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- Stochastic Modeling for Pollution Tracking
- Simulations
- Filtering in a Random Environment
- Overview of Center

1. Description Of Water Pollution Problem

- Factories along river or groundwater system
- Undesired chemicals or bacteria released by each factory at random times
- Chemicals initially distributed according to some proportional function
- Contaminants react, drift, and disperse through water sheet
- Quality of water ? Location of chief polluters ? How to predict the transport of the contaminants?

... where :

- u(t,x) - concentration of contaminants
- D > 0 - dispersion rate
- V - water velocity or drift rate
- R - birth and death of bacteria or adsorption of chemicals
- Aij - random contaminant deposits
- ij - random release times
- u0(x) - initial contaminant concentration
- b.c. - dispersive flux across the boundary = contaminant concentration does not change at boundary

3. Filtering Problem for Signal Process u(t,x)

- Observations - sample pollution at discrete well sites or average of lake: Yk = h(utk , Vk)
- Optimal filter: i.e best guess E [f(u(tk))|Yk]
Yk= σ {Y1 ,...,Yk}

- Find uN approximating u
- Calculate the approximate filter:
- E [f(uN(tk))|Yk]

4. Construction Of Markov Chain

- Divide the region [0,L1] x [0,L2] into L1N x L2N cells
- Construct discretized operators N and N
- nk(t) - the number of particles in cell k at time t
- {nk (t)} is modeled as a Markov chain
- Particles evolve according to births and deaths from reactions, random walks from diffusion and drift, area dependent births from Poisson noise source
- l-1 is mass of each particle

The Approximate Markov Process

- is given by :
- Semi-group and martingale theories can be used to analyze the mathematical structure of the Markov chain

5. The Law Of Large Numbers

For both quenched and annealed approaches:

uN(t) converges to u(t)and N uN(t) converges to u(t)

in distribution sense as N

Quenched approach: evolving Markov chains for each fixed path of the driving Poisson source

Annealed approach: considering the Poisson source as a random medium for the Markov chains

Results in filtering theory state that we can construct approximate filters from uN

The estimation of u(t, x) can help locate the polluters

- Thomas Kurtz introduced and studied this type of Markov chain approximation for ODE’s. (1971)
- Arnold and Theodosopulu extended model to the partial differential equations of chemical reactions. (1980)
- Peter Kotelenez established high density limits for model. (1988)
- Douglas Blount established a general technique for establishing crucial estimates in these models. (1991, 1994)
- Kouritzin and Long revised model to i) speed up the implementation, ii) allow convection, iii) allow more general nonlinearities, and iv) Poisson-measure driving noise. New analysis methods were required.

THE DINGHY PROBLEM

- A dinghy is lost at sea
- The dinghy moves randomly
- We know the underlying stochastic model
- We only have very noisy observations from a high-altitude sensor
- We want to track the location and the orientation of the dinghy
- The goal rescue the dinghy’s occupants

Nonlinear Filtering for Diffusions in Random Environments

- Background for Signal Process
- • Motivation: tracking problem of a dinghy lost at sea.
- • Formal SDE for the motion of the dinghy:

dXt = - 1 a (Xt) W (Xt)dt + b(Xt)dt + (Xt)dBt,

2

where W is a real-valued random field on Rd that can be nowhere differentiable d

a = T and b = (b1,…,bd) with bi = jaij.

j =1

• B,W are independent random sources

Example: dXt = b(Xt) dt + (Xt) dBt models motion of dighy itself –½ a(Xt)W(Xt)dt brings in effect of the waves.

• Find Solution (Xwt, Pwt)via Dirichlet form theory,

u, vD(w)

w (u,v) = Rd < a(x) u (x), v(x) > e-w(x)dx/2,

on the Hilbert space H =L2 (Rd; e-w(x)dx) with domain D(w) = {u H: | a 1/2u| H}.

• (W,(W),Q) models the random environment. Consider the law Pwf of Xwt with initial law f(x)dx as a probability measure on C (R+, Rd). The mapping (W,f) W L1+ (Rd) Pwf is measurable.

·

Example: W(x) = W((0,x]) is a Brownian sheet = a zero mean Gaussian field with W(A), W (B) independent when A B = , W (AB) = W (A) + W (B) and E(W(A))2 = 2 (A).

- Construction of diffusion in random environments:
- Let f be our initial random probability measure on Rd.
- We define a new probability measure on C (R+, Rd) by Pwf (A) = EQ [Pwf (A)].
- Then (Xt, t ) is a diffusion in the random medium W with initial law f if its law is Pwf .

- (Xt, t ) is not Markov!
- Can we come up with an equation for Pwf(X[0,t]dx| observations up to t)? i.e. Is there a useful filtering equation?

• The signal process is the “diffusion” in a random medium defined as above.

• The observation model:

Yt = t0 h(Xs)ds + Vt.

• To calculate the optimal filter E [ (X[0,t])|t], t = {Ys, st}]

3. Filtering for Historical Processes: Quenched Approach

• Fix W W , consider the historical process Xw[0,t] (s)= Xwts , s R+ and calculate the optimal pathspace filter E [(Xw[0,t])| wt].

