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Numerical Methods for Computational Science and Engineering

Numerical Methods for Computational Scienceand EngineeringLecture 1, Sept 18, 2014: IntroductionPeter ArbenzComputer Science Department, ETH ZurichE-mail: arbenz@inf.ethz.ch

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Numerical Methods for Computational Science and EngineeringIntroduction

Outline of todays lectureI

What is numerical methods for CSE

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Survey of the lecture

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Organization of the lecture (exercises/examination)

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References

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Start of the lecture

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Numerical Methods for Computational Science and EngineeringIntroduction

Scientific Computing

NumCSE, Lecture 1, Sept 18, 2014

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Numerical Methods for Computational Science and EngineeringA CSE example

A CSE example: particle acceleratorsObserved phenomenonI

Charged particles are accelerated in electromagnetic fields.I

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Cathode ray tubes: beam of electrons deflected by e-field tocreate image (classic TV set).

Particle accelerator is a device to propel charged particles tohigh speeds by applying electromagnetic fields.I

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Paul Scherrer Institut: Numerous accelerator to investigatematerial properties, or for cancer treatment.CERN: Large hadron collider (LHC) to generate newelementary particles.Linear accelerators vs. cyclotrons. Demo: cyclo.m

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Numerical Methods for Computational Science and EngineeringA CSE example

A CSE example: particle accelerators (cont.)

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Numerical Methods for Computational Science and EngineeringA CSE example

A CSE example: particle accelerators (cont.)Model1. Electric fields are modeled by the time-harmonic Maxwellequations (periodic solution (E(x, t) = e(x)e it ), magneticfield eliminated, all pysical quantities = 1)curl curl e(x) = e(x),n e = 0,

x ,

div e(x) = 0,

x .

is accelerator cavity.2. Particles move according to Newtons law of motiondx(t)= v,dt

dv(t)q=(E + v B) .dtm0

where E = Eext + Eself . Eext is E-field from 1. (q charge, m0rest mass.)

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Numerical Methods for Computational Science and EngineeringA CSE example

A CSE example: particle accelerators (cont.)Discretization1. e-field in the Maxwell equation is a continuous function.Approximate it by a finite element function.This leads to a large sparsegeneralized eigenvalue problemAx = Mx.Size can be in millions.2. Particles are particles.Big issue is the computation of the self-field, i.e.,particle-particle interaction.NumCSE, Lecture 1, Sept 18, 2014

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Numerical Methods for Computational Science and EngineeringA CSE example

A CSE example: particle accelerators (cont.)Discretization (cont.)I

Computing particle-particle interactions costs O(np2 )operations in the number of particles np .

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Remedy: Approximately compute the electric potential (x)that is due to the charged particles. To that end we have tosolve the Poisson equation(x) =

1(x)0

with some boundary conditions.The quantity (x) on the right is the charge density which isin fact approximated on a regular grid.

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Numerical Methods for Computational Science and EngineeringA CSE example

A CSE example: particle accelerators (cont.)Also the solution (x) is obtained on this regular grid.

The gradient of (x) provides the force that acts on theparticle at position x,Eself = grad (x).NumCSE, Lecture 1, Sept 18, 2014

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Numerical Methods for Computational Science and EngineeringA CSE example

A CSE example: particle accelerators (cont.)Solution algorithmsI

We need to solve systems of linear equationsAx = b,

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where A is large and sparse, maybe positive definite.In a time-dependent problem we may have to solve such asystem in every time step.For the Poisson problem on regular grids there are fastPoisson solvers that are related to the fast Fourier transform.We need to solve eigenvalue problemsAx = Mx,where A and M are large and sparse to get the driving e-field.

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Numerical Methods for Computational Science and EngineeringA CSE example

A CSE example: particle accelerators (cont.)Efficiency, accuracyI

These problems are to be solved efficiently w.r.t. time andmemory.

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Linear vs. nonlinear models.

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Direct vs. iterative solution of linear systems.

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High vs. relaxed accuracy.

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Repeated solution of equal/similar problems may amortizeexpensive solution procedures.

