Courses Details


Location:

All morning activities will be held at RLM 7.124. All afternoon activities will be held at RLM 7.118. Computer accounts will be provided and you can use the computer lab on the 7th floor of RLM.

Topics and material:

Introduction to Multiscale Modeling
by Bjorn Engquist (University of Texas at Austin)
Lectures 1-5
Homework
References: 1, 2

High Frequency Methods for the Wave Equations
by Nick Tanushev (University of Texas at Austin)
Homework

Fast and Multiscale Algorithms for Integral Equations and Transforms
by Lexing Ying (University of Texas at Austin)
Lectures 1-2, 3, 4-5
Homework
References: 1

Stochastic Multiscale Modeling
by Gil Ariel (University of Texas at Austin)
Lectures and Homework

Multiscale modeling for Ordinary Differential Equations and Level Sets
by Richard Tsai (University of Texas at Austin)
Homework
References: 1

Introduction to Wavelets and Homogenization
by Olof Runborg (KTH)
Lectures
Slides: 1 2 3
Homework

Tutorial:
Notes on ODE by Elizabeth Thoren.
Demo on Matlab by Nicholas Leger.
Notes on Fourier analysis by Emanuel Indrei.
Notes on Probability by Ricardo Alonso

Projects:

Project 1.
Project 2.
Project 3.
Project 4.
Project 5.

Schedule:

First week (7/21 - 7/25)

Monday
10:00-12:00
Registration
12:00-14:00
Lunch break
14:00-15:00
Engquist (Lecture 1)
15:00-16:00
Engquist (Lecture 2)
16:00-17:00
Tutorial on ODE
17:00-18:00
Tutorial on PDE

Tuesday
9:00-10:00
Engquist (Lecture 3)
10:00-11:00
Engquist (Lecture 4)
11:00-12:00
Tanushev (Lecture 1)
12:00-14:00
Lunch break
14:00-15:00
Tanushev (Lecture 2)
15:00-16:00
Tutorial on Matlab
16:00-17:00
Tutorial on Fourier Analysis
17:00-18:00
Tutorial on Probability

Wednesday
9:00-10:00
Engquist (Lecture 5)
10:00-11:00
Tanushev (Lecture 3)
11:00-12:00
Tanushev (Lecture 4)
12:00-14:00
Lunch break
14:00-18:00
Work on homework problems


Thursday
9:00-10:00
Tanushev (Lecture 5)
10:00-11:00
Ying (Lecture 1)
11:00-12:00
Ying (Lecture 2)
12:00-14:00
Lunch break
14:00-18:00
Work on homework problems

Friday
9:00-10:00
Ying (Lecture 3)
10:00-11:00
Ying (Lecture 4)
11:00-12:00
Ying (Lecture 5)
12:00-14:00
Lunch break
14:00-18:00
Work on homework problems

Second week (7/28 - 8/1)

Monday
9:00-10:00
Ariel (Lecture 1)
10:00-11:00
Ariel (Lecture 2)
11:00-12:00
Tsai (Lecture 1)
12:00-14:00
Lunch break
14:00-18:00
Work on homework problems


Tuesday
9:00-10:00
Ariel (Lecture 3)
10:00-11:00
Tsai (Lecture 2)
11:00-12:00
Tsai (Lecture 3)
12:00-14:00
Lunch break
14:00-18:00
Work on homework problems


Wednesday
9:00-10:00
Tsai (Lecture 4)
10:00-11:00
Tsai (Lecture 5)
11:00-12:00
Runborg (Lecture 1)
12:00-14:00
Lunch break
14:00-18:00
Work on homework problems


Thursday
9:00-10:00
Ariel (Lecture 4)
10:00-11:00
Runborg (Lecture 2)
11:00-12:00
Runborg (Lecture 3)
12:00-14:00
Lunch break
14:00-18:00
Work on homework problems


Friday
9:00-10:00
Ariel (Lecture 5)
10:00-11:00
Runborg (Lecture 4)
11:00-12:00
Runborg (Lecture 5)
12:00-14:00
Lunch break
14:00-18:00
Work on homework problems

Course descriptions:

Introduction to Multiscale Modeling
by Bjorn Engquist (University of Texas at Austin)

We will start by presenting a few practical multiscale modeling examples and giving a survey of analytic multiscale methods. This introduction will include classical topics such as the theory of singular perturbations, boundary layers, homogenization and geometrical optics. We will focus on material that illuminates basic principles and also is important in the design and analysis of computational techniques. The computational cost of solving multiscale problems will be discussed.

