Reduced Basis Method for Poisson-Boltzmann Equation Workshop in - PowerPoint PPT Presentation

Reduced Basis Method for Poisson-Boltzmann Equation Workshop in Industrial and Applied Mathematics, WIAM16 Cleophas Kweyu, Lihong Feng, Matthias Stein, Peter Benner September 01, 2016 Partners:

Outline 1. Motivation 2. Introduction 3. Finite Difference Discretization 4. Essentials of Reduced Basis Method (RBM) 5. Numerical Results 6. Conclusions and Outlook Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 2/24

Motivation Electrostatic Interactions [ Holst ’94 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complexity of a charged particle in solution surrounded by other charged particles. Figure: 2-D view of the 3-D Debye-H¨ uckel model. Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 3/24

Introduction Poisson-Boltzmann Equation (PBE) [ Holst ’94 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PBE N m k 2 ( x ) sinh( u ( x )) = (4 π e 2 � ∇ u ( x )) + ¯ − � ∇ . ( ǫ ( x ) � c z i δ ( x − x i ) , in Ω ∈ R 3 , k B T ) i =1 Nm u ( x ) = ( e 2 z i e − k ( d − a i ) � c ǫ w (1 + ka i ) d on ∂ Ω , d = | x − x i | , k B T ) (1) i =1 u ( ∞ ) = 0 . k 2 = 8 π e 2 � N c I 1000 ǫ k B T , ( I = µ ) = 1 i =1 c i z 2 i , 2 u ( x ) = e c ψ ( x ) k B T . Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 4/24

Introduction Poisson-Boltzmann Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . � � ǫ 1 ≈ 2 if x ∈ Ω 1 0 if x ∈ Ω 1 or Ω 2 ¯ ǫ ( x ) = , k ( x ) = √ ǫ 3 k ǫ 2 (= ǫ 3 ) ≈ 80 if x ∈ Ω 2 or Ω 3 if x ∈ Ω 3 Figure: PBE coefficients Source: Introduction to Molecular Electrostatics with APBS, Robert Konecny Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 5/24

Introduction Linearized PBE (LPBE) [ Fogolari et al ’99,Holst ’94 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumption: ψ ( x ) ≪ 1. LPBE N m k 2 ( x ) u ( x ) = (4 π e 2 � − � ∇ . ( ǫ ( x ) � ∇ u ( x )) + ¯ c k B T ) z i δ ( x − x i ) , (2) i =1 Applications of the PBE and LPBE potential at the surface of a biomolecule - docking sites, potential outside the molecule - free energy of interaction, free energy of a biomolecule - biomolecular stability. Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 6/24

Finite Difference Discretization Centered finite differences of LPBE [ Simakov2013 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 − 2 , j , k ( u i +1 , j , k − u i , j , k ) + 2 , j , k ( u i , j , k − u i − 1 , j , k ) − 2 , k ( u i , j +1 , k − u i , j , k ) dx 2 ǫ i + 1 dx 2 ǫ i − 1 dy 2 ǫ i , j + 1 1 1 1 + dy 2 ǫ i , j − 1 2 , k ( u i , j , k − u i , j − 1 , k ) − dz 2 ǫ i , j , k + 1 2 ( u i , j , k +1 − u i , j , k ) + dz 2 ǫ i , j , k − 1 2 ( u i , j , k − u i , j , k − 1 ) + ¯ k 2 i , j , k u i , j , k = Cq i , j , k . (3) (a) Discretization of continuous variables (b) Molecular surfaces and volumes Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 7/24

Essentials of Reduced Basis Method (RBM) Introduction [ Benner et al ’2015, Eftang ’2011 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Reduction: FOM to ROM Replace FOM Au N ( µ ) = f N ( µ ) , µ ∈ D , u N ( µ ) ≈ u N ( µ ) , N ≪ N . with ROM Au N ( µ ) = f N ( µ ) , RBM is a parametrized model order reduction (PMOR) technique, exploits an offline/online procedure, powerful tools - greedy algorithm and a posteriori error estimation, assumption - typically low dimensional solution manifold, M N = { u N ( µ ) : µ ∈ D } . (4) RB space V is built upon 4 - generated by greedy algorithm, range ( V ) = span { u N