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

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

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 ª [ u ] + x 2 © : n = . CMS-2009w

Some asymptotic narrow-escape MFPT results ¹ v = 1 j ­ Z ( x ) d ; Average MFPT: Unit sphere: ¹ v » j ­ 4 " D h 1 ¡ ¼ 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) ¹ v » j ­ 4 " D h 1 ¡ ¼ H l o g + O ( ) i CMS-2009w

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: ¹ v » j ­ 4 " D h 1 ¡ ¼ 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. ¹ v = j ­ 4 " D · 1 + ¼ l o g µ 2 ¶ ¡ 9 5 3 O ( ) ¸ CMS-2009w

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 µ ¡ ) Á R d : x j ° Solution: G s ( x ; j ) = 1 2 ¼ ¡ + 8 ³ ´ 4 l o g µ c ° ¶ 7 : Angle between x ; j : ° CMS-2009w

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 µ 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 CMS-2009w

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 ¼ Trap locations: x 1 ; : N Green function matrix: G s ´ B @ R 1 2 ¢ N . ; ¡ C A = 9 ¼ i j ( x ) : Interaction term: p c ( x 1 ; : N ) ´ C T G s CMS-2009w

Method of solution: Matched asymptotic expansions 3. Main result 1: MFPT for N different circular traps MFPT: v = j ­ 2 ¼ " D N ¹ c £ 1 + l o g ¡ ¢ P G s ( x ; ) p : · O ¤ Average MFPT: v = j ­ 2 ¼ " D N ¹ c £ 1 + l o g ¡ ¢ P p ( x ; : ) · O ¤ CMS-2009w

Unit sphere with N identical traps. Case of N identical traps Average MFPT: ¹ v = j ­ 4 " D N £ 1 + ¼ l o g ¡ 2 ¢ 9 5 3 H ( x ; : ) O ¤ Interaction energy: H ( x 1 ; : N ) = X i j + µ ¡ 2 l o g ¶ CMS-2009w

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 CMS-2009w

Laplacian eigenvalue optimization Diffusion equation: u t = D ¢ ; Separation of variables: u = X i e ¡ ¸ D t ª ( r ; µ Á ) ¢ ª i = ¡ ¸ u t = D ¢ ; Main result 2: principal eigenvalue for N identical circular traps ¸ » 2 ¼ " N c j ­ £ 1 + l o g ( a ) ¡ 3 ¢ 9 H x ; : ¤ c = 2 a ¼ ; j ­ 4 3 To within 3 terms, ¹ v » 1 = ( D ¸ ) CMS-2009w

Minimization of the interaction energy Interaction energy for N identical circular traps: H ( x 1 ; : N ) = X i j + µ ¡ 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

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

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 ¼ + R d : D t ª [ u ] + x 2 © : n = . CMS-2009w

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 ª [ u ] + x 2 © : n = . CMS-2009w

Volume traps: MFPT asymptotic solution Expression for MFPT: v » j ­ 4 ¼ N ¹ C D " 2 1 ¡ X = G ( x ; ) + c p : O 3 5 Average MFPT: ¹ v » 1 D j ­ ¸ + O ( " ) = 4 ¼ N C · p c x ; : 2 D t ª [ u ] + x 2 © : n = . CMS-2009w

Volume traps: MFPT asymptotic solution Example: MFPT minimization for unit sphere with N identical traps G ( x ; » ) = 1 4 ¼ j ¡ + l o g ³ 2 c s µ ´ 6 ­ ¢ 7 Energy: H b a l ´ P N i = 1 j e G ; ³ ( ¡ ± ) + R p x : 4 ¼ 7 2 D t ª [ u ] + x 2 © : n = . e G i ; j = 4 ¼ ( ¡ B ) CMS-2009w

Volume traps in unit sphere: optimization Example: MFPT minimization for unit sphere with N identical traps G ( x ; » ) = 1 4 ¼ j ¡ + l o g ³ 2 c s µ ´ 6 ­ ¢ 7 Energy: H b a l ´ P N i = 1 j e G ; ³ ( ¡ ± ) + R p x : 4 ¼ 7 2 D t ª [ u ] + x 2 © : n = . e G i ; j = 4 ¼ ( ¡ B ) CMS-2009w

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 ; ³ ( ¡ ± ) + R using ECAM and DSO methods. Result: traps are located on concentric spheres! Some examples: D t ª [ u ] + x 2 © : n = . CMS-2009w

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)? CMS-2009w

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