Course number: | MATH-GA 2011.002 |

Semester: | Fall 2017 |

Time & Location: | Thurs, 1:25pm - 3:15pm in WWH 512 |

Instructor: | Mike O'Neil (oneil@cims.nyu.edu) |

Office hours: | By appointment |

This course will be an introduction the theory and application of integral equations in classical mathematical physics, as well as the numerical methods required for their efficient and accurate solution. These numerical methods include quadrature for singular functions, analysis-based fast algorithms (e.g. fast multipole methods), iterative and fast-direct solvers (for the resulting dense linear systems). Methods from potential theory, applied analysis, functional analysis, numerical linear algebra, complex analysis, and asymptotic analysis are central to the construction of almost all of these algorithms.

There is no one textbook for this course. Instead, there will be continually updated lectures notes available. These lecture notes will contain many useful references for each of the topics and algorithms covered in class. As they become relevant, original journal articles and textbooks will be listed below in the table of lecture topics.

Relevant code examples will be posted on gitlab.com/oneilm/integralequations.

Lecture notes, updated throughout the semester, can be downloaded here: int_eq_notes_2017.pdf.

The grades in the course will be determined by a course project and classroom interactions. A description of the course project can be found here:

Important information for the course will appear below as necessary.

- Class on September 7th is CANCELLED. The first class on September 14

Below is an updated list of lecture topics along with any documents that were distributed, or relevant code.

Date | Topics | Materials |
---|---|---|

September 7 | No class - cancelled. | |

September 14 | Overview, electrostatics and the Laplace equation | Jackson, Classical Electrodynamics |

September 21 | 2D Laplace boundary value problems, layer potentials | Colton, Partial Differential Equations |

September 28 | Fredholm theory | Riesz and Nagy, Functional AnalysisPorter and Stirling, Integral equations |

October 5 | Trapezoidal discretization and quadrature | Atkinson, The Numerical Solution of Integral
Equations of the 2nd KindKress, 1991 Kapur and Rokhlin, 1997 Alpert, 1999 Hao, et. al., 2014 |

October 12 | Adaptive discretizations, 3D Laplace BVPs | |

October 19 | 3D Laplace, surface discretizations | Bremer & Gimbutas, 2012 |

October 26 | Barnes-Hut and higher-order tree codes | Appel, 1985 Barnes & Hut, 1986 |

November 2 | The 3D Laplace FMM, multipole translations | Greengard, 1987 Epton & Dembart, 1995 |

November 9 | Accelerating translation operators | |

November 16 | The 3D Helmholtz FMM | Cheng, et. al., 2006 |

November 23 | No class - Thanksgiving. | |

November 30 | Integral equations in electromagnetics | Colton and Kress, Integral Equation Methods
in Scattering Theory |

December 7 | Butterfly algorithms | Michielssen & Boag, 1996 Candès, et. al., 2009 O'Neil, et. al., 2010 |

December 14 | Kernel independent methods and fast direct solvers |
Gimbutas & Rokhlin, 2002 Ying, Biros, & Zorin, 2004 Gillman, Young, & Martinsson, 2012 |