations: finding dinner

posted Jul 1, 2011, 3:23 AM by Ecmi Milano   [ updated Jul 7, 2011, 11:56 PM ]

Instructor: Derek Moulton

 Many spiders use webs to capture their prey. These spiders typically have very poor vision, and rely almost entirely on their sensation of vibrations in the web to detect stimuli. When a spider feels vibrations from something in its web, it has two main problems to solve: (1) what is causing the vibration? and (2) where on the web is the source of the vibration? If the spider detects prey, it is important for the spider to be able to locate the prey quickly and efficiently. How the spider accomplishes prey localization depends on the species of spider and the type of web they weave. In this project we will consider sheet web spiders, whose webs have strands of silk so dense they resemble a continuous sheet. The aim of the study is to develop a mathematical model of spider web vibrations and to try to understand how spiders are able to achieve prey localization.

Preferred mathematical background: ODE's, PDE's, basic numerical analysis.

7) Stochastic Algorithms for Capacitance Analysis in Integrated Circuits

posted Jul 1, 2011, 3:21 AM by Ecmi Milano   [ updated Jul 7, 2011, 11:57 PM ]

Instructor: Francisco Bernal

An integrated circuit can be modelled as a two-dimensional domain where the voltages are piecewise constant. On an insulating platform (the waffer) several electronic, conducting components are placed which stand at various voltages. Estimation of the electric fields in the integrated circuit require that a Laplace equation be solved in the whole domain. It is critical, from the point of view of design, that the electric field does not 'spike' anywhere, since this would give rise to spurious electric currents which would yield the integrated circuit useless. The industry is therefore interested in cheap estimates of the electric field at specific locations in the integrated circuit -in the design phase, when many different possible layouts are considered prior to manufacture. Since the number of components is in the order of tenths of millions (or more), standard tools such as finite elements are ruled out for this task. This kind of problem, however, is ideally suited for stochastic solution methods of PDEs, because: a) the solution is only required at specific points, and b) a modest accuracy suffices. Moreover, the typically rectangular shape of electronic components allow for improvements to the standard stochastic method, such as the so-called 'walk on cubes' algorithm.
COMMENTS: This problem is not only of industrial relevance but very topical (Intel announced recently the first 3D chip ever).

Preferred mathematical background:
  1. a basic background in electrostatics (basically the connection between the electric field, the electric potential, and Laplace equation)
  2. basic notions of probability and statistics
  3. basic notions of programming and basic numerical schemes such as Euler's method for IVPs.
  4. there is plenty of room for modelling and imagination here, because since just a rough solution is required, it has to be decided which features are the most relevant (and which may be left out) in order to speed up the algorithm.

6) Modelling of glaciers

posted Jul 1, 2011, 3:20 AM by Ecmi Milano   [ updated Jul 7, 2011, 11:58 PM ]

Instructor: Siri-Malén Høynes

Glaciers all over Europe are retreating and the glaciers in Norway are not exceptions. Many people think that if the temperatures keep going up, some of the Norwegian glaciers will melt away. One glacier in Norway, the Briksdal glacier, has experience both retreat and growth in the last 10-15 years. This glacier is close to where I grew up, so I have seen the rapid changes this glacier has been through in the last years. Since it is one of the more interesting Norwegian glaciers, I want to focus on this glacier in this modelling problem.
How glaciers retreat and grow does not only depend on temperature, more important is the amount of precipitation, and also the size and slope of the glacier. The simplest models of glaciers lead to solving a first-order hyperbolic PDE, which in most cases will be non-linear. We will look at the basics behind the simplest glacier model and see how far we can get in making a good model for the melting of the Briksdal glacier. This will require knowledge in solving PDE's numerically (using MATLAB or octave).
In the end, if there is time, it would be nice to test some future weather scenarios and see if we can expect the glacier to melt away or grow back to it's former size.

Preferred mathematical background: Hyperbolic differential equations, solving differential equations numerically

5) How mathematicians predict the future?

posted Jul 1, 2011, 3:19 AM by Ecmi Milano   [ updated Jul 7, 2011, 11:59 PM ]

Instructor: Agnieszka Wylomanska

This project aims at developing and validating suitable mathematical model for the interpretation and description of financial data, like interest rates. It is very important to find the proper system that describes real data sets and it is well known that it can be useful especially in the prediction problem. Because data from financial markets indicate very characteristic behavior therefore it is not easy to find the appropriate model that takes into consideration such many aspects as: seasonality, deterministic trend and relationship observed in the data.The work will be divided into four steps:1.Analysis of three data sets from financial markets - seasonality and deterministic trend detection.2. The common analysis of the mentioned time series – relationship between real financial data.3. Modeling the data sets on the basis of points 1 and 2.4. Validating the obtained model and prediction.In our analysis we will use the techniques based on the least squares method that provide to detection and removing the seasonality and deterministic trend from the real data sets. Moreover to find the relationship between two or more time series we will take into consideration the multiple regression method that assumes the linear relation between observed data sets and is also based on the least squares technique. On the basis of the obtained results we will validate the model and predict the real financial data.

Preferred mathematical background:
fundamentals of statistics, fundamentals of probability, differential equations.

4) Fair voting in parliamentary elections

posted Jul 1, 2011, 3:18 AM by Ecmi Milano   [ updated Jul 8, 2011, 12:00 AM ]

Instructor: Phil Knight 

Most countries in Europe use some form of proportional representation (PR) as a means of allocating members of parliament at both local and national level. The UK has joined in recently and a number of elections (e.g., European, Scottish) use forms of PR; although there is strong resistance to bringing it into the election for members of parliament (MPs) to Westminster, as the recent referendum on the Alternate Vote system indicates. Currently, UK elections are fought on the first past the post (FPTP) system. In a multi-party election, this can skew results significantly. For example, in the 2001 general election, the Labour party won 65% of the seats but only 40% of the votes. Conversely, in 2010 the Liberal Democrats received 23% of the votes but only 9% of the seats.FPTP is also alleged to be responsible for effectively disenfranchising many voters, as the demographics of some constituencies means that they almost never change hands. For example, Gower, Normanton and Makerfield have elected Labour MPs without exception since 1906. Voters in these areas know that there is almost no chance of their vote making a difference, turnout can be very low. A similar problem in Switzerland led to a successful legal case by a disgruntled voter, and a canny lawyer may be able to use Human Rights legislation to alter the voting system elsewhere.One of the main objections to PR for UK parliamentary elections is that it breaks the link of MPs with individual constituencies: as well as being members of a party, MPs have traditionally represented the interests of individual voters in the towns or districts they have been elected to.The task here is to devise a model of PR to solve the problems of a FPTP system while retaining a constiuency link. Ideally, the system should be straightforward to implement and not subject to anomalous allocation of representatives. In particular, it should be able to accommodate the nationalist parties in the UK which have strong regional representation. The applicability of any proposed system outside of the UK should be investigated.

Preferred mathematical background: Some exposure to linear algebra. Preferrably some numerical analysis: numerical linear algebra and optimisation would be helpful but not essential. Some discrete mathematics would be helpful, too.

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