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Abstracts and Talk Materials
Mathematical Modeling in Industry XIV - A Workshop for Graduate Students (CIMAT in Guanajuato, Gto., MÃ©xico)
August 2 - 11, 2010

Project Description:

In mathematical finance, financial risk is nearly always a statistic of the underlying assets in question. That is, some salient property of the joint density function of a basket of assets is chosen to proxy risk. One of the first such formulations was the Capital Asset Pricing Model, wherein assets were assumed to be jointly normal, and volatility was therefore assumed as a risk metric. As time has informed us, though, the normal distribution may not be the best choice to describe equity returns on all time scales. Alternate risk measures have been created, all with varying degrees of success and at least a modicum of failure.

One promising risk metric is Conditional Value at Risk, or CVaR. This measure has many appealing properties, one of which seems paramount: it leaves the task of defining the joint density to the practitioner. We will study CVaR as an objective function in a set of optimization problems. Rockafellar and Uryasev (1999) suggested a linear programming formulation to solve such a problem. However, we note a curse of dimensionality issue with their proposal. We will examine alternative methods for solving the basic CVaR optimization problem with linear constraints; viz., we will consider a smooth approximation to the objective as well as a reformulation of the problem. We will also, of necessity, be concerned with modeling a joint density for the assets we will concern ourselves with. In the end, we will test our methodology ex-post on real market data.

References:

R.T. Rockafellar and S. Uryasev, Optimization of Conditional Value-At-Risk. The Journal of Risk, Vol. 2, No. 3, pp. 21-41, 2000.

S. Alexander, T. F. Coleman, and Y. Li, Minimizing VaR and CVaR for a Por tfolio of Derivatives, Journal of Banking and Finance, Vol. 30, no. 2, pp. 583-605, 2006.

G. Iyengar and A.K.C. Ma, Fast gradient descent method for mean-CVaR optimization, 2009, preprint.

Prerequisites:

Familiarity with mean-variance optimization, constrained optimization methods, and some statistics.

Desired: Coursework in mathematical finance, statistics and optimization; Matlab programming.

Project Description:

Our project is a flow modelling project. It would be a project using similar calculations to those used in Computational Fluid Dynamics or Sedimentation Engineering.

We have a solution mining process, that has specific application to potash mining. The process uses interconnected horizontal wells drilled through a high grade potash zone to develop a horizontal development that should eventually dissolve out all the potash in the zone, while leaving the salt in the caverns created.

Potash will be dissolved out of the ore using a brine saturated in sodium chloride, but substantially undersaturated in potassium chloride. Sodium Chloride will collect in the well bore and the intersections similar to sand and rock in a developing meandering stream. In fact the whole process is similar to the mechanics of a meandering stream, except that the flow is bounded on the vertical surface by ore (rather than air).

The calculations involve consideration of the dissolution rate (we can provide dissolution rate curves related to temperature, flow velocity, and brine saturation). Actual flow patterns will need to be modelled based on the tendency of a fluid to flow in a sinusoidal pattern even in a clean pipe, but amplified as the sine flow erodes the cavern wall at the extremities of the sine wave. Sedimentation occurs inside the curve on the point bar.

One well will intersect the next well at angles of 30 to 45 degrees. This will cause a larger case of the meander in the well bore. This will be the major extension of the cavern action. Sedimentation occurs in the vortex created.

While the meander and intersections will add to the horizontal extension, the low flow area over the point bars will continue to add to the cavern height and will eventually be the major production zone. Salt liberated will simply fall to the point bar as the cavern height increases.

Note:

The project can be handled as a simple computational problem, determining the amount of potash dissolved down the length of the horizontal well bore and around the intersection cavern based on the changing cross sectional area over time, the change accompanying change in flow velocity in the meander, the change in saturation of the brine and change in temperature of the brine (due to heat of solution of the potash). I will supply curves and data from literature to provide equations and experimental curves for these factors. The more complicated project would involve predicting the form of the flow from CFD engineering models as well as the calculations described.

