Multilevel Compression of Linear Operators
Descendants of Fast Multipole Methods (FMMs)
and CalderónZygmund Theory
Logistics and Syllabus: Yale, Fall 2005 Semester
Catalog Number:
AMTH510a
Instructor:
Mark Tygert (Yale),
in collaboration
with PerGunnar Martinsson (U. of Colorado)
and Vladimir Rokhlin (Yale)
Location:
A. K. Watson (51 Prospect St.) Room 400
Times:
16.0017.30 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:

Volume and Boundary Integral Equations
 Laplace Equation
 Yukawa/Screened Coulomb/Modified Helmholtz Equation
 Bilaplace Equation
 TimeDependent Wave Equation
and Scalar Helmholtz/TimeHarmonic Wave Equation
 TimeDependent Maxwell Equations
and Vector Helmholtz/TimeHarmonic Maxwell Equations
 Applications in Classical Mechanics:
Electro and Magnetostatics,
Elasticity Theory and Very Low Reynolds Number 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)
 Interpolation/Skeletonization

Fast Methods for Applying Imbeddedly Separable Linear 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 Plane Wave TimeDomain (PWTD) Algorithm for the Solution
of TimeDependent Wave Equations

Direct/LocallyAdaptive Solution Techniques

DivideandConquer Diagonalization and Singular Value Decomposition (SVD)
Techniques and Fast Algorithms for Special Function Expansions