Numerically, many points may behave like eigenvalues when an operator is highly nonnormal.
Like cogs in a clock, numerical linear algebra allows the rest of applied mathematics to operate smoothly. Like those cogs, it deserves care and tuning. Researchers at CAAM elucidate the numerical properties of known routines and develop new methods to meet the challenges presented by computational mathematics. Of particular interest at CAAM are the topics of eigenvalue computation and iterative methods.
Danny Sorensen put eigenvalue computation at CAAM on the map with his implicitly restarted Arnoldi method, which has been implemented in Matlab as the "eigs" command. The method allowed for the computation of a few eigenvalues of matrices of previously untouchable size.
One of the fascinating topics of Mark Embree's research is the analysis of eigenvalues of nonnormal matrices and operators. (Such operators do not commute with their adjoints.) Traditional methods can be very misleading, as small perturbations to a nonnormal matrix can move eigenvalues far away from the eigenvalues of the original matrix. Pseudospectra analysis has proven to be a promising tool on many applications.
Eigenvectors pick out the fundamental traits common to a set of faces.
In modern applications, often we find ourselves working with large quantities of data. How to store, analyze, and extract information from it frequently poses great challenges.
For example, graduate student Ed González, under the supervision of Yin Zhang, studies human face detection. Can we pick out the salient features common to group of photographs? Of course, faces of two random people typically have many differences: blue eyes vs. brown, blonde hair vs. black, a mole here, a double chin there. However, as anybody who has been to a party knows, faces start to blur together when you have seen a lot of them.
Ed has been working on finding the fundamental attributes in common to a set of faces. Are the common attributes something we may easily identify, such as "chubby cheeks" or "crooked noses?" The commonalities turn out to be a little more nebulous, resembling murky clouds of ghost-like images
We call these clouds eigenfaces. They are a set of eigenvectors derived from a covariance matrix relating the set of faces. When properly weighted, eigenfaces can be summed together to create an approximate gray-scale rendering of a human face.
Remarkably few eigenfaces are needed to give a fair likeness of most people's faces, so eigenfaces provide a means of applying data compression to faces for identification purposes. With work like Ed's, computer face recognition may soon be no longer the stuff of science fiction.
Find out how Mili Shah has been using matrix factorizations to more accurately portray the motion of molecules.
Scientists and engineers often need to simulate or control physical processes. Whether the phenomena are as gigantic as global weather patterns or as tiny as magnetic self-inductance in a microchip, the number of differential equations needed to accurately describe the process can be dauntingly huge. The goal of model reduction is to describe the phenomenon just as accurately with far fewer equations. The smaller models allow for quicker simulations and faster responses to changing systems, which in turn save time and money.
The people working in model reduction at CAAM are Mark Embree, Danny Sorensen, and John Sabino. Find out more about model reduction at Rice.
