# CS448J: Concepts and Algorithms of Scientific and Visual Computing

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**Autumn 2015, TTh 10:30-11:50, R 200-107.**

## Summary

This course covers a selection of fundamental concepts and algorithms for scientific and visual computing. Based on prior knowledge in basis calculus, linear algebra, numerical interpolation and optimization, this course introduces the concept of the phase space, variational principles, methods for ordinary and partial differential equations, Fourier analysis, and multiscale modeling. The lecture is algorithmically oriented, aiming to enable the students to develop efficient solutions for practically relevant problems, based on solid theoretical foundations and mathematically precise modeling. It covers practical applications, like the simulation of rigid and deformable objects, fibers, fluids, molecular dynamics, signal/image analysis and processing, as well as wavelet-based modeling on different scales.

## Requirements

The course will assume basic knowledge such as taught in MATH41, MATH42, CS103, or CS205A. It is suitable for undergraduate as well as for graduate students.

## Syllabus

**(1) Phase Space**

Application: Phase Space Analysis

**(2) Variational Principles**

Application: Systems of Coupled Oscillators, Particle Systems

Application: Variational-based Image Segmentation

**(3) Ordinary Differential Equations (ODEs)**

Application: Simulation of Deformable Objects

Application: Rigid Body Dynamics

Application: Molecular Dynamics

**(4) Partial Differential Equations (PDEs)**

Application: Fiber Simulation (Cosserat Equations)

Application: Fluid Simulation (Navier-Stokes Equations)

**(5) Fourier Analysis**

Application: Signal Analysis and Filtering

Application: Image Compression

**(6) Multiscale Modeling**

Application: Hierarchical Spacetime Control

Application: Wavelet Importance Sampling

**(7) Outlook**

## Lecture Notes / Slides

The topics are illustrated on the blackboard as well as using the following slides. Please keep in mind, that the slides do not cover the whole content of the lecture.

09-22: Phase Space

09-24: Variational Principles

09-29: Canonical Equations

10-01: Image Segmentation

10-06: Ordinary Differential Equations

10-08: Symplecticity

10-13: Stiff Differential Equations

10-15: Constraint Methods

10-20: Partial Differential Equations

10-22: Finite Element Method

10-27: Lie Theory

10-29: Fourier Analysis

[11-03/05: Signal Analysis (by D. Hyde)]

11-10: Discrete Fourier Transforms

11-12: Fast Fourier Transform

11-17: Wavelets

11-19: Multiresolution Analysis

[11-24/26: (Thanksgiving Recess)]

12-01: Advanced Computation Models

12-03: Quantum Algorithms

## Student Work

There will be a problem set assigned each week, which will be graded. This homework track is mostly theoretical, but it will include a final project and smaller programming tasks along the way. The final project will consist of writing a simulator for one of the main types of phenomena discussed in the course. The students may collaborate on the assignments provided each student writes up his or her own solutions and clearly lists the names of all the students in the group. There will be no midterm or final exam.

## Exercise Sheets

The Homework is due every Tuesday. Written solutions should be handed in before the lecture. Programming assignments must be submitted by email to your tutor.

00: Warming-up (0 Points, not graded)

01: Variational Principles (20 Points, due Oct 6)

02: Classical Variational Problems (18 Points, due Oct 13)

03: Integrating Ordinary Differential Equations (20 Points, due Oct 20)

04: Fermi Pasta Ulam Problem (10 Points, due Oct 27)

05: Heat Equation (15 Points, due Nov 3)

06: Fourier Analysis (12 Points, due Nov 10)

07: Fourier Transforms (17 Points, due Nov 17)

08: Fast Fourier Transform (22 Points, due Nov 24, electronic submission)

09: Multiresolution Analysis (16 Points, due Dec 1)

## Solutions

Solutions to selected exercises are provided for the students.

00-5: Analysis of Algorithms

01-1: Double Pendulum

01-3: Fundamental Lemma

02-X: Classical Variational Problems

02-X: Classical Variational Problems (by W. Lin)

03-1: Foucault Pendulum

03-2: Verlet Integration

04-1: Fermi Pasta Ulam Problem

05-1: Heat Equation (HeatEquation.m, hls2rgb.m, stanford.png)

06-1: Fourier Analysis

07-X: Fourier Transforms

08-X: Fast Fourier Transform

09-X: Multiresolution Analysis

## Grading Policy

Assignments: 25%, Final Project: 75%.

## Honor Code

The university expects both faculty and students to respect and follow Stanfordâ€™s Honor Code.

## Literature

S. Arora, B. Barak. Computational Complexity: A Modern Approach. Cambridge University Press, 2009.

J.-L. Basdevant. Variational Principles in Physics. Springer, 2007.

I. Daubechies. Ten Lectures on Wavelets. SIAM, 1992.

W. Hackbusch. Multi-Grid Methods and Applications. Springer, 2010.

J.M. Haile. Molecular Dynamics Simulation: Elementary Methods. Wiley, 1997.

E. Hairer, S.P. Nørsett. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, 2009.

E. Hairer, C. Lubich. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, 2010.

E. Hairer, G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, 2010.

G. Kaiser. A Friendly Guide to Wavelets. Birkhäuser Classics, 2011.

L.D. Landau, E.M. Lifshitz. Mechanics, Third Edition, Course of Theoretical Physics, Volume 1. Butterworth-Heinemann, 1982.

R.H. Landau, M.J. Páez, C.C. Bordeianu. Computational Physics: Problem Solving with Computers. Wiley, 2007.

A. Mitiche, I.B. Ayed. Variational and Level Set Methods in Image Segmentation. Springer, 2011.

R.M. Murray, Z.Li, S.S. Sastry. A Mathematical Introduction to Robotic Manipulation. CRC Press, 1994.

S. Osher, R. Fedkiw. Level Set Methods and Dynamic Implicit Surfaces. Springer, 2002.

E.M. Stein. Fourier Analysis: An Introduction. Princeton University Press, 2003.

G. Strang. Computational Science and Engineering. Wellesley-Cambridge Press, 2007.

O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, 2013.

## Course Staff

Instructor: Prof. Dominik L. Michels

Email: michels@cs.stanford.edu

Office: 208 Gates CS Bldg.

Office Hours: F 10-12

Course Assistant: David Hyde

Email: dabh@stanford.edu

Office: 209 Gates CS Bldg.

Office Hours: F 10-12