Dan Crisan - rtraba · PDF fileDan Crisan (Imperial College London) ... Feynman-Kac formulae...

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......A mathematician’s view on Asimov’s psychohistory

Dan Crisan

Imperial College London

Dan Crisan (Imperial College London) A mathematician’s view on psychohistory 23 January 2013 1 / 22

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Acknowledgements Acknowledgements

Collaborators:Alan Bain Alexandros Beskos Tom CassJean-Francois Chassagneux Martin Clark Pierre Del MoralFrancois Delarue Josha Diehl Peter DjuricArnaud Doucet Peter Friz Jessica GainesMalte Grunwald Kari Heine Ajay JasraNick Kantas Michael Kouritzin Tom KurtzChristian Litterer Terry Lyons Kai LiKonstantinos Manolarakis Joaquin Miguez Colm NeeOlasunkanmi Obanubi Harald Oberhauser Yoshiki OtobeSzymon Peszat Boris Rozovskii Sylvain RubenthalerNizar Touzi Nick Whiteley Jie Xiong

Phd students:Saadia Ghazali Konstantinos Manolarakis Olasunkanmi ObanubiColm Nee Kai Li Sean ViolanteWang Han Ivo Mihaylov Eamon McMurrayKollias-Liapis Spyridon Adam Persing Andrea GranelliMarcel Ogrodnik

My family:Alexander, Andrea and Cosette


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A simple example Heat transfer

A variety of large scale phenomena can be analyzed through their small scalecounterpart. A “simple” example:

Heat transfer

Macroscopic scale: Heat transfer through mechanisms grouped in threebroad categories: conduction, convection and radiation. Conduction is thetransfer of heat between substances that are in direct contact. The flow ofheat always occurs from a region of high temperature to another region oflower temperature.

Heat flow in paper

Microscopic scale: Thermal energy is related to the kinetic energy ofmolecules. The greater a material’s temperature, the greater the thermalagitation of its constituent molecules (manifested both in linear motion andvibrational modes). Kinetic energy is transferred from regions with greatermolecular kinetic energy towards regions with less kinetic energy.

Simulation of molecular kinetic energy evolution


MVI_2479.avi

bmplanar.avi

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Feynman-Kac formulae The Heat Equations

Microscopic level Macroscopic level

Brownian motionW = {Wt , t ≥ 0}

Heat equationu = {ut(x), t ≥ 0}

{∂tut = 1

2∆utu0 = Φ

Feynman-Kac Formula:

ut(x) = E [Φ(x + Wt)]


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Feynman-Kac formulae The Heat Equations


ut(x) = E [Φ(x + Wt)].

Microscopic level Macroscopic level

Numerical approximation: Exact solution:

ut(x) ≃n∑

i=1

Φ(x + W it ). ut(x) =

∫Φ(y)

1√2πt

e−∥y−x∥2

2t dy .

Malliavin differentiation (d=1): Standard differentiation (d=1):

dut(x)dx

=1t

E [ϕ(x + Wt)Wt ].dut(x)

dx=

1t

∫Φ(y)

(x − y)√2πt

e−∥y−x∥2

2t dy .


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Feynman-Kac formulae Semilinear PDEs

Can do this for a more general equation:∂tu = V0u +

12

d∑i=1

V 2i u + f (t , x , u,V1u, ...,Vdu) , t ∈ (0,T ], x ∈ Rm

u(0, x) = Φ(x), x ∈ Rm

, (1)

Viu, i = 0, 1...,d is the directional derivative of u in the direction Vi thatsatisfy the UFG condition.W the C∞

b (Rd )-module generated by the vector fields {Vi , i = 1, ...,N}within the Lie algebra generated by {Vi , i = 1, ...,N} is finitely generatedas a vector space and {V[α], α ∈ A0(m)} is a finite set of generators.


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Feynman-Kac formulae Semilinear PDEs


u(t , x) = E [Λt,x(W )] =

∫ω∈C([0,∞),Rd )

Λt,x(ω)dPW (ω).

Microscopic level Macroscopic levelNumerical approximation: No exact solution• replace PW with PW = 1

n

∑ni=1 δωi

• approximate Λt,x with a simple version Λt,x• control the computational effortMalliavin differentiation No standard differentiation

Viut(x) =1t

E [Λit,x(W )]

V0u =12

d∑i=1

V 2i u + f (t , x , u,V1u, ...,Vdu) V0 := ∂t − V0

• u not necessarily time differentiable• temporal variable cannot be distinguished from the space one• u remains differentiable in the direction V0,V1....Vd .


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Feynman-Kac representations Numerical Implementation

Let u ∈ C1,2([0,T ]× R4) be the solution of the Cauchy problem∂tu =

4∑i=1

(µixi

dudxi

+σ2

i x2i

2d2udx2

i

)+ f (t , x , u,∆u), t ∈ (0,T ], x ∈ R4

u(0, x) = Φ(x), x ∈ R4

,

wheref (t , x , y , z) = −ry −

∑4i=1(µi − r)zi , x , z ∈ R4.

