Multilevel Compression of Linear Operators

Descendants of Fast Multipole Methods (FMMs) and Calderón-Zygmund Theory

Logistics and Syllabus: Yale, Fall 2006 Semester

Course Homepage

Catalog number: AMTH510a

Instructor: Mark Tygert (Yale), in collaboration with Per-Gunnar Martinsson (U. of Colorado)

Location: Leet Oliver Memorial Hall (12 Hillhouse Ave.) Room 200

Times: 2:30 P.M. to 4:00 P.M. Mondays and Wednesdays

Grading: If you're taking the course for credit, your grade will be determined wholly by a "chat" with the instructor at the end of the course.

Syllabus:

1. Volume and boundary integral equations
   • Laplace equation
   • Yukawa/screened-Coulomb/modified-Helmholtz equation
   • Bi-Laplace equation
   • Time-dependent wave equation and scalar-Helmholtz/time-harmonic-wave equation
   • Time-dependent Maxwell equations and vector-Helmholtz/time-harmonic-Maxwell equations
   • Applications in classical mechanics: electro- and magnetostatics, elasticity theory and very low-Reynolds-number fluid dynamics, acoustics, electrodynamics, many-body dynamics, and dynamics on lattices and in spherical coordinates
2. Iterative/not-locally-adaptive solution techniques
   • Simplest/stationary: Neumann/Born series and Chebyshev approximations
   • Krylov-subspace-based/non-stationary: Generalized Minimum RESidual (GMRES) and Conjugate Gradient (CG) methods
3. Numerical representations of function spaces and linear operators based on algebra
   • Singular Value Decompositions (SVDs)
   • Interpolation
4. Fast methods for applying low-RUS linear operators
   • Numerical representations based on algebra
   • Numerical representations based on exponentials/plane-waves
5. Fast methods for applying the Green operators of time-harmonic wave equations
   • Bessel functions and partial wave expansions
   • Low frequency
   • High frequency
   • Wideband
   • Directional/windowed translation operators
6. Fast Plane-Wave Time-Domain (PWTD) algorithm for the solution of time-dependent wave equations
7. Direct/locally-adaptive solution techniques
8. Divide-and-conquer diagonalization and Singular Value Decomposition (SVD) techniques and fast algorithms for special function expansions