Normally Offered:

Each Fall

Campus/Online:

On-campus only

Requisites:

ECE 30411 and ECE 61800

Requisites by Topic:

Electromagnetics and numerical electromagnetics; familiarity with a computer programming or scripting language (eg, Python, MATLAB, Fortran, C++);

Catalog Description:

This course presents advanced integral equation formulations for electrostatics, magnetostatics, and full-wave electromagnetics, with emphasis on engineering modeling and large-scale computational analysis. The course begins with layer-potential representations and classical electromagnetic integral equations used in antenna, scattering, and interconnect problems, followed by discussion of discretization, singular-integral treatment, and numerical considerations arising in practical software implementations. The focus then shifts to iterative solvers and techniques for accelerating dense matrix-vector operations, including conjugate-gradient methods combined with fast Fourier transforms (CG-FFT), adaptive integral methods, and fast multipole methods. Both physics-based acceleration approaches using multipole and plane-wave expansions and algebraic techniques such as kernel-independent fast multipole methods are covered. Emphasis is placed on understanding algorithmic assumptions, accuracy-performance trade-offs, and applicability to realistic electromagnetic engineering problems.

Learning Outcomes

A student who successfully fulfills the course requirements will have demonstrated an ability to:

• Explain the role of single- and double-layer potentials in electrostatics and describe the associated jump relations and boundary operators used in boundary integral formulations.
• Formulate classical integral equation representations for electromagnetic boundary-value problems (e.g., EFIE, MFIE, CFIE), and articulate the assumptions and limitations underlying each formulation.
• Identify numerical challenges in integral equation discretizations, including singular and near-singular integrals, and explain standard strategies for self-term treatment, quadrature correction, and precorrected schemes.
• Recognize and interpret low-rank structure in integral equation matrices and explain how techniques such as SVD, matrix sketching, ACA, and related factorizations exploit this structure.
• Describe the fundamental principles of the Fast Multipole Method (FMM), including multipole and local expansions, translation operators, and hierarchical domain decomposition.
• Differentiate between classical, kernel-independent, and plane-wave FMM variants, and identify the regimes in which each approach is most effective.
• Explain how oscillatory kernels and high-frequency problems alter the structure of fast algorithms and motivate the use of rotations, directional methods, and plane-wave representations.
• Discuss the challenges posed by layered Green's functions, including Sommerfeld integrals, and explain how these challenges influence the design of fast solvers.
• Compare FMM-based acceleration techniques with FFT-based approaches, such as the precorrected-FFT and Adaptive Integral Method, and identify the trade-offs associated with each.
• Critically read and interpret research literature in integral equations and fast algorithms, identifying key assumptions, algorithmic choices, and sources of approximation or breakdown.
• Develop informed judgment about method selection, enabling students to assess which numerical techniques are appropriate for a given physical problem, kernel type, and geometric setting.

Lecture Outline:

Assessment Method:

Homework, projects, class participation (4/2026)