Φ(x) = (x1x2x3x4 − k)+.


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Feynman-Kac representations Common Feature

Common feature of many PDEs: their solutions can be represented asintegrals of certain nonlinear functionals with respect to the Wiener measure.

Feynman-Kac formula

u(t , x) = E [Λt,x(W )] =

∫ω∈C([0,∞),Rd )

Λt,x(ω)dPW (ω)

Microscopic level Macroscopic level Timeline

Brownian motionW = {Wt , t ≥ 0}

Heat equation

{∂t ut = 1

2 ∆utu0 = Φ

Feynman 1948 Kac 1949

Zakai equation dut = Lut + hut dYt Duncan, Mortensen, Zakai 1970

McKean-Vlasov PDEs∂t ut =

∑di,j=1 aij (ut )∂i∂j ut

+∑d

i=1 bi (ut )∂i ut + c(ut )utGartner 1988

Semilinear PDEs{

∂t ut = Lut + f(t, x, ut ,∇ut

)u0 = Φ

Pardoux & Peng 1990, 1992

Fully Nonlinear PDEs F(t, x, ut ,∇ut ,∆ut ) = 0 Soner, Touzi & Victoir 2007

3 − d incompressibleNavier − Stokes equation

{∂t ut + (ut · ∇)ut − ν∆ut + ∇p = 0

∇ · ut = 0 Constantin & Iyer 2008

K-S equation dut = Lut dt + ut (h)dYt − ut (h)ut (h)dt Crisan & Xiong 2009

viscous Burgers equation ∂t ut + ut∂x ut − ν∂2x ut = 0 Novikov & Iyer 2010


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The Stochastic filtering Problem The filtering problem

Stochastic Filtering

(X ,Y ) = {(Xt ,Yt), t ≥ 0}X the signal process - “hidden component”Y the observation process - “the data” - Yt = ft(X , “noise”).

The filtering problem: Find the conditional distribution of the signal Xt givenYt = σ(Ys, s ∈ [0, t ]), i.e.,

ϑt (φ) = E[φ(Xt)|Yt ], t ≥ 0, φ ∈B(Rd ).

The model

Xt = X0 +

∫ t

0b (Xs)ds +

∫ t

0σ (Xs) dVs

Yt =

∫ t

0γ (Xs) ds + Ut ,


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The Stochastic filtering Problem The SPDE

The Kallianpur-Striebel formula

ϑt (φ) =ρt (φ)

ρt(1),

ρ = {ρt , t ≥ 0} satisfies the Duncan-Mortensen-Zakai equation:

dρt(x) = A∗ρt(x)dt + ρt(x)

(m∑

k=1

γk (x)dY kt

)(2)

where

Aφ (x) =12

d∑i,j=1

aij (x) ∂i∂jφ (x) +d∑

i=1

bi (x) ∂iφ (x) a = σσ⊤

The Feynman-Kac formula

ρt (φ) = E

[φ(Xt) exp

(∫ t

0

m∑k=1

γk (Xs) dY ks − 1

2

∫ t

0

m∑k=1

γk (Xs)2 ds

)∣∣∣∣∣Yt

](3)


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The Stochastic filtering Problem The SPDE

Car tracking: Extracting information from video images

Signal (vt , vt , θt , xt , yt , τt) :• (xt , yt) be the coordinates of the car (the center of the rear axle),• vt be its tangential velocity (vt is the rate of change of vt ),• τt its orientation (the angle between the direction of the car and the xt -axis)• θt the steering angle (the angle between the direction of the car and thedirection of the front wheels).

dvt = αdW 1t

dvt = vtdtdθt = −βθtdt + γdW 2

tdxt = vt cos(τt)dtdyt = vt sin(τt)dtdτt = a−1vtθtdt

(4)

Observation (xt , yt , τt) rough estimations of (xt , yt , τt) obtained by a visualsearching algorithm.

dxt = vt cos(τt)dt + δ3dW 3t

dyt = vt sin(τt)dt + δ4dW 4t

d τt = a−1vtθtdt + δ5dW 5t

(5)


new.mov

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Navier-Stokes equation Framework

2D Stochastic Navier-Stokes equationTorus T2 , [0, L)× [0, L) with periodic boundary conditions:

∂u∂t

− ν∆u + u · ∇u +∇p = f + W (t , x) for all (x , t) ∈ T2 × (0,∞), (6)

∇ · u = 0 for all (x , t) ∈ T2 × (0,∞),

u(x , 0) = u0(x) for all x ∈ T2.

