Math: the art in problem solving - UCLA Department of ...mikel/UGrad_Colloquium_2014.pdf · Math: the art in problem solving Michael Lindstrom Introduction Techniques Modelling Nondimensionalization

Math: the artin problem

solving

MichaelLindstrom

Introduction

TechniquesModelling

Nondimensionalization

Asymptotics

Numerical Analysis

ProblemsElectrodialysis

Magnetized TargetFusion

Mass Spectrometry

Summary

Math: the art in problem solving

Michael Lindstrom

October 22, 201415:00-16:00

email: [email protected]: math.ubc.ca/∼MLRTLM


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Math is everywhere

quantifiable descriptionsglacier motion, spreading of diseases, market inflation,digitally recorded music, ...real world → model → abstractify → analyze → interpretmath is useful


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Outline

applied math technique introselectrodialysis applicationfusion applicationmass spectrometry applicationconcluding remarks


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Modelling Overview

“sketch” relevant physics, biology, etc.assign variables, write equationsinsightful results


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Modelling Example

finding the dimensions of a cylindrical tank with open tophaving a volume of 1000 cm3 with least possible surfacearealet h =height, r =radius; need πr2h = 1000 withminimum value of 2πrh + πr2

cheaper than building thousands of cylinders!


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Modelling Example

finding the dimensions of a cylindrical tank with open tophaving a volume of 1000 cm3 with least possible surfacearealet h =height, r =radius; need πr2h = 1000 withminimum value of 2πrh + πr2

cheaper than building thousands of cylinders!


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Nondimensionalization Overview

make “diorama”remove dimensions, reduce constantskey relationshipsimproved numericsall variables comparable: velocity, pressure, time, etc.


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Nondimensionalization Example

rabbit population P has carrying capacity C satisfiesP ′(T ) = rP(1− P/C) at time T , initial populationP(0) = P0

units: [P] = [C ] = [P0] = rabbits, [r ] = 1/time, [T ] =timenondimensionalize to

p′(t) = p(1− p), p(0) = p0

qualitative behaviour: p0


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Asymptotics Overview

mathematical “impressionism”exploit relative smallness of parameter(s)approximate model problem solution accurately

solution = estimate + correction + smaller correction + ...

analytic insight


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Asymptotics Example

consider 0.0001x3 − x + 1 = 0; let ε = 0.0001� 1asymptotic prediction:

x ∼ 1 + ε+ ...

x ∼ 1√ε− 1

2 + ...

x ∼ −1√ε− 1

2 + ...

Asymptotic Exact1.0001... 1.0001...

99.5... 99.496...-100.5... -100.496...


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Numerical Analysis Overview

“digital snapshot”arbitrarily high precisioncontrol error termsmay require high computational power


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Numerical Analysis Example

evaluate I =∫ 1

0√

x5 + 1dxRiemann sum: let N > 0, h = 1/N and xj = jh so thatI = h

∑Nj=1

√x5

j + 1 + error

theory: |error| ≤ some constantN → 0 as N →∞

N Sum10 0.999

100 1.0661000 1.0738

10000 1.0746100000 1.0747

better methods exist!


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Electrodialysis Motivation

clean water and CO2 sequestration goodmany pollutants ionicelectrodialysis: filter out impure ions from water, makeuseful products with electric field and selective membranes


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Problem

UBC chemical engineers building prototypes, operationalconditions important

Given experimental data of input concentrations, the totalvoltage and the total current drawn across the series ofchannels, what losses can be attributed to each individualchannel and what is taking place in each channel?


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Problem

UBC chemical engineers building prototypes, operationalconditions importantGiven experimental data of input concentrations, the totalvoltage and the total current drawn across the series ofchannels, what losses can be attributed to each individualchannel and what is taking place in each channel?


