Industrial Mathematics - Alexiades
M475 - Spring 2018
Modeling, Analysis, and Computation of interesting
scientific / technological / industrial problems
commonly known as Computational Science
Prof. Vasilios ALEXIADES Ayres 213 974-4922 email@example.com
Office Hours: TR 2:30-3:30 and by arrangement
Attendance is mandatory.
No textbook to buy!
Work and Grading: No exams!
8-10 Lab/Homework assignments: 40% ,
Project assignments: 40% ,
Term/Team Project: 20%
Do not hesitate to came talk to me if you are facing difficulties.
All incidents of academic misconduct will be reported to the Student Judicial Affairs office.
If you need an accommodation based on the impact of a disability, please contact me privately.
Contact the Office of Disability Services (2227 Dunford Hall, 974-6087) to coordinate reasonable
accommodations for documented disabilities.
Computational Science : doing Science by means of
computation ("in silico").
Involves: scientific problem → math problem
→ computational algorithm →
numerical solution →
implications for original scientific problem.
It has become the 3rd pillar of Science,
complementing Theory and Experiment.
Real scientific/technological/managerial problems canNOT be
Need to be solved numerically (approximately),
so need effective approximations/algorithms
and understant the effects of errors in the calculations.
Want algorithms to be: effective, accurate, reliable,
efficient and robust !
These aims often play against each other, so trade-offs need to be made...
Issues of verification, validation, uncertainty quantification
are becoming increasingly important.
The course will simulate the core aspects of
Computational Science including:
modeling and computational simulation of physical phenomena,
writing reports, writing proposals, collaborating with colleagues
on a research project, and presenting your work.
I. Crystal precipitation
- physical model leading to ODE system
- about ODEs - well posedness of IVP
- equilibria - root finding (Newton method) - plotting
- analysis of the model
- Euler scheme - computational errors
- classical RK4 and other numerical schemes
II. Air pollution: Advection and Diffusion Processes
- linear advection - wave propagation
- 1st order PDEs - method of characteristics
- the general consrvation law ut + div F = 0
- derivation from first principles
- conservation of species
- advective and diffusive fluxes
- continuity equation
- finite volume discretization of ut + Fx = 0 - explicit/implicit
- advection ( F = uV ) - explicit upwind scheme
- CFL condition - implementation
- diffusion ( F = −Dux ) - parabolic PDEs - boundary conditions
- explicit scheme - CFL condition
- advection-diffusion ( F = uV − Dux )
- explicit scheme - CFL condition
- effect of small/large Peclet number
- super-time-stepping acceleration
- a few words about Lax-Wendroff and other schemes
III. Chemical reactions via mass action kinetics
IV. Uncertainty Quantification and parameter estimation
Some other possible topics:
V. Melting and Freezing
- phase-change basics, moving boundary problems
- Stefan Problem, exact solution, analytic approximations
- enthalpy formulation, explicit scheme
VI. The catalytic converter
- diffusion-reaction model
- control problem
- calculus of variations - Euler-Lagrange equation
- numerical scheme for the forward model
VII. Electron beam lithography (inverse problems)
- forward scattering (dose to exposure)
- inverse problem (exposure to dose) - ill posed problem
- Fourier-Poisson integral solution of diffusion equation
- Fourier series solution of diffusion equation
- Fourier series approximation of the inverse problem
- Discrete Fourier Transform, FFT
------------ Some comments from happy students -------------
"Thank you for a very interesting and informative class. I looked
forward to taking it and am incredibly glad I did."
"You made this class very interesting, challenging, and (dare i say it)
fun ... I REALLY enjoyed the final project and feel more
confident in my abilities because of this class."
"This class was one of the best, if not the best, of my college career. I really enjoyed it."
"Extremely relevant course material, broken down in a very understandable method by instructor"
"... the best math class I've had so far.... I really learned a lot and plan to use it."
"Loved it. It's the best class I've ever taken"
"... For someone who enjoys programming, and has a real
desire to see what all this math can be used for, it has
been a terrific course."