References:

National Centre For Computational Hydroscience And Engineering, University of Mississippi http://www.ncche.olemiss.edu/

COMPUTATIONAL METHODS FOR FLUID DYNAMICS Ferziger, Joel H., Peric, Milovan 3rd, rev. ed., 2002, XIV, 423 p. 128 illus., Softcover ISBN: 978-3-540-42074-3

Sedimentation Engineering Theory, Measurements, Modeling, and Practice (ISBN: 0784408238 /0-7844-0823-8) Vanoni Vito A.

Prerequisites:

Required: differential equations, computing skills.

Keywords: Stream meander modelling, sedimentation engineering

Project Description:

The management of water resources is one of the big future challenges of our world. Besides the "big picture" how to provide the humankind with a sufficient amout of high quality potable water, there are a lot of problems to be solved on an operational level. For example, local water suppliers have to make sure that water is available when needed in sufficient quantity and quality. They have to deal both the requirement for the lowest utility prices and the fiscal requirements of their municipalities which might possibly be in a precarious financial situation. Usually, the energy expended for pumps in wells and pumping stations is their major cost factor. The complex rates of electricity providers have high-cost and low-cost periods as well as limitations of total energy consumption; violations result in considerable additional fees.

The goal of the project is to develop a software which is able to find an tradeoff between mimimum energy costs and the reduction of pump switches, where this last aspect minimizes service and maintenance efforts. The degree of freedom for the optimization is essentially an intelligent water storage management.

The project requires basically three steps for the team to be performed:
1. Modelling of water nets as well as the corresponding production and delivery problem, based on resources and demand forecasts
2. A concept how to deal conflicting goals, like cost optimization and reduction of pump switches
3. Definition and implementation of solution algorithms. This includes to decide which libraries of basic optimization algorithms should be used.

Besides mathamatics, it is intended to introduce some basic aspects of project management. For example, there will be assigned "roles" of the team members, like project manager, systems architect, or sales (I am the customer). A project schedule has to be set up in the beginning.

References:

G.L. Nemhauser, L.A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, 1988

R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows - Theory, Algorithms, and Applications, Prentice Hall, 1993

http://zibopt.zib.de/ The optimization suite of the Zuse Zentrum Berlin is one possibility to attack the optimization problems of the project. The advantage is that it is free for academic purposes. However, the team members are free to choose any other tool for linear and integer optimization, provided they will have solved the licensing requirements. (I strongly recommend not to start from scratch, by implementing the Simplex algorithm.)

Prerequisites:

Methods of Linear Optimization as well as programming skills are a must (preferably in C/C++, especially if the team will decide to choose the ZIB Optimization Suite), experience in Discrete Optimization and Graph Theory a real advantage.

The correct and accurate modeling of two phase-flow inside a Proton-Exchange Membrane Fuel Cell (PEMFC) is one of the key elements of a realistic simulation. A schematic drawing of the cross section of a PEMFC can be seen in Figure 1.

Figure 1: Schematic cross-section of a PEMFC.

The reactant gases are supplied to the fuel cell through the gas channels and diffuse into the layered porous medium to the chemically active catalyst layers on the anode and cathode side, where they are consumed. The Proton-Exchange Membrane is (PEM) as gas tight as possible, and supposed to function purely as a proton conductive medium. In the cathode catalyst layer, oxygen reacts with electrons (supplied through the electrically conductive porous medium) and protons (transported through the membrane and the catalyst layer) and water is produced. The water is produced in either liquid or vapor form, depending on the local conditions, and is transported through the porous network structure and removed by the flow in the gas channels.

