## Overview

See the relevant section of the graduate handbook for details of the timing and procedures of the comprehensive exam.

Letters to the candidate specifying the topic for part I of the comprehensive exam should also communicate the date when the candidate received the question; the date it must be sent to the committee (four weeks later); and the date and time of the defense (1-2 weeks after that, i.e. 5-6 weeks after the initial assignment).

## Examples

### Port-Hamiltonian Modeling

In recent years, the port-Hamiltonian approach has become popular for modeling and simulation of multi-physics systems as subsystems of electrical, mechanical, etc. components. A system is modeled by its Hamiltonian function, the system’s total energy, and is viewed as the interconnection of three types of ideal components: energy-storing elements, energy-dissipating elements, and energy-routing elements. The subsystems are linked by ports through which energy flows. This approach is compositional: power-conserving connection of several pH systems via ports results in a new pH system.

Write a report that overviews the fundamentals of port-Hamiltonian systems and outlines recent developments in this area. Based on this review, identify between 1 and 3 interesting open problems. A problem should be concrete, in the sense that it could become the topic of a future journal article. You should motivate why a problem is worthy of study, and why it is interesting, non-trivial, and not hopelessly difficult. You should briefly outline approaches that could be used to solve the proposed open problem(s), highlighting their computational aspects; however, you do not need to solve it completely.

The entire document should not be longer than 20 pages.

While we expect you to find the relevant references yourself, we propose the following sources as a point of departure for your investigations:

- A. van der Schaft, Port Hamiltonian Modeling and Control
- A. van der Schaft, Port-Hamiltonian Systems Theory: An Introductory Overview

### Event Location In Numerical Methods For ODEs/DAEs

Locating events when integrating systems of ordinary differential equations (ODEs) or differential-algebraic equations (DAEs) is an important problem in scientific computing, with many applications, and in particular, in simulating hybrid dynamical systems. These are systems that combine discrete and continuous behavior. The continuous behavior is typically given by systems of ODEs or DAEs. Although there is a substantial amount of work in this area, reliable event location is still a challenging problem.

You should produce an overview of the state-of-the-art methods for event location for ODEs and DAEs. You are expected to

- identify the difficulties that arise in event location
- identify the most reliable and efficient methods and how they deal with such difficulties
- propose three research topics, such that progress on them would lead to important contributions

Regarding task (c), these research topics should be concrete, in the sense that they could be the subject of a future journal article. You should motivate why these problem(s) are worth of studying, and why they are interesting, non-trivial, and not hopelessly difficult. You should also briefly outline approach(es) that could be pursued to solve the proposed open problems highlighting their computational aspects; however, you do not need to solve these problems.

While we expect you to find the relevant references yourself, we propose the following sources as a point of departure for your investigations:

Taeshin Park and Paul I. Barton. 1996. State event location in differential-algebraic models. ACM Trans. Model. Comput. Simul. 6, 2 (April 1996), 137-165. DOI=10.1145/232807.232809 http://doi.acm.org/10.1145/232807.232809

L. F. Shampine, I. Gladwell, and R. W. Brankin. 1991. Reliable solution of special event location problems for ODEs. ACM Trans. Math. Softw. 17, 1 (March 1991), 11-25. DOI=10.1145/103147.103149

### Computationally Efficient Track-Before-Detect Algorithms

Track-Before-Detect (TBD) algorithms are commonly used in target tracking problems under low signal-to-noise ratio conditions. The idea is to use raw (or unthresholded) data (i.e., without declaring detections) to gain more information using one or more frames of sensor data. A number of algorithms using dynamic programming, nonlinear optimization (e.g., Maximum-Likelihood Probabilistic Data Association (ML-PDA) algorithm), Hough transform, particle filters and Probability Hypothesis Density (PHD) filters have been proposed in the literature.

- Present the results of your extensive literature search on track-before-detect using the aforementioned algorithms.
- Formulate two research problems related this topic. Two possible problems are a) track-before-detect with correlated clutter; b) coherent vs. non-coherent techniques for TBD in correlated non-Gaussian clutter. However, you may suggest alternative problems.

