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Center for Nonlinear Studies |
It is well known that representing functions using optimal nonlinear approximations is much more efficient than using more standard linear methods (e.g., Fourier series, wavelets, ect.), especially for functions with singularities or sharp transition regions. However, standard algorithms for computing such optimal approximations don't always converge, and often require extended precision if high accuracy is desired. In this talk I will discuss a new fast and accurate algorithm for computing optimal rational approximations. A key tool behind computing such approximations is a new algorithm for computing small eigenvalues of certain structured matrices with high relative accuracy, which is impossible using standard eigenvalue methods. I will also present numerical applications of using optimal approximations for solving viscous Burgers equation with large Reynolds number, which demonstrate that optimal approximations can be a viable alternative to more standard linear methods in numerical analysis. Finally, I will discuss ongoing work for using optimal approximations to solve equations in quantum chemistry, where preliminary experiments suggest that using such approximations require a factor of 100-1000 fewer parameters than competing linear methods.
Host: Beth Wingate