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Numerical Methods for Computational Science and EngineeringSurvey of the lecture

Survey of the lecture1. Introduction2. Roundoff errors3. Nonlinear equations in one variable (2 lectures)4. Linear algebra basics (Read sections 4.14.4 in AscherGreif)5. Direct methods for linear system, pivoting strategies, sparsematrices (2)6. Linear least squares problems (2)7. Iterative methods for linear system (2)8. Eigenvalues and singular values (2)9. Nonlinear systems and optimization (3)10. (Piecewise) polynomial interpolation (3)NumCSE, Lecture 1, Sept 18, 2014

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Numerical Methods for Computational Science and EngineeringSurvey of the lecture

Survey of the lecture (cont.)11. Best approximation12. Filtering algorithms, Fourier transform13. Numerical differentiation14. Numerical integration (2)15. Ordinary differential equations, initial value problems (3)

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Numerical Methods for Computational Science and EngineeringAbout this course

About this courseFocusIII

on algorithms (principles, scope, and limitations),on (efficient, stable) implementations in Matlab,on numerical experiments (design and interpretation).

No emphasis onI

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theory and proofs (unless essential for understanding ofalgorithms)hardware-related issues (e.g. parallelization, vectorization,memory access)(These aspects will be covered in the courseHigh Performance Computing for Science and Engineeringoffered by D-INFK)

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Numerical Methods for Computational Science and EngineeringAbout this course

Goals Knowledge of the fundamental algorithms in numericalmathematics Knowledge of the essential terms in numerical mathematicsand the techniques used for the analysis of numericalalgorithms Ability to choose the appropriate numerical method forconcrete problems Ability to interpret numerical results Ability to implement numerical algorithms efficiently inMatlab

Indispensable: Learning by doing ( exercises)NumCSE, Lecture 1, Sept 18, 2014

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Numerical Methods for Computational Science and EngineeringAbout this course

LiteratureUri Ascher & Chen Greif: A First Course in NumericalMethods. SIAM, 2011.http://www.siam.org/books/cs07/

NumCSE, Lecture 1, Sept 18, 2014

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Excellent reference.

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Main reference for large parts of thiscourse.

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Target audience: undergraduate studentsin computer science.

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I will follow this book quite closely.

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Numerical Methods for Computational Science and EngineeringAbout this course

Literature (cont.)W. Dahmen & A. Reusken: Numerik fur Ingenieure undNaturwissenschaftler, Springer, 2006.A lot of simple examples and good explanations, but also rigorous mathematical treatment. Targetaudience: undergraduate students in science and engineering.

H.-R. Schwarz & N. Kockler: Numerische Mathematik.Teubner, 2006. 6. Auflage.Easy to read. Target audience: undergraduate students in science and engineering.

C. Moler: Numerical Computing with Matlab. SIAM 2004.Good reference for some parts of this course; Target audience: Matlab users and programmers. Seehttp://www.mathworks.ch/moler/.

W.Gander, M.J. Gander, & F. Kwok: Scientific Computing.An introduction using Maple and Matlab. Springer 2014.

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Numerical Methods for Computational Science and EngineeringAbout this course

PrerequisitesEssential prerequisite for this course is a solid knowledge in linearalgebra and calculus. Familiarity with the topics covered in thefirst semester courses is taken for granted, see K. Nipp and D. Stoffer, Lineare Algebra, vdfHochschulverlag, Zurich, 5 ed., 2002. M. Gutknecht, Lineare algebra, lecture notes, SAM, ETHZurich, 2009.http://www.sam.math.ethz.ch/~mhg/unt/LA/HS07/. M. Struwe, Analysis fur Informatiker. Lecture notes, ETHZurich, 2009.

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Numerical Methods for Computational Science and EngineeringOrganization

OrganizationLecturer:Prof. Peter Arbenz

arbenz@inf.ethz.ch

Assistants:Daniel HuppChristian SchullerAlexander LobbeSharan JagathrakashakanAlexander BohnManuel MoserFabian ThuringTimo Welti

huppd@inf.ethz.chschuellc@inf.ethz.chalobbe@student.ethz.chsharanj@student.ethz.chabohn@student.ethz.chmoserman@student.ethz.chthfabian@student.ethz.chweltit@student.ethz.ch

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Numerical Methods for Computational Science and EngineeringOrganization

VenueClasses:Tutorials:

Mon 10.15-12.00 (CAB G11); Thu 10.15-12.00 (HG G5)Mon 13.15-15.00Thu 8.1

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