Compression of Linear Operators

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

Logistics and Syllabus: NYU, Fall 2009 Semester

Course Homepage

Catalog number: G63.2011.001(MATH)/G22.2945.001(CS)

Instructor: Mark Tygert

Location: Room 512, Warren Weaver Hall

Times: 1:25 P.M. to 3:15 P.M. Mondays

Grading: This course will be graded as a seminar 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)
   • Interpolative Decompositions (IDs)
4. Fast methods for applying non-oscillatory linear integral 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