Compression of Linear Operators
Descendants of Fast Multipole Methods (FMMs)
and CalderónZygmund Theory
Logistics and Syllabus: NYU, Fall 2009 Semester
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:

Volume and boundary integral equations
 Laplace equation
 Yukawa/screenedCoulomb/modifiedHelmholtz equation
 BiLaplace equation
 Timedependent wave equation
and scalarHelmholtz/timeharmonicwave equation
 Timedependent Maxwell equations
and vectorHelmholtz/timeharmonicMaxwell equations
 Applications in classical mechanics:
electro and magnetostatics,
elasticity theory and very lowReynoldsnumber fluid dynamics,
acoustics,
electrodynamics,
manybody dynamics,
and dynamics on lattices and in spherical coordinates

Iterative/notlocallyadaptive solution techniques
 Simplest/stationary: Neumann/Born series and Chebyshev approximations
 Krylovsubspacebased/nonstationary:
Generalized Minimum RESidual (GMRES)
and Conjugate Gradient (CG) methods

Numerical representations of function spaces and linear operators
based on algebra
 Singular Value Decompositions (SVDs)
 Interpolative Decompositions (IDs)

Fast methods for applying nonoscillatory linear integral operators
 Numerical representations based on algebra
 Numerical representations based on exponentials/planewaves

Fast methods for applying the Green operators of timeharmonic wave equations
 Bessel functions and partial wave expansions
 Low frequency
 High frequency
 Wideband
 Directional/windowed translation operators

Fast PlaneWave TimeDomain (PWTD) algorithm for the solution
of timedependent wave equations

Direct/locallyadaptive solution techniques