2d N-SDan Crisan (Imperial College London) A mathematician’s view on psychohistory 23 January 2013 13 / 22

chaotic4.avi

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u : T2 × [0,∞) → R2 - the velocityp : T2 × [0,∞) → R2 - the pressuref : T2 → R2 - the forcingW (t , x) - noise

H ,{

L − periodic trig. pol. u : [0, L)2 → R2∣∣∣∇ · u = 0,

∫T2

u(x)dx = 0}L2(T2))2

P : (L2(T2))2 → H - the Leray-Helmholtz orthogonal projector. Anorthonormal basis for H is given by

ψk (x) ,k⊥

|k |exp

(2πik · x

L

)k = (k1, k2)

⊤ ∈ Z2 \ {0} k⊥ = (k2,−k1)⊤.


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For u ∈ H .u =

∑k∈Z2\{0}

uk (t)ψk (x).

W (t , x) =∑

k∈Z2\{0}

εkψt(x)W kt ∈ H

{W kt }(t≥0, k∈Z2\{0}) i.i.d. Brownian motions and∑

k∈Z2\{0}

(4π2|k |2)sε2k <∞ for s ∈ R,

and then W (t , ·) ∈ H.The stochastic Navier-Stokes equation can be written as

dudt

+ νAu + B(u, u) = f + W (t , x). (7)

• A = −P∆ is the Stokes operator• B(u,u) = P(u · ∇u)• f is the original forcing projected into H.


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The equations for the modes:

duk (t) =

−νλk uk (t)− αl,jk

∑l+j=k

ul(t)uj(t) + fk

dt + εk dW kt .

αl,jk =

{2πi(l2 j1−l1 j2)(k1 j1+k2 j2)

L|k||l||j| if k = l + j ,0 otherwise;

Define the projection operators Pλ : H → H and Qλ : H → H by

Pλu =∑

k∈Z2\{0}|2πk|2<λL2

uk (t)ψk (x), Qλ = I − Pλ;

and consider the projected eigenvalues, we obtain the following evolutionequation for the approximation of uk (t), which is denoted by uk (t), for eachk ∈ Z \ {0} with |2πk |2 < λL2:

duk (t) =

(−νλk uk (t)− αl,j

k

∑Γ

ul(t)uj(t) + fk

)dt + εk dW k

t ; (8)

where the set Γ ,{(l , j)

∣∣∣l + j = k and |2πl |2 < λL2 and |2πj |2 < λL2}

.


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Model parameters

• we use k1, k2 = −32, ...,0, ...32 (i.e. a 642 grid for the discrete fouriercomponents).• Smoothing problem approximate p(x0|y1:5) where each yi is a 4x4 grid onthe torus and

yi(j) = u(xj , ti) + N(0, 0.2).

• the dynamics are initialised by a random sample from the prior N(0, δAα)• for the prior, δ = 5 and α = 2.2.• torus size is 2π.• forcing is ∇cos(κ · x) with κ = (1, 1) for the stationary regime and κ = (5, 5)for the chaotic• ν is 1/50 for chaotic and 1/10 for stationary

MCMC plot: computation cost roughly 105 iterations per day, i.e. with the slowmixing need more than 10 days for a decent but not super-reliable answer.

SMC plots: computational cost around 18 hours for N=500 particles and 5intermediate steps.


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Asimov′s phsychohistory

“Phychohistory: Branch of mathematics which deals with the reactions ofhuman conglomerates to fixed social and economic stimuli. With refinedknowledge of group psychology, the future can be extrapolated from thepresent. Phychohistory cannot work with individuals over any lengths of timeany more than you can apply the kinetic theory of gases to single molecules.”

Nobel laureate Paul Robin Krugman said that his interest in economics beganwith Asimov’s Foundation novels, in which the social scientists of the futureuse ’Psychohistory’ to attempt to save civilization. Since ’Psychohistory’ inAsimov’s sense of the word does not exist, Krugman turned to economics,which he considered the next best thing.


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It would be a wonderful achievement to be able to study the present and thenaccurately predict what will happen in the future.

Mathematics has come of age:good understanding of how to model noise/system uncertainty:stochastic processes (Brownian motion, Levy process), rough controlsmodels for “noisy” dynamical systems: SDEs, SPDEs, RDEsrigorous analysis beyond classical results: stochastic calculus, Malliavincalculus, stochastic geometry, theory of rough pathsaveraging over individual evolutions to give the overall picture: strong lawof large numbers, central limit theorems, large deviationscan incorporate observed behaviour: stochastic filteringcan model interactive behaviour individual-individual, individual-society:stochastic networks, McKean-Vlasov processescan handle multi-scale (slow/fast) phenomena: homogenization theory...


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A mathematics model for the evolution of the human society:

how do we model individual/family/nation behaviour evolutionwhat are the human characteristics that influence societyhow do we distinguish between common traits and unique qualitieshow do we identify social and political patterns......

Not a mathematics-only problem: Understanding the complexities of humansociety requires major input from many other disciplines: sociology, history,economics, politics, etc.

The ingredients are available, all that we need now is the recipe.


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