We will then discuss well-established numerical multiscale methods. Again there will be some classical topics as, for example, multigrid but also more recent techniques like the multiscale finite element method. The emphasis will be on what multiscale features are important in the problems for these methods to work.

Finally we will study the heterogeneous multiscale method (HMM) and related numerical algorithms and applications. One such example is Continuum simulations of solids or fluids for which some atomistic information is needed. This is a typical multiscale problem with very large ranges of scales. For such problems it is necessary to restrict the simulations on the microscale to a smaller subset of the full computational domain. HMM is a framework for developing and analyzing numerical methods that couple computations from very different scales. Local microscale simulations on small domains supply missing data to a macroscale simulation on the full domain.


High Frequency Methods for the Wave Equations
by Nick Tanushev (University of Texas at Austin)

The minicourse on ``High frequency methods for wave equations'' will begin with some physical ideas about the wave equation. We will then move on to the high frequency limit, where the wave equation reduces to a simpler form and explore asymptotic solutions in this limit. We will also talk about some applications of these asymptotic solutions.


Fast Multiscale Algorithms for Integral Equations and Transforms
by Lexing Ying (University of Texas at Austin)

The purpose of these lecture is to discuss fast algorithms for computing integral transforms and equations. In the first lecture, we will begin with the famous fast Fourier transform (FFT) algorithm and its application in the numerical solution of linear PDEs. The second lecture will be devoted to the non-uniform fast Fourier transform (NFFT) and its application in data interpolation and signal processing.

In the third lecture, we will talk about the boundary integral formulations for Laplace equations and Helmholtz equations. This will setup the analytical background for the topics to be discussed in the last two lectures.

In the fourth lecture, we will focus on the famous fast multipole method. This algorithms evaluates the pairwise interaction between a large set of particles in linear time. Applications include a lot of problems in molecular dynamics and astrophysics.  In the fifth lecture, we will turn our attention to the multidirectional algorithm for computing oscillatory integral transforms which appear in the boundary integral formluation of the Helmholtz equation.


Stochastic Multiscale Modeling
by Gil Ariel (University of Texas at Austin)

The purpose of the lectures is to explore the time evolution of stochastic systems in which some aspects of the dynamics occur on a much faster time scale than others. We begin with a brief review of some of the elementary properties of stochastic processes, Brownian motion and Ito calculus. These ideas are further developed for several example systems which evolve on two or more time scales. We study both analytical and computational methods and find that separation of time scales can lead to an effective slow behavior that is either deterministic of stochastic.

Some elementary background in probability, such as discrete and continuous
random variables is assumed.


Multiscale modeling for Ordinary Differential Equations
by Richard Tsai (University of Texas at Austin)

The lecture will aim at constructing efficient numerical methods for oscillatory solutions of ordinary differntial equations. I will start by reviewing conventional numerical methods for ODEs, continued by discussing a selective aspects of geometric integration, certain asymptotic approaches to handle fast oscillations, averaging theorems, and end by describing the latest development in multiscale numerical methods.


Introduction to Wavelets and Homogenization
by Olof Runborg (KTH)

Wavelets (2.5 hours): I will talk about the general theory of wavelets, how they are constructed and some of their properties. Topics will include multiresolution analysis, filter banks,
approximation properties and the fast wavelet transform. I will also discuss some applications in image compression and numerical homogenization for partial differential equations.

Subdivision (1 hour): Subdivision is a type of scheme for constructing smooth curves and surfaces through successive refinement of an initial coarse mesh approximation. These are widely used in computer games and animated movies.  I will go through some basic schemes and analyze their mathematical properties, like convergence and regularity.

Multiresolution and normal meshes (1.5 hour): By combining the ideas of subdivision and wavelets we can construct multiresolution meshes, which describe curves and surfaces at varying levels of detail. Normal meshes is one example with a particularly efficient representation. I will present some analysis of normal meshes and also show how a multiresolution mesh representation can be used in the dynamic case to compute the movement of curves and surfaces in a velocity field at a low computational cost.