Math 8441 Numerical Analysis and Scientific Computing Matthias - - PowerPoint PPT Presentation

Math 8441 – Numerical Analysis and Scientific Computing

Matthias Maier

Introduction

Wednesday September 7, 2016

1 Introduction | Math 8441

Math 8441 – Syllabus

E-mail: msmaier@umn.edu Office: 331 Vincent Hall Office hours: We 1:30 – 2:30, Fr 1:30 – 3:30 Website: http://www.math.umn.edu/~msmaier/math8441

• Dr. Arnold’s lecture notes:

https://www.ima.umn.edu/~arnold/8441-8442.15-16/class-notes/notes.pdf › Bi-weekly homework assignments

(due on 9/16, 9/30, 10/14, 10/28, 11/11, 11/23, 12/9)

› Midterm (in class), November 4, 11:15 AM – 12:05 PM › Take-home Final due December 14 (hand-out December 7)

HW / Midterm / Final – 600 / 150 / 250

2 Introduction | Math 8441

Math 8441 – Syllabus

. . . and Scientific Computing

A part of this course will also be about numerical algorithms and concrete numerical computations.

› We will use Python in class. › Does someone want to use something else?˜

This Friday & first homework assignment

› (Install a Python environment) › A first example

3 Introduction | Math 8441

So, what is it all about?

»Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis.« 1 »[Scientific computing] is [. . . ] the application of computer simulation and other forms of computation from numerical analysis and theoretical computer science to solve problems in various scientific disciplines.« 1

1Wikipedia

4 Introduction | Math 8441

Roadmap / Example

The next two semesters in 20 minutes. . .

5 Introduction | Math 8441

Roadmap / Example

An example – the diffusion equation

Find u : [0; 1]2 ! R such that ` 1 »

„ @2

@x 2

1

+ @2 @x 2

2

«

u = f ; (boundary conditions. . . )

6 Introduction | Math 8441

Roadmap / Example

. . . analysis tells us

In a certain sense equivalent

Find u 2 [: : : ] such that 1 » ˆ

[0;1]2

„ @u

@x1 @’ @x1 + @u @x2 @’ @x2

«

d2x = ˆ

[0;1]2 f ’ d2x

8’ 2 [: : : ] ` ! (Analysis, existence and uniqueness)

7 Introduction | Math 8441

Roadmap / Example

Instead of [0; 1]2 approximate on a grid with finitely many vertices: ` ! Discretization What to do with

› functions?, g ! gh; G

` ! interpolation, approximation

› integrals?,

´

[0;1]2 g ! I g

` ! numerical integration

8 Introduction | Math 8441

Roadmap / Example

. . . after a bit . . .

Discretized problem

Find uh 2 [: : : ] such that 1 » I

„@uh

@x1 @’h @x1 + @uh @x2 @’h @x2

«

= I

“

f ’h

”

8’h 2 [: : : ] That’s a system of linear equations! ` ! numerical linear algebra

9 Introduction | Math 8441

Roadmap / Example

Solution uh with 64 unknowns:

10 Introduction | Math 8441

Roadmap / Example

Solution uh with 256 unknowns:

10 Introduction | Math 8441

Roadmap / Example

Solution uh with 1024 unknowns:

10 Introduction | Math 8441

Roadmap / Example

Solution uh with 4096 unknowns:

10 Introduction | Math 8441

Roadmap / Example

Solution uh with 16384 unknowns:

10 Introduction | Math 8441

Roadmap / Example

Solution uh with 65536 unknowns:

10 Introduction | Math 8441

Roadmap / Example

Solution uh with 262144 unknowns:

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Roadmap / Example

Solution uh with 1048576 unknowns: ` ! error analysis

10 Introduction | Math 8441

So, what is it all about?

Numerical analysis

› Error and error propagation › Lagrange interpolation and approximation › Numerical integration › Systems of nonlinear equations and optimization › Numerical solution of ordinary differential equations › Numerical solution of partial differential equations › Iterative methods of numerical linear algebra › ( . . . )

11 Introduction | Math 8441