Course 8200 - Topics in Applied Mathematics: Multiscale Modeling and Methods - Fall 2024

Official Information
Course Number:	Math 8200.001
CRN:	50548
Course Title:	Topics in Applied Mathematics: Multiscale Modeling and Methods
Times:	MW 10:30-11:50
Places:	617 Wachman Hall

Instructor:	Benjamin Seibold
Instructor Email:	seibold(at)temple.edu
Instructor Office:	518 Wachman Hall
Instructor Office Hours:	M 1:00-2:00, W 2:00-3:00

Course Textbooks:	Materials will be drawn from a variety of resources, both for lectures and for additional reading: Weinan E, Principles of Multiscale Modeling, Lecture Notes, 2011 Mark H. Holmes, Introduction to Perturbation Methods, Second Edition, Springer Martin Frank, Introduction to Kinetic Theory, Lecture Notes, 2024 Martin Frank, Uncertainty Quantification, Lecture Notes, 2020 Alexandre J. Chorin and Ole H. Hald, Stochastic Tools for Mathematics and Science, Springer, 2013 Dennis M. Kochmann, Multiscale Modeling, Lecture Notes, 2023
Official:	Course Syllabus
Prerequisites:	none
Topics Covered:	Many real-world systems possess a variety of scales, with the micro-scale dynamics of the constitutive particles (such as atoms in materials, biological cells in organisms, or vehicles in traffic flow) shaping emergent structures on the macroscopic “laboratory” scale in non-trivial ways. This course provides a trip into the world of mathematical multiscale methods that enable the systematic traversing of these scales. Besides introducing applications in materials, traffic flow, and the life-sciences, the course covers analytical multiscale methods (such as continuum dynamics from molecular dynamics, averaging methods, homogenenization, Mori-Zwanzig formalism, kinetic theory, moment methods, and uncertainty quantification) as well as discusses important computational multiscale methodologies (such as multigrid methods, the fast multipole method, and adaptive mesh refinement).
Course Goals:	Students will obtain a perspective of many aspects related to mathematical models and methods. They will learn about modeling, mathematical analysis, and computational methods that effectively handle problems that exhibit multiple scales.
Attendance Policy:	Students are expected to attend every class. If a student cannot attend a class for some justifiable reason, he or she is expected to contact the instructor before class.
Course Grading:	Assignments: 50%; final examination: 50%
Final Exam Date:	12/14/2024
Course Schedule
08/26/2024 Lec 1	Examples of multiscale problems, examples of methods Read: Multiscale modeling, Galileo Galilei
08/28/2024 Lec 2	Discrete vs. continuum, micro vs. macro, density estimation Read: Continuum mechanics, Density estimation, KDE, Sampling
09/04/2024 Lec 3	Sampling, continuity equation, traffic flow Read: Rejection sampling, Metropolis-Hastings, Continuity equation, Traffic flow
09/09/2024 Lec 4	Basic traffic models, from micro to macro Read: Traffic model (microscopic, macroscopic)
09/11/2024 Lec 5	Microscale in continuuum models, asymptotic analysis Read: Asymptotic analysis, Asymptotic expansion
09/16/2024 Lec 6	Singular perturbations, boundary layers, Burgers' equation Read: Singular perturbation, Boundary layer, Matched asymptotics, Burgers' equation
09/18/2024 Lec 7	WKB method Read: WKB method
09/23/2024 Lec 8	Homogenization (Henry Brown): Lecture notes Read: Homogenization
09/25/2024 Lec 9	Cellular models Read: Cellular automaton, Rule 184
09/30/2024 Lec 10	Random walks and diffusion (Jacob Woods) Read: Random walk, Lévy flight, Fick's law, Diffusion
10/02/2024 Lec 11	Continuum limits of cellular models Read: Method of characteristics
10/07/2024 Lec 12	Multiscale cellular automata (Madison Shoraka) Read: Directed graph, Adjacency matrix, Network theory
10/09/2024 Lec 13	Cell transmission model, hierarchy of physical models Read: Cell transmission model, Continuum mechanics, Kinetic theory, Molecular dynamics, Quantum mechanics, Hartree-Fock method
10/14/2024 Lec 14	Density Functional Theory (Afrina Meghla) Read: Planck constant, Uncertainty principle, Schrödinger equation, Born-Oppenheimer approx., Kohn-Sham equations, Density functional theory
10/21/2024 Lec 15	Molecular dynamics Read: Molecular dynamics, Lennard-Jones potential
10/23/2024 Lec 16	Euler equation limit from Boltzmann equation (Nicole Zalewski) Read: Boltzmann equation, Chapman-Enskog theory
10/28/2024 Lec 17	Statistical mechanics Read: Statistical mechanics, Liouville equation, Detailed balance, Canonical ensemble, Langevin equation
10/30/2024 Lec 18	Kinetic traffic models (Blessing Nwonu) Read: Review Vehicular Traffic 2009, Paper Physica A 2013
11/04/2024 Lec 19	Molecular chaos, H-theorem, BGK and Vlasov equation Read: Molecular chaos, Loschmidt's paradox, H-theorem, Vlasov equation
11/06/2024 Lec 20	Fast summation methods (Youmna Layoun) Read: Low-rank approximation, Fast multipole method, Octree, Barnes-Hut
11/07/2024 Lec 21	Radiation transport, diffusion approximation Read: Radiative transfer, Diffusion approximation
11/11/2024 Lec 22	Moment methods Read: Moment, Method of moments
11/13/2024 Lec 23	Domain decomposition methods (Logan Reed) Read: Domain decomposition, Schwarz alternating method
11/18/2024 Lec 24	Asymptotic preserving methods: concept and non-AP method Read: Jin paper 2022, Godunov's scheme
11/20/2024 Lec 25	Asymptotic preserving methods: AP scheme Read: Jin paper 2022, staggered grids
12/02/2024 Lec 26	Uncertainty quantification, Monte Carlo, generalized polynomial chaos Read: Uncertainty quantification, Monte Carlo, Quasi-Monte Carlo, Karhunen-Loève expansion, Polynomial chaos
12/04/2024 Lec 27	Mori-Zwanzig formalism Read: Mori-Zwanzig formalism, Duhamel's principle, Conditional expectation, C0-Semigroup
12/09/2024 Lec 28	Optimal prediction, Outlook on adaptive mesh refinement and multigrid Read: Adaptive mesh refinement, Multigrid method
12/14/2024	Final Examination
Matlab Programs
Test of two numerical methods for the wave equation with stiff relaxation in second component (modeling the P1 model of radiation transport is the strong scattering regime). One method is asymptotic preserving, while the other is not: temple8200_ap_methods.m
Homework Problem Sets
Problem set 1 (due 09/30/2024) Problem set 2 (due 10/07/2024) Problem set 3 (due 10/14/2024) Problem set 4 (due 10/21/2024) Problem set 5 (due 10/30/2024) Problem set 6 (due 11/06/2024) Problem set 7 (due 11/13/2024) Problem set 8 (due 11/20/2024)