Student thesis topics

Pre-requisite skills:

• C/C++ programming knowledge: All the topics have a significant programming component. Most of my current software is written in C, and you may need to understand them in order to work on topics which continue my previous work. I plan to migrate to C++ for newer software, and so you should be able to do this too.

• Mathematics background: All the topics require at least a moderate amount of Mathematics background. Even in the simplest case, you will need to implement numerical methods, for which familiarity with Mathematics and its terminology is required. I will mention the specific requirements along with each topic.

• A yearning for learning: You should be interested in maximizing your learning, rather than in minimizing your effort! You should also be interested in doing good quality work.

Skills that you will acquire:

• Software development: You will get experience in writing a significant piece of software. Furthermore, some of these software will probably have users outside FSU, and so you will get exposure to user support, creating a user-friendly interface, etc.

• Mathematics: All the projects have some mathematical component to them, and so you will learn something new.

• Research skills: There is a creative aspect to research that is hard to teach in a classroom. Working on these research topics will expose you to real research. You will also read journal articles, interact with other researchers, etc.

• Algorithm analysis: Most of the topics will require some non-trivial analysis of new algorithms, and so you will acquire this skill.

• Written communication: I will ask you to submit short reports of your current activities regularly, in order to give you practice in professional writing. You will also probably need to write an article for a conference or a journal publication, apart from your thesis or project report.

• Oral presentations: I will ask you to give frequent oral presentations, and perhaps occasionally have others too attend and give suggestions, so that you develop confidence in your oral presentation skills.

Miscellaneous matter

• Time frame: I expect that a project or a thesis will take around a year (two regular semesters and a summer). However, you should aim to finish your thesis about half a semester before you graduate, in order to give enough time for formalities. Of course, this is just an estimate, and assumes that you put in sufficient effort of sufficient quality.

• Why do a thesis?: All the topics I mention are really research topics suited to a thesis. A project will just emphasize the software development component a little more. So the effort required will be around the same. You might as well have a thesis on your resume. Furthermore, if you do good work, then I may send you to a conference, to give a presentation perhaps. This will be a professionally enriching experience for you. A thesis may also help you in this tight job market.

• Funding: You can discuss this with me, in case you plan to do a thesis. However, I have limited funds and want to be careful about spending it, and so will want to observe your work for a sufficient period of time before giving any commitment.

Topics

1. Parallelization of Monte Carlo linear algebra algorithms

   • Level: Master's thesis, PhD

   • Description: Monte Carlo is particularly suited to parallel and distributed computing, since it is "embarrassingly parallel". However, when the state space is very large, one may need to resort to domain decomposition techniques that destroy this embarrassingly parallel nature. We need to develop parallel algorithms that are still latency tolerant, and so will be suitable for a distributed heterogeneous cluster. This will be an inter-disciplinary collaboration, involving the development of better Monte Carlo techniques, combining Monte Carlo with conventional techniques, and parallelization. We will concentrate on parallelization and software development, while the others concentrate on the other aspects.

   • Mathematics pre-requisites: Familiarity with linear algebra: Matrices, matrix-vector multiplication, solving a linear system, and, ideally, eigenvalues too. Some knowledge of probability too will be required.

   • References:http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0T-3VFBH8Y-24&_user=130899&_coverDate=08%2F01%2F1998&_rdoc=4&_fmt=full&_orig=browse&_srch=%23toc%235655%231998%23999529997%2344885!&_cdi=5655&_sort=d&_acct=C000010879&_version=1&_urlVersion=0&_userid=130899&md5=23637667713a15cdb56c22e2c9ccfac2. We will not use this technique, but it will give you an idea of the field. Read the parallelization part in particular.

2. Problem solving environment of quasi-Monte Carlo integration

   • Level: Master's thesis/project, PhD

   • Description: At the Master's level, you will implement existing and new parallelization techniques for low discrepancy (quasi-random) sequences, and make the current software I have more user friendly and portable. You will also need to read the literature, and for a thesis, suggest improvements to existing techniques. At a PhD level, you will extend this to a problem solving environment for Monte Carlo integration by including automatic variance reduction techniques, error estimation, and GUI. You may also specialize it to specific applications, such as derivative pricing in finance. This work will involve interaction with other researchers in QRN generation.

   • Mathematics pre-requisites: Knowledge of integration, some probability.

   • References: http://www.cs.fsu.edu/~asriniva/papers/mcqmc.ps

3. Monte Carlo techniques for graph partitioning

   • Level: Master's thesis/project

   • Description: Graph partitioning has applications in parallel computing to minimize communication cost, in VLSI design, etc. The spectral method for graph partitioning uses an eigenvector to partition the graph. It lost favor over the past several years, because it is very slow, though the quality of the partitioning is good. However, it is still be used as a part of multi-level methods. We want to make the spectral method faster using Monte Carlo techniques. We will also test a different "normalized Laplacian", instead of the conventional one. We will compare our schemes against two popular graph partitioning packages.

   • Mathematics pre-requisites: Familiarity with linear algebra: Matrices, matrix-vector multiplication, solving a linear system, and eigenvalues. Some elementary knowledge of probability too may required.

4. Experimental study of selection algorithms

   • Level: Master's thesis/project

   • Description: Randomized selection algorithms generally outperform deterministic schemes. In a certain financial risk application, we need to select the n th largest element of a large set of numbers quickly, where n is often the highest 1% or 5%. I have suggested an algorithm that is very effective for such selection from the tails. However, we need to analyze it better and determine suitable parameters to optimize its performance, implement several algorithms, and compare them exhaustively. We may also study parallel selection.

   • Mathematics pre-requisites: Algorithm analysis, and ideally some elementary probability.

5. Determining large components of matrix vector products

   • Level: Master's thesis/project

   • Description: Large components of matrix-vector products have important applications in certain financial risk problems, and in information retrieval. I derived a linear algebra bound that proved very useful in the finance application, and improved it using computational geometry techniques. I would like you to implement other ideas that I have, and come up with your own too. I would also like to test the effectiveness of these techniques in information retrieval, and perhaps pattern recognition. (I have not tried them in these two applications, and so you would need to do a literature search to find out the current state of the art.) Applications of some Monte Carlo techniques too would be needed.

   • Mathematics pre-requisites: Familiarity with linear algebra: Matrices, and matrix-vector multiplication. Some knowledge of probability and algorithm analysis too will be required.

6. Parallel Monte Carlo techniques in finance

   • Level: Master's project

   • Description: We will apply parallel pseudo and quasi-random number generation techniques, and different automatic and application specific variance reduction techniques to finance problems, such as option pricing. Your work will be primarily implementation and experimentation, and so I suggest it only as a Master's project, and not as a thesis. But some publication (either conference or journal) should still come out of it.

   • Mathematics pre-requisites: Some probability and differential equations would be ideal, though not required.