Multilevel Compression of Linear Operators

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

Logistics and Syllabus: Yale, Fall 2005 Semester

Course Homepage

Catalog Number: AMTH510a

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

Location: A. K. Watson (51 Prospect St.) Room 400

Times: 16.00-17.30 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:

1. Volume and Boundary Integral Equations
   • Laplace Equation
   • Yukawa/Screened Coulomb/Modified Helmholtz Equation
   • Bilaplace 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/Skeletonization
4. Fast Methods for Applying Imbeddedly Separable 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