The optimal performance of a fuel cell depends to a high degree on maintaining optimal saturation (liquid water content) and relative humidity (water vapor content). A certain humidity level is required to guarantee high proton conductivity of the Proton- Exchange Membrane (PEM), but if the humidity is too high, the pores of the network begin to fill with water, and the transport of the gases becomes more and more difficult. One way to mitigate the flooding problem is to assign part of the pore network to the liquid phase and the other part to the gas phase by the introduction of hydrophobic and hydrophilic compounds the the porous structure, and by the introduction of multi-layer diffusion media with varying material properties. Simple model simulation do not agree well with experimental values of the liquid water saturation, see Figure 2.

Figure 2: Experimental vs. theoretical water content (Image taken from Adam Z. Webers conference presentation). It is therefore necessary to describe the porous diffusion media in more detail and with greater understanding of their structural and chemical properties.

The function of the porous structure of a PEMFC can be split in a structural part, i.e. the distribution of the pore sizes, and a chemical part, i.e. the wettability of the material depending on the contact angle. The chemical treatment leads to multiple contact angles, idealized by a prescribed contact angle distribution. A further complication arises from the fact that the contact angle is different for the advancing and the receding edge of an interface, and therefore different whether water is drained or produced. Given a pore size distribution and the contact angle distribution, the saturation profile depending on the capillary pressure can be calculated and compared to experimental values. Due to the complex interaction in this kind of porous network, a hysteresis in the capillary pressure vs. saturation curve can be observed, as shown in Figure 3.

Figure 3: Contact angle hysteresis (Image taken from Adam Z. Webers conference presentation).

Figure 4: Lattice Boltzmann simulation of liquid water flow through a porous medium (Movie obtained from Palabos image galery).

The objective of this project is to study the two-phase flow through a porous network with the previously stated structural and chemical properties, to obtain the characteristic saturation curves and to properly quantify the energy stored in the hysteresis loop.

References:

1. Fuel cell systems explained (2nd edition), Larminie, James; Dicks, Andrew, 2003 John Wiley & Sons
2. Wettability and capillary behavior of fibrous gas diffusion media for polymer electrolyte membrane fuel cells, J.T. Gostick et al., Journal of Power Sources 194/1(2009), dx.doi.org/10.1016/j.jpowsour.2009.04.052
3. Characterization of internal wetting in polymer electrolyte membrane gas diffusion layers, P. Cheung et al., Journal of Power Sources, 187/2, 2009, 487-492, dx.doi.org/10.1016/j.jpowsour.2008.11.036

Prerequisites:

Required: 1 semester of ordinary differential equations, 1 semester of partial differential equations, 1 semester numerical analysis, computing skills (python/scipy/sage or Matlab or C/C++).

Desired: 1 semester of physics or fluid dynamics in general.

Keywords: Fuel cell, two-phase flow, hysteresis, porous media

Keywords: Geodesy, special and general relativity, gravimetry, GPS.

Background:

The exact computation of the gravitational field of the Earth is the point of departure to obtain a complete three-dimensional image of the surface of the Earth, which is used in various applications. The latter include geology, hydrology, civil engineering, global positioning systems, among others. The ideal surface obtained through these techniques is called the geoid, and it provides the best reference model for the above mentioned applications.

Geoid determination depends on accurately measuring the values of gravitational forces at different points on Earth. To understand the precision of existing instruments let it be known that phenomena as the following have to be taken into account:

1. The rotation of the Earth that generates a centrifugal force that reduces the gravity force,
2. The tides which change the distribution of the planet's mass and thus the values of gravitational forces.

Problem: It is a fairly recent technique to perform gravimetric measurements on moving platforms, for example with gravimeters mounted on aircrafts and satellites. Considering the above mentioned required accuracy, one has to consider the following factors:
1. Even the smallest external forces on aircraft mounted gravimeters,
2. The non-inertial nature of the reference frames that correspond to moving platforms which is due to the Earth's curvature.

There already exist some models and formulas that take into account such effects, but we do not know of a complete model that fully considers them. Also, it does not seem to exist a model that has taken care of the relativistic effects of a non-inertial platform. We recall that global positioning systems (GPS) already incorporate relativistic effects to achieve the required accuracy of about 5 meters for non-military users. In contrast, an important goal is to determine the geoid with high precision (centimeters).