These projects should be concrete, in the sense that they could be the topic of a future journal article. You should motivate why these problems are worthy of study, and why they are interesting, non-trivial, and not hopelessly difficult. You should briefly outline approaches that could be used to solve the proposed open problems highlighting their computational aspects; however, you do not need to solve these problems.

The entire document should not be longer than 20 pages.

### Generalized Additive Models

Your research proposal topic involves the review of the statistical and computational methods of Generalized Additive Models. You need to review key approaches that have been proposed in the literature, and describe some of the important application areas. Your discussion should include

- Formulation of the key statistical ideas on which the models are based, and a detailed discussion of the algorithms for fitting them
- A detailed discussion of smoothers including splines and LOESS, how to compare them, and how to choose degrees of freedom and smoothing parameters.
- The implementation of these models in various statistical software packages
- A literature review which includes the state-of-the-art in this field

Based on your review, you are to propose one or two concrete research questions relevant to computing, which are suitable for further research. These questions should be concrete, in the sense that they could be the topic of a future journal article. You should motivate why these problem(s) are worthy of study, and why they are interesting, non-trivial, and not hopelessly difficult. You should also briefly outline approach(es) that could be used to solve the proposed open problems highlighting their computational aspects; however, you do not need to solve these problems.

While we expect you to find the relevant references yourself, we propose the following sources as a point of departure for your investigations:

Buja, A., Hastie, T., and Tibshirani, R (1989) Linear Smoothers and Additive Models, The Annals of Statistics 17, 453-510.

Hastie, T. and Tibshirani R. (1987). Generalized additive models: some applications, Journal of the American Statistical Association 82, 371-386.

Hastie, T. and Tibshirani R. (1986). Generalized additive models, Statistical Science 1, 297-310.

### Stochastic Partial Differential Equations

Your research proposal topic involves the review of the computational methods for the solution of stochastic partial differential equations (SPDEs). You need to review key approaches that have been proposed in the literature, and describe some of the important application areas. Your discussion should include

- precise statement of the problem together with definitions of the different stochastic and deterministic quantities typically associated with solutions of SPDEs,
- discussion of Monte Carlo approaches,
- discussion of techniques based on polynomial chaos expansions, and
- a literature review which includes the state-of-the-art in this field.

Based on your review, you are to propose one or two concrete research questions relevant to computing, which are suitable for further research. These questions should be concrete, in the sense that they could be the topic of a future journal article. You should motivate why these problem(s) are worthy of study, and why they are interesting, non-trivial, and not hopelessly difficult. You should also briefly outline approach(es) that could be used to solve the proposed open problems highlighting their computational aspects; however, you do not need to solve these problems.

While we expect you to find the relevant references yourself, we propose the following source as a point of departure for your investigations:

G. J. Lord, C. E. Powell and T. Shardlow, “An Introduction to Computational Stochastic PDEs”, Cambridge University Press, 2014.

### Parameter Estimation For Nonlinear Stochastic Dynamical Systems

Nonlinear stochastic dynamical systems are used to address many scientific research questions. A key question is how to use observations to estimate parameters for such systems. You should produce an overview of state-of-the-art methods for fitting nonlinear stochastic dynamics systems to observational data, focusing on methods that account for observation error as well as underlying dynamical stochasticity (process error). You should focus on:

- Particle filtering methods, including modern versions (e.g., the IF2 algorithm)
- Markov Chain Monte Carlo methods (MCMC), including modern versions (e.g., Hamiltonian MCMC, and no-U-turn samplers (NUTS))

Other approaches (approximate Bayesian computation, optimization-based methods) may be discussed briefly, but you should not focus on these. You should also propose two concrete areas of possible research in the applicability of these methods. Each proposal should involve:

- a scientific research question that can be studied with nonlinear stochastic dynamical systems
- discussions of obstacles or uncertainties in how parameters can efficiently be estimated for this system, and
- suggestions for how existing methods could be improved or compared for the purposes of application to the question

The proposed projects should be practical, but you do not need to solve them. You are responsible for reviewing the relevant literature. The following links are provided only as possible starting points.

- Inference for dynamic and latent variable models via iterated, perturbed Bayes maps
- The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo

The work should be done entirely by you, and the submitted document should not exceed 20 pages, in 12-point type with reasonable margins.