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Carbonic Acid Channel

given voltage drop V , determine the current I


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Models

model 0: uniform [H+] and [HCO−3 ], current driven byvoltage gradient - wrong species cross membranes

model 1: spatially varying [H+] and [HCO−3 ] withoutreactions - too few ionsmodel 2: with reactions

but ions not produced fast enough #fail


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Models

model 0: uniform [H+] and [HCO−3 ], current driven byvoltage gradient - wrong species cross membranesmodel 1: spatially varying [H+] and [HCO−3 ] withoutreactions - too few ions

model 2: with reactions



solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Models

model 0: uniform [H+] and [HCO−3 ], current driven byvoltage gradient - wrong species cross membranesmodel 1: spatially varying [H+] and [HCO−3 ] withoutreactions - too few ionsmodel 2: with reactions



solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Models

model 3


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Models

numerics on asymptotic model more consistent withexperimental datadiscovered device doesn’t operate the way intended!

“... all models are wrong, but some are useful.” - GeorgeBox


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Models

numerics on asymptotic model more consistent withexperimental datadiscovered device doesn’t operate the way intended!“... all models are wrong, but some are useful.” - GeorgeBox


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Fusion Motivation

fusion energy in sun31H + 2

1H → 42He + 1

0n + 17.6 MeV and other reactionsclean energysustain high particle density and temperaturefusion on earth?magnetized target fusion at General Fusiondesign: compress plasma in imploding shell of lead-lithium,driven by fast high-pressure impulses


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Problem

How much will the plasma compress under such a design?


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Problem

How much will the plasma compress under such a design?


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Simple Dimensionless Model

spherical symmetryhomogeneous plasmanonlinear, coupled system of PDEs: mass and momentumconservationfree boundaries - need to solve for radii within problemnondimensionalization

small initial plasma radiusvery small initial density disturbancevery small impulse timevery, very small initial plasma pressure


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Insights Obtained

wave-like behaviour: sound speed dominatespressure amplitude grows like 1/ralmost all input energy reflected!fusion may be within reach - Lawson condition


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Insights Obtained

minimum radius R∗L :

R∗L ≈C4

s Pplasma,0R7inner,0%

30

πP4maxR4

outer,0T 20

= 4mm

agrees well with numerical methodsParameter Meaning

Cs lead sound speed at atmospheric pressurePplasma,0 initial plasma pressureRinner,0 initial plasma radius%0 lead density at atmospheric pressure

Pmax piston impulse pressureRouter,0 initial lead radius

T0 impulse time scale


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Spectrometry Motivation and Problem

detect chemical species/isotopes with masscharge-to-mass separation in mass spectrometersome current means

magnetic fields costlytime of flight discretequadrupole too specific

PerkinElmer: come up with a design for amass-spectrometer without magnetic fields, withcontinuous measurement and which can detect any mass.


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Spectrometry Motivation and Problem

detect chemical species/isotopes with masscharge-to-mass separation in mass spectrometersome current means

magnetic fields costlytime of flight discretequadrupole too specific

PerkinElmer: come up with a design for amass-spectrometer without magnetic fields, withcontinuous measurement and which can detect any mass.


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Concepts

acceleration in electric field proportional to charge-to-massratio and field strengthpositive charge oscillates between fixed positive charges(one-dimensional)same amplitude and different masses, different periods


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Idea

fire ion beam into tube of chargespredict qualitative trajectories via asymptoticsoscillation wavelength mass-dependentnumerics of ODEs verify distinct masses spatially separate


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Results


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Art

each technique has rolecombination often usefulresults extremely beautiful/satisfying


solving

MichaelLindstrom

Introduction

TechniquesModelling


Asymptotics

Numerical Analysis



Mass Spectrometry

Summary

Final Remarks

thanks to many collaborators on these projects:- Arman Bonakdarpour, Saad Dara, and David Wilkinson

(UBC Chemical and Biological Engineering Department)- Sandra Barsky and Aaron Froese (General Fusion)- Iain Moyles, Michael Ward, and Brian Wetton (UBC Math

Department)- Samad Bazargan (PerkinElmer)- others

some suggested applied courses:- MATH 215, 316, 400, 401 (differential equations)- MATH 361 (modelling, math-bio)- MATH 210, 405 (numerics)- MATH 450 (asymptotics)- some non-math science courses!

slides posted at: math.ubc.ca/∼MLRTLM/#talks

Documents

Math: the art in problem solving - UCLA Department of ...mikel/UGrad_Colloquium_2014.pdf · Math: the art in problem solving Michael Lindstrom Introduction Techniques Modelling Nondimensionalization