Project:

Our main goal is to study the models used for gravimetric measuring and geoid computation on moving and non-inertial platforms. We will first estimate the effective accuracy of the results obtained with the use of classical mechanics. Next, we will consider the models that take into account special and general relativistic effects. Our main tool will be the already known theory and formulas as well as some that we can develop as the project advances. Another relevant component will be the use of software and graphing tools to understand and visualize the phenomena involved. These should suggest corrections and improvements to the current techniques.

Feasibility

As mentioned above, we already have at our disposal theoretical models in the classic mechanics setup. Also, we have obtained some preliminary theoretical computations that evidence the influence of relativistic effects in the gravimetric measurements on moving platforms. Finally, we can compare the modeling results obtained with the measurements available at some government instances.

References:

1. Geodesy 3rd Edition, Torge, Wolfgang, 2001, de Gruyter, ISBN 3110170728
2. Gravitation, Misner, C., Thorne, K., Wheeler, J., 1973, Freeman, ISBN 0716703440
3. Satellite Geodesy 2nd Edition, Seeber, Gunter, 2003, de Gruyter, ISBN 3110175495

Prerequisites: Required: 1 semester course of differential geometry, computing skills (e.g. C/C++).

Desired: (one or more of the following) 1 semester course of introductory physics or classical mechanics, 1 semester course of Riemannian geometry, 1 semester course of special relativity.

Keywords: Image stabilization, Image segmentation, Classification, Tracking, Particle filters.

Project Description:

The project consists in detecting moving objects in suburban and forest firing areas that could indicate people in danger. Automatic monitoring systems can help in this situation, either, making a fully automatic analysis or sending a pre-alarm, to indicate situations that require operator's attention. The data are video sequences acquired, supposedly, with an unmanned aerial vehicle (UAV), or remotely piloted vehicle.

There are several advantages of using UAV in fire monitoring with respect to manned planes (or helicopters); they have a relative the lower cost of operation; they can fly in dangerous situations; they can fly for longer period of time and they can be equipped with video cameras for performing autonomous scene analysis.

Challenges of the project are:
1. We have an unstable source video. The UVS motion introduces shifts and shakes of the scene in the video. Hence, a preprocessing for video stabilization could be useful [2].
2. The fire's smoke is moving and must not be confused with objects' motion. We will require classifying moving objects between smoke and interesting blobs. Color, texture and motion features can support such a classification [4,5].
3. Static objects in the scene, as trees or houses, can occlude the moving objects (MOs). Moreover smoke can partially or fully occlude such MOs. For managing this situation, a motion analysis could alert us for occlusions and allows tracking partially occluded object [3].
4. Noise may produce false positive. Thus, the motion analysis could provide us from the information for classifying among noise, smoke and MOs and then to eliminate false positives [5].

During the development of our algorithms, we need to take into account that the monitoring system is pretended to be implemented in real time and executed autonomously in the UAV, i.e. we will prefer computationally efficient methods.

References:
1. Handbook of Mathematical Models in Computer Vision, N. Paragios, Y Chen, O. Faugeras Eds. Springer NY, 2006
2. R. Szeliski, "Image Alignment and Stitching," Chap 17 in [1], pp 273-292.
3. A. Blake, "Visual tracking: a short research roadmap," Chap 18 in [1], pp 293-307.
4. M. Rivera, O. Ocegueda, J.L.Marroquin, "Entropy controlled quadratic Markov measure fields for efficient image segmentation," EEE Trans. on Image Process., vol 16.(12), 3047-3057, 2007.
5. C.M. Bishop Pattern Recognition and Machine Learning, Springer NY, 2006. In particular chapters: 7,9,12

Prerequisites:

Required: 1 semester of image processing, 1 semester numerical methods, 1 semester numerical optimization, computing skills (Matlab or C/C++).

Desirable: 1 semester pattern recognition.

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