• Two State Broadwell Model
Let v >0 be a constant velocity, F(x) be a flip rate function, w1(x,t) and w2(x,t) represent exit time distributions conditioned on the particle initially being at position x and having positive (+v) and negative (-v) velocity respectively. The Broadwell PDEs are
$$
\frac{\partial w_1}{\partial t}=v\frac{\partial w_1}{\partial x}+F(x)(w_2-w_1)\\
\frac{\partial w_2}{\partial t}=-v\frac{\partial w_2}{\partial x}+F(x)(w_1-w_2)
$$
subject to initial conditions
\begin{equation}
w_1(x,0) = 0, \qquad w_2(x,0) = 0. \label{eqn:wic}
\end{equation}
and boundary conditions
\begin{equation}
w_1(x=L/2,t) = \delta(t), \qquad w_2(x=-L/2,t) = \delta(t). \label{eqn:wbc}
\end{equation}


 

BroadwellProcess3-eps

 Figure 1. Random walk for Broadwell process 

  • Monte Carlo Simulation of Broadwell Process

The following figure represents the exit time distribution w1(x,t) of the random walk described in Figure 1.
P3 (Pak-Wing Fok's conflicted copy 2012-02-13)

 Figure 2. Simulated exit times distribution of a Broadwell process. (a) F(x)=10x3+5ex+1, v=1/2 (b) F(x)=1+x2, v=1.
Insets show Laplace Transformed data. N=40,000 realizations was used in each case.

 

  • Projection Method to Solve Inverse Problem
Let W1,2(x,t) be the cumulative density functions. The inverse problem of interest is, given  Monte Carlo simulated data w1,data(x,t) and w2,data(x,t) for t>0 and fixed -L/21(x,s) and \tilde w2(x,s) be Laplace transforms of w1 and w2.  Two methods are developed to reconstruct F(x). The first method(t-method) uses exit time data directly, and the second method(s-method) uses Laplace-transformed exit time data.
Let F*(x) and FM(x) be the target and the reconstructed flip rate functions respectively. FM(x) is written as a linear combination of M Legendre polynomials on [-L/2, L/2] with coefficients a=[a0,a1,...,aM-1]. Then objective function takes either form

\begin{equation}
\Pi_1(\mathbf{a}) =
\int_0^{L/v} |W_1(x_0,t;\mathbf{a}) - W_{\textrm{1,data}}(x_0,t)|^2 \, dt
+ \int_0^{L/v} |W_2(x_0,t;\mathbf{a}) - W_{\textrm{2,data}}(x_0,t)|^2 \, dt
\\
\Pi_2(\mathbf{a}) = \int_0^{\infty} |\tilde{w}_1(x_0,s;\mathbf{a}) -
\tilde{w}_{\textrm{1,data}}(x_0,s)|^2 \, ds + \int_0^{\infty}
|\tilde{w}_2(x_0,s;\mathbf{a}) - \tilde{w}_{\textrm{2,data}}(x_0,s)|^2 \, ds,
\end{equation}

target function
 

  • Reconstruction Results
6-panel-reconstruction-perfect-jpg

 Figure 3. Reconstructed approximations to flip rate functions F*(x) from noisy exit time data.
(a,b) F*(x)=1-0.7x-0.3x2+6x3, M=4; (c,d) F*(x)=0.1(x2e-x+1), M=5; (e,f) F*(x)=1+x+3x2, M=3.

  • Two State Broadwell Model

Let v >0 be a constant velocity, F(x) be a flip rate function, w1(x,t) and w2(x,t) represent exit time distributions conditioned on the particle initially being at position x and having positive (+v) and negative (-v) velocity respectively. The Broadwell PDEs are
PDEs

 

BroadwellProcess3-eps

 Figure 1. Random walk for Broadwell process 

  • Monte Carlo Simulation of Broadwell Process

The following figure represents the exit time distribution w1(x,t) of the random walk described in Figure 1.
P3 (Pak-Wing Fok's conflicted copy 2012-02-13)

 Figure 2. Simulated exit times distribution of a Broadwell process. (a) F(x)=10x3+5ex+1, v=1/2 (b) F(x)=1+x2, v=1.
Insets show Laplace Transformed data. N=40,000 realizations was used in each case.

 

  • Projection Method to Solve Inverse Problem
Let W1,2(x,t) be the cumulative density functions. The inverse problem of interest is, given  Monte Carlo simulated data w1,data(x,t) and w2,data(x,t) for t>0 and fixed -L/21(x,s) and \tilde w2(x,s) be Laplace transforms of w1 and w2.  Two methods are developed to reconstruct F(x). The first method(t-method) uses exit time data directly, and the second method(s-method) uses Laplace-transformed exit time data.
Let F*(x) and FM(x) be the target and the reconstructed flip rate functions respectively. FM(x) is written as a linear combination of M Legendre polynomials on [-L/2, L/2] with coefficients a=[a0,a1,...,aM-1]. Then objective function takes either form
target function
 

  • Reconstruction Results
6-panel-reconstruction-perfect-jpg

 Figure 3. Reconstructed approximations to flip rate functions F*(x) from noisy exit time data.
(a,b) F*(x)=1-0.7x-0.3x2+6x3, M=4; (c,d) F*(x)=0.1(x2e-x+1), M=5; (e,f) F*(x)=1+x+3x2, M=3.