at Helsinki University of Technology. This type of seminar can not take place at KTH.
We compare
- OLD: before the IT-age: without computer
- OLD: unreasonable effectiveness of analytical mathematics in sciences: FALSE
- OLD: not understandable + not useful
- OLD = math education of today
- NEW: in the IT-age: with computer
- NEW: reasonable effectiveness of computational mathematics in sciences: TRUE.
- NEW: understandable + useful
- NEW = reform math of tomorrow
- Death and Rebirth of the Computing Engineer 1, 2, 3, 4.
We consider one major area of technology: fluid mechanics = experimental + mathematical:
- mathematical fluid mechanics = Navier-Stokes equations: turbulent solutions.
Facts:
- OLD: Cannot solve NS by analytical mathematics: turbulence
- NEW: Can solve NS using computational mathematics: turbulence.
Result:
- OLD: Incorrect cook-book formulas: useless: lift/drag mystery: flight mystery
- NEW: Correct computational solutions: useful: no mystery.
What:Why:How:Value
- OLD: analytical formulas: solve NS: pen/paper: useless
- NEW: simple analytical formulas + computation : solve NS : pen/paper + computer: useful.
NEW: Solve
(DE) du/dt = f(u,t) (NS)
by time-stepping. To learn:
- calculus: derivative
- integral = solution to du/dt = f(t)
- construct by time-stepping: u(new) = u(old) + k f(old)
- elementary functions = solutions of simple DE: exp, log, sin, ...
- linear algebra: solve AX=B
- discretization: Galerkin finite elements
- programming.
Mathematical fluid mechanics part of NEW mathematics education.
NEW mathematics education:
- condensation: fewer formulas
- expansion: general differential equations: richer applications
- effectivization: Ockhams Razor
- understanding: principles
- proof by construction step by step
- ability to simulate complex systems
- = simulation technology = technology for/with simulation
- = solve equations: convection/diffusion/reaction/wave...
- = computational fluid/solid/quantum/electromagnetics...
- construct - compute - analyze - understand - construct -
NEW calculus:
- constructive -- computational
- Lipschitz continuity: | u(x) - u(y) | < L | x - y |
- Derivative du/dx by linearization: | u(x) - u(y) - du/dx (x-y) | < C| x - y | | x - y |
- definitions of continuity + derivative without using limits
- bisection: limit
- fixed point iteration x = g(x): limit
- Newton f(x) = 0: limit
- Solve DE: du/dt = f(u,t) by time-stepping: universal
NEW linear algebra:
- vector, matrix, linear combination, basis
- analytic geometry
- Gaussian elimination
- Jacobi iteration
- eigenvalue
- orthogonality.
Mathematics = ??
- NEW: formulate + solve equation = simulation technology = engineering mathematics
- OLD :mathematics = art + beauty + logic + theorem + proof = ?
Theorem -- proof vs constructive proof - construction = theorem:
- NEW: proof by construction: theorem
- OLD: theorem: proof by ??
Constructive mathematics = computer game:
- engineering mathematics = simulation technology
- simulation technology = computer games
- math education = how to construct computer games.
Mathematical computer games:
- bouncing ball: geometry in 2d and 3d + visualization + interaction
- spring-mass system: Newton + Hooke law
- fluid mechanics: Newton law + Eulerian description
- solid mechanics: Hooke's law + Lagrangian description
- fluid-structure interaction: action games, sports.
- science games: quantum .... cosmology.
Summary:
- global competition requires engineering education reform: NOKIA?
- computer game math: active, interactive, think, do, learn, understand, open
- strong resistance to math education reform: passive, sit, look, confusion, closed
- now: collapse of traditional math education+ birth of reform math
- horrible time for classical analytical mathematics
- wonderful time for constructive computational mathematics!
- What: calculus + linear algebra + computer + programming
- Why: computer simulation of real world
- How: construct computer games.
Realization:
Inlägg
Professor,
SvaraRaderaI've had your Computational Differential Equations text on my desk for nearly a decade - with the Body and Soul books added more recently. Probably my favorite mathematical texts ever. (John Boyd's text on spectral methods comes in close behind.)
Every day I learn to appreciate further the general application of numerical and analytial mathematics to attack problems of mathematical physics in a generalized way rather than the narrow cook-book methods still taught in university today. (I think of this as I dread a viscous fluid flow lecture I will attend this afternoon where we are scheduled to review a handful of analytical solutions to the NS equations...)
Thanks for your work!
Good to hear. Thanks/Claes.
SvaraRadera