1. Narrow Escape Problems:
Asymptotic Optimization of Locations of Small Traps for the Unit Sphere
Alexei F. Cheviakov University of Saskatchewan
Michael J. Ward, University of British Columbia
CMS-2009w

2. Talk plan
Problem for the mean first passage time (MFPT) with traps located on the unit sphere
Applications, current results
Asymptotic solution talking into account trap locations (outline)
Relation with principal Laplacian eigenvalue optimization
Global optimization of trap locations: numerical results
Case of spherical traps inside the unit sphere
Applications, current results
Asymptotic solution talking into account trap locations (outline)
Relation with principal Laplacian eigenvalue optimization
Global optimization of trap locations inside a unit sphere
Open problems
CMS-2009w

3. 4 v = ¡ ; x 2 ­ v = ; x 2 @ ­ [ @ v = ; x 2 ­ :
BVP for the mean first passage time (MFPT) 4 v = 1 D ; x 2 v = ; x 2 @ a [ N j 1
" j = 1 ; : N @ n v = ; x 2 r : 4 v = 1 D ; x 2
Applications:
Chemistry and biochemistry;
Cell biology; escape through pores on membranes of cell and cell nucleus. D t [ u ] + x 2 : n = .
CMS-2009w

4. Some asymptotic narrow-escape MFPT resultsv = 1 j Z ( x ) d ;
Average MFPT:
Unit sphere: v j 4
" D h 1 l o g + O ( ) i
One circular absorbing window:
One elliptic absorbing window.
Two circular absorbing windows. 4 v = 1 D ; x 2
Arbitrary bounded domain:
One circular absorbing window: (H – mean curvature) v j 4
" D h 1 H l o g + O ( ) i
CMS-2009w

5. New MFPT results ¹ v = 1 j ­ Z ( x ) d ; ¹ v » j ­ 4 " D h 1 ¡ ¼ l o g
Average MFPT:
Unit sphere: v j 4
" D h 1 l o g + O ( ) i
One circular absorbing window:
The new results: 4 v = 1 D ; x 2
Three-term asymptotic expansion of MFPT and average MFPT for N different small circular traps on a unit sphere. E.g.: One trap:
For N>1, the third term depends on relative trap locations. v = j 4
" D 1 + l o g 2 9 5 3 O ( )
CMS-2009w

6. Method of solution: Matched asymptotic expansions
1. The surface Neumann Green's function for the sphere
Problem: x 4 G s = 1 j ; x 2 @ r ( c o ) Á R d : x j
Solution: G s ( x ; j ) = 1 2 + 8 4 l o g c 7 :
Angle between x ; j :
CMS-2009w

7. x : Method of solution: Matched asymptotic expansions ´ s s v » " w +
2. Local curvilinear coordinates
Near the trap x j : s 2 y
" 1 ( x j ) ; r s i n Á 2 : s 1
Solution:
Near-field: v
" 1 w + l o g 2 :
Far-field: v
" 1 + l o g 2 3 :
Substitute in the problem; match as j y ! 1
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8. Method of solution: Matched asymptotic expansions
3. Main result 1: MFPT for N different circular traps
Capacitance vector: C T ( c 1 ; : N ) i = 2
Trap locations: x 1 ; : N
Green function matrix: G s B @ R 1 2 N . ; C A = 9 i j ( x ) :
Interaction term: p c ( x 1 ; : N ) C T G s
CMS-2009w

9. Method of solution: Matched asymptotic expansions
3. Main result 1: MFPT for N different circular traps
MFPT: v = j 2
" D N c 1 + l o g P G s ( x ; ) p : O
Average MFPT: v = j 2
" D N c 1 + l o g P p ( x ; : ) O
CMS-2009w