• Let P0 be a new probability. measure that turns Y into a Brownian motion and w[0,t] () = E0 [ (Xw[0,t]) At| wt],

At = exp{t0h(Xs)dYs – ½ t0|h(Xs) |2ds}

• Obtain an Zakai equation for the unnormalized pathspace measure valued filtering process.

3. Filtering for Historical Processes: Quenched Approach (continued)

• Solution is given in terms of multiple Wiener-Ito’s integrals.

w[0,t] () = Rd Ttw, 0(x)f(x)dx+ t0[ Rd Tt1w, 0 (Uw,t1t-t1)(x)f(x)dx]dYt1

+ t0t10[ Rd Tt2w, 0 (Uw,t2t1-t2 (Uw,t1t-t1)) (x) f(x)dx] dYt2 dYt1 + …

• Here Ttw, 0 is the evolution operator for X under P0 and U is an associated operator.

• Measurability of the joint distribution of (Xw, Yw) with respect to W.

4. Filtering for Diffusions in Random Environments: Annealed Approach.

• There is no known stochastic evolution equation for the filtering process associated with the diffusion in random medium based on the noisy observation.

• The random environments are not accessible, which must be averaged. out (i.e annealed approach):

[0,t] () = w [0,t] ( )Q(dW) T0tg = Tw,0tg Q(dW)

• From the measurability we get the chaos expansion for the filtering process associated to our filtering model:

[0,t] () = Rd T0t(x) f(x)dx + t0[ Rd T0t1(Uw,t1t-t1 ) (x) f(x)dx] dYt1

+ t0 t10 [ Rd T0t2(Uw,t2t1-t2(Uw,t1t -t1 ))(x)f(x)dx]dYt2dYt1+ …

• Truncate expansion and approximate the stochastic integrals Implementation method

5. Mathematical Background. Annealed Approach.

- Brox and Schumacher independently introduced the one-dimensional diffusion in random medium using Ito-McKean time change techniques. (1986)
- Tanaka showed this model is recurrent. (1993)
- Many authors have established properties like self-similarity.
- Mathieu introduced the multidimensional model using Dirichlet forms and studies the behaviour as the amplitude of B goes to zero. (1994)
- Kouritzin, Long, and Sun estimate the paths of such processes based on corrupted, distorted, partial observations.

Brief Introduction to the Alberta Branch of MITACS-PINTS Center

- Postdoctoral Fellow and Graduate Students
- Dr. Hongwei Long (PIms-MITACS Industrial PDF)
- Dr. Wei Sun (PIms-MITACS Industrial PDF)
- David Ballantyne (Graduate Student)
- Calvin Chan (Undergraduate Student)
- Hubert Chan (Graduate Student)
- Michelle Prefontaine (Graduate Student)
- Paul Wiebe (Graduate Student)

- Simulation Front Center
- Branching particle filtering; path-space filter, combination of filtering and parameter estimations, application to advanced historical tracking of dinghy lost at sea. (By David Ballantyne, Hubert Chan and Michael Kouritzin).
- Convolutional filters: application to a filtering model for mean reverting stochastic volatility using Levy driven prices as observation. (By Paul Wiebe, Michelle Prefontaine and Calvin Chan)
- Markov chain approximation: application to water pollution model which is characterized by a stochastic reaction-diffusion equation. (By David Ballantyne and Hubert Chan).

- Theoretical Front Center
- Uniqueness and weak convergence of solutions of the nonlinear filter equations (by Andrew Heunis and Vladimir Lucic, U. Waterloo): study the distributional uniqueness and weak convergence of the (normalized) filter equations.
- Convergence of Markov chain approximation to stochastic reaction diffusion equations. (by Michael Kouritzin and Hongwei Long).
- Nonlinear filtering for diffusions in random environments (by Michael A. Kouritzin, Hongwei Long and Wei Sun).

3. Theoretical Front (continued) Center

(iv) Holder continuity for spatial and path processes via spectral

analysis. (by D. Blount and Michael Kouritzin).

(v) On a class of discrete generation interacting particle systems.(by P. Del Moral, Micheal Kouritzin and L. Miclo).

(vi) A pathspace branching particle filter. (by Michael Kouritzin).

(vii) Rates for branching particle approximation of continuous-

discrete filters. (by D. Blount an Michael Kouritzin).

(viii) Convolutional filters. (by Micheal Kouritzin and Paul Wiebe).

- Sponsors and their interests Center
(i) Lockheed Martin Naval Electronics and Surveillance System: surveillance and tracking, search and rescue, anti-narcotic smuggling, air traffic management, and global positioning.

(ii) Lockheed Martin Canada, Montreal: same interests as above.

(iii) VisionSmart, Edmonton: Quality control of industrial processes such as real time analysis of oriented strand board (OSB) density variations using thermography techniques and pattern recognition of naturally occurring substances etc.

(iv) Acoustic Positioning Research Inc., Edmonton: Track stage performers using acoustic techniques and adjust lighting, sound effects. Create performer-movement- controlled.

(v) Stantec (future): environmental monitoring, pollution tracking.

5 Center. Ideas for the future

(i) Filtering by Markov chain approximation.

(ii) Tracking and estimation of bacteria and other species

(iii) Filtering when observations in random environments.

(iv) Implement chaos method and try out with random

environments.