A nonlinear approximation of a function (in some norm) refers to finding the best (e.g. the shortest) approximation via a linear combination of functions from a specific but typically wide class of functions (i.e., significantly wider than a basis set). One of the early and useful examples is an optimal (in the uniform norm) approximation via rational functions.

It has been shown that nonlinear approximations are far superior to approximations using a linear subspace of fixed functions (e.g., bases, point values and the like) in that the approximation error decays much more rapidly (often exponentially fast) as a function of the number of elements participating in the approximation.

For example, we recently observed this behavior for a wide class of signals using a near optimal approximation of their Fourier transform via decaying exponentials. This approach results in rational approximation of the function itself and, inter alia, allows one to identify the level of noise in the signal.

The talk will give an overview of several applications where the introduction of nonlinear approximations (in one form or another) has led to a significant improvement in accuracy and/or speed of algorithms.

Nonlinear approximations stand to have a wide impact on scientific computing. The key challenge resides in developing algorithms for their construction, which is one of the focal points of the research of our group at University of Colorado.Understanding swimming at low Reynolds numbers: successes and challenges