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A Feynman-Kac Formula Implementation for the
Linearized Poisson-Boltzmann (LPB) Equation
Inc., 7406 Alban Station Court, Suite A112, Springeld, VA 22150
Department of Computer Science
Florida State University
Tallahassee, FL 32306-4530 USA
E-mail: chwang@cs.fsu.edu
URL: http://www.cs.fsu.edu/chwang
Chi-Ok Hwang, Michael Mascagni, and James A. Givena
a Angle
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Table of Contents
First passage time probability distribution
Dirichlet Problem for the Linearized Poisson-Boltzmann (LPB)
equation
\Walk on Spheres" (WOS)
Elepov-Mihailov solution and Feynman-Kac representation
Implementation
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Chi-Ok Hwang
Slide 1
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w.r.t.
Table of Contents
Runtime
-absorption layer
Errors
{ Errors from -absorption layer
{ -absorption Error Analysis
Future Work
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Chi-Ok Hwang
Slide 2
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WOS for 2-D
@
( )
X @ first−passage location
x; starting point
Ω
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Chi-Ok Hwang
Slide 3
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u x0
=
n
Qi i
N
1
X
i=1
n
di i
1 ) ;
1
sinh(dini
di i
n
(1)
(2)
(3)
Slide 4
= d(Pini ): (4)
Dirichlet Problem for LPB
n
Qi i
r2 (x) = 2 (x); x 2 ;
(x) = 0(x); x 2 @ ;
Elepov-Mihailov solution:
N
( ) = 1 Qini 0(Xni );
where
0 = 1;
Qi
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Dirichlet Problem for LPB
p d
Slide 5
Elepov-Mihailov solution:
{ N : total number of diusing random walkers
{ i refers to ith random walker
{ Xni : position where the ith random walker is absorbed in
the -absorption layer after ni WOS steps
{ dini : radius of ni th WOS of the ith random walker
! survival probability of a random walker in a continuous
and free diusion region
( ) = d= sinh(d);
(5)
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Chi-Ok Hwang
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@ 2
d
g];
Dirichlet Problem for LPB
0
Feynman-Kac representation of the solution:
Z
( ) = E [ 0(X (@
)) expf
u x
exp(
2 2
6);
d =
(6)
Slide 6
(7)
{ @ = f : X ( ) 2 @ g: the rst passage time
{ X (@ ): the rst passage location on the boundary, @ ! survival probability of a random walker in a continuous and
free diusion region using the rst moment of the rst-passage
time distribution
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Chi-Ok Hwang
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Survival Probabilities
1
0.9
0.8
0.7
Feynmann−Kac
Elepov−Mihailov
P(dκ)
0.6
0.5
0.4
0.3
0.2
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Chi-Ok Hwang
0.1
0
0
1
2
3
4
5
dκ
6
7
8
9
10
Slide 7
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Chi-Ok Hwang
Z1
0
@t0
0
Z1
0
2 2)
n X1
0 2 2 )dt0
t d
( 1)n
1 + 2d2=(n2 2)
exp[ (n22 + 2 d2)t0]dt0
@P (t )
exp(
Equivalence of FK to EM
X1
( 1)n(
n=1
= 2
n=1
d
;
=
sinh(
d)
(8)
(9)
(10)
(11)
Slide 8
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Equivalence of FK to EM
( 0) = 1 + 2
P t
X1
n=1
( 1)n exp(
n t :
2 2 0)
where the rst passage time probability distribution, P (t0), is
t0: D =d2
{
{
D : diusion constant
: rst passage time
: radius of WOS
{ r
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Chi-Ok Hwang
(12)
Slide 9
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First-passage Time Probability
Distribution
1.0
0.8
P(t’)
0.6
0.4
0.2
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0.0
0.0
0.2
0.4
0.6
0.8
1.0
t’
Slide 10
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Feynman-Kac Implemetation for LPB
Slide 11
For each WOS step,
Sampling t0 using a random number in (0; 1)
Corresponding survival probability, exp( t0d22 ), is obtained
Survival probability is compared with a random number in
(0; 1) and if the random number is greater than the survival
probability, the random walker is removed at this WOS step.
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Dirichlet Problem for LPB
SN
X
i=1
(13)
Slide 12
N : number of trajectories
Ns : number of survived-and-absorbed random walkers
Xni : nal position of the walker on the boundary when it is
absorbed after ni WOS steps
N
An estimate for the solution at x0 :
Ns
=1
0 (Xni )
{
{
{
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Sphere
1.1
1
0.9
0.8
ψ/ψ0
0.7
0.6
0.5
0.4
0.3
0.2
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0.1
0
0
0.5
1
1.5
−1
r (in units of κ )
2
2.5
3
Slide 13
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Cylinder
1.1
1
0.9
0.8
Ψ/Ψ0
0.7
0.6
0.5
0.4
0.3
0.2
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0.1
0
0
0.5
1
1.5
2
2.5
−1
r (in units of κ )
3
3.5
4
Slide 14
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Plate
1.1
1
0.9
0.8
ψ/ψ0
0.7
0.6
0.5
0.4
0.3
0.2
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0.1
0
0
1
2
3
−1
r (in units of κ )
4
5
Slide 15
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Parallel Plates
1.1
1.0
0.9
ψ/ψ0
0.8
0.7
0.6
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0.5
0.4
−1.5
−1.0
−0.5
0.0
0.5
−1
r (in units of κ )
1.0
1.5
Slide 16
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Runtime
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-absorption Layer
w.r.t.
running time (secs)
20
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15
10
running time data
logarithmic regression
5 −7
10
10
−6
−5
10
−4
−3
10
10
ε−absorption layer
−2
10
10
−1
Slide 17
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Errors
Slide 18
Errors associated with the number of trajectories
{ can be reduced by increasing the number of trajectories
Errors associated with the -absorption layer
{ It is possible to investigate empirically using enough number
of trajectories
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Errors from -absorption Layer
0.02
absolute error
0.015
0.01
0.005
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0
simulation error
linear regression
0
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
ε−absorption layer
0.1
Slide 19
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Future Work
Slide 20
Solving (x) 2 (x) = g(x)
Extension to a method for the time-independent Schrodinger
equation
Quasirandom walks
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