10. Unit sphere with N identical traps.
Case of N identical traps
Average MFPT: v = j 4
" D N 1 + l o g 2 9 5 3 H ( x ; : ) O
Interaction energy: H ( x 1 ; : N ) = X i j + 2 l o g
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11. Unit sphere with N identical traps.
Example: MFPT vs. trap radius for N=1, 2, 4.
Solid: 3-term
Dotted: 2-term
Triangles: full numerical simulation
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12. Laplacian eigenvalue optimization
Diffusion equation: u t = D ;
Separation of variables: u = X i e D t ( r ; Á ) i = u t = D ;
Main result 2: principal eigenvalue for N identical circular traps 2
" N c j 1 + l o g ( a ) 3 9 H x ; : c = 2 a ; j 4 3
To within 3 terms, v 1 = ( D )
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13. Minimization of the interaction energy
Interaction energy for N identical circular traps: H ( x 1 ; : N ) = X i j + 2 l o g u t = D ; H C = N X i 1 j + x ;
Coulomb (repelling electrons on sphere): H L = N X i 1 j + l o g x :
Logarithmic:
Optimal arrangement - ? (2N-3 degrees of freedom)
N=2: poles; N=3: triangle; N=4: tetrahedron; N>4 -- ?
CMS-2009w

14. Minimization of the interaction energy
Numerical global optimization software:
The Extended Cutting Angle method (ECAM). Deterministic global optimization technique, applicable to Lipschitz functions. (“Ganso” library).
Dynamical Systems Based Optimization (DSO). A dynamical system is constructed, using a number of sampled values of the objective function to introduce “forces”. The evolution of such a system yields a descent trajectory converging to lower values of the objective function. (“Ganso” library).
Lipschitz-Continuous Global Optimizer (LGO). Commercial global optimization software, based on a combination of rigorous (theoretically convergent) global minimization strategies, as well as a number of local minimization strategies.
CMS-2009w

15. 4 v = ¡ ; x 2 ­ n v = ; x 2 @ ­ [ @ v = ; x 2 ­ :
Volume traps: BVP for the mean first passage time 4 v = 1 D ; x 2 n a v = ; x 2 @ a [ N j 1
" j = 1 ; : N @ n v = ; x 2 : 4 v = 1 D ; x 2
Can be solved for ‘any’ domain, in terms of the corresponding Green’s function: G = 1 j ( x ) ; 2 @ n 4 + R d : D t [ u ] + x 2 : n = .
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16. Volume traps: BVP for the mean first passage time
Solution: denote G B @ R 1 ; 2 N . C A c : G i ; j ( x ) R
Interaction term: p c ( x 1 ; : N ) T G = X i j C D t [ u ] + x 2 : n = .
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17. Volume traps: MFPT asymptotic solution
Expression for MFPT: v j 4 N C D
" 2 1 X = G ( x ; ) + c p : O 3 5
Average MFPT: v 1 D j + O (
" ) = 4 N C p c x ; : 2 D t [ u ] + x 2 : n = .
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18. Volume traps: MFPT asymptotic solution
Example: MFPT minimization for unit sphere with N identical traps G ( x ; ) = 1 4 j + l o g 2 c s 6 7
Energy: H b a l P N i = 1 j e G ; ( ) + R p x : 4 7 2 D t [ u ] + x 2 : n = . e G i ; j = 4 ( B )
CMS-2009w

19. Volume traps in unit sphere: optimization
Example: MFPT minimization for unit sphere with N identical traps G ( x ; ) = 1 4 j + l o g 2 c s 6 7
Energy: H b a l P N i = 1 j e G ; ( ) + R p x : 4 7 2 D t [ u ] + x 2 : n = . e G i ; j = 4 ( B )
CMS-2009w

20. Volume traps in unit sphere: optimization
Optimization: N traps, 3N-3 spherical coordinates.
Numerical global minimization of H b a l P N i = 1 j e G ; ( ) + R
using ECAM and DSO methods.
Result: traps are located on concentric spheres!
Some examples: D t [ u ] + x 2 : n = .
CMS-2009w

21. Discussion Open problems / future work:
Rigorous justification of asymptotic results.
Results for arbitrary 3D domain with surface traps.
Relation to homogenization theory.
Computation of higher-order asymptotic terms.
Volume traps: compute arrangements for large N in sphere.
Scaling law for optimally located volume traps (large N)?
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22. Thank you for your attention!
The end
Thank you for your attention!
CMS-2009w