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 1 Wikipedia 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 „ @ 2 + @ 2 ` 1 « u = f ; @ x 2 @ x 2 » 1 2 (boundary conditions. . . ) 6 Introduction | Math 8441
Roadmap / Example . . . analysis tells us In a certain sense equivalent Find u 2 [ : : : ] such that „ @ u 1 + @ u ˆ @’ @’ ˆ « d 2 x = [ 0 ; 1 ] 2 f ’ d 2 x 8 ’ 2 [ : : : ] @ x 1 @ x 1 @ x 2 @ x 2 » [ 0 ; 1 ] 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 ! g h ; G ! interpolation, approximation ` › integrals?, ´ [ 0 ; 1 ] 2 g ! I g ` ! numerical integration 8 Introduction | Math 8441
Roadmap / Example . . . after a bit . . . Discretized problem Find u h 2 [ : : : ] such that 1 „ @ u h @’ h + @ u h @’ h « “ ” » I = I f ’ h 8 ’ h 2 [ : : : ] @ x 1 @ x 1 @ x 2 @ x 2 That’s a system of linear equations! ` ! numerical linear algebra 9 Introduction | Math 8441
Roadmap / Example Solution u h with 64 unknowns: 10 Introduction | Math 8441
Roadmap / Example Solution u h with 256 unknowns: 10 Introduction | Math 8441
Roadmap / Example Solution u h with 1024 unknowns: 10 Introduction | Math 8441
Roadmap / Example Solution u h with 4096 unknowns: 10 Introduction | Math 8441
Roadmap / Example Solution u h with 16384 unknowns: 10 Introduction | Math 8441
Roadmap / Example Solution u h with 65536 unknowns: 10 Introduction | Math 8441
Roadmap / Example Solution u h with 262144 unknowns: 10 Introduction | Math 8441
Roadmap / Example Solution u h 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
Recommend
More recommend
