Applied Math Colloquium

The two most commonly used methods for sparse LU factorization are the left-looking method and the right-looking method. Both are used internally in MATLAB. In this talk, specific examples of these two methods will be discussed and compared. In the left-looking method, the kth step of factorization computes the kth column of L and U from the kth column of A and columns 1 to k-1 of L and U. In the sparse case, it has a particularly elegant implementation (due to Gilbert and Peierls), and is the only known algorithm whose run time is O(flop count). The right-looking method is based on outer products. In the multifrontal method, these outer products are held in small square or rectangular dense submatrices. The method is much more complex than the left-looking method, and it is not guaranteed to run in time O(flop count). However, it can exploit dense matrix operations (the BLAS) better than the left-looking method. For each method, specific classes of problems generate matrices that are best factorized with that method.

Decompositions of random matrices arise continuously in the study of the performance of information systems, communication technology and statistical signal processing methods. The scientific communities studying these areas have slowly but steadily developed a great interest in the study of random matrices, but the prevalent approach is to use known results, mostly borrowed from the Physics' literature. The research on array processing and the study of multi-user communications and space-time coding would enormously benefit from a deeper understanding of how one can derive the statistics of random matrices' decompositions in general.

In this talk, as specific examples, we will show how the application of Random Matrix Theory allows to analyze the performance of random multi-input multi-output (MIMO) fading channels, modeling a wireless system with transmit and receive diversity, and of direct sequence code division multiple access channels (DS-CDMA) with “random” codes.

The talk will cover methodological aspects as well as applications with the intent of highlighting a field of engineering pullulating of good random matrix problems to attack.

In the last few years, a large fraction of new results on the asymptotic convergence of the eigenvalues of random matrices has been obtained using the tools of free probability. The power of free probability does not only rely on its new tools (like the R-transform or S-transform), but also on its fresh view on established results.

In this talk, I will give an idea what free probability is about and why and how it is connected to random matrices. In particular, I will also talk about recent joint work with J. Mingo and P. Sniady, where we provide a fresh new look on fluctuations of random matrices from the "free" perspective.

We study an optimal investment problem under incomplete information for an investor with constant relative risk aversion. We assume that the investor can only observe asset prices, but not the instantaneous returns. Furthermore, we assume that the instatantaneous returns follow an Ornstein--Uhlenbeck process, and that their initial distribution is Gaussian. We analytically solve the Bellman equation for this problem, and indentify the optimal investment strategy under incomplete information. We explore the relationship between the value function under partial observation and the value function under full observation, and derive a formula for the economic value of information. Furthermore, we discuss how the optimal strategy under partial observation can be computed from the optimal strategy for an investor with full observation. Explicit solutions are presented in a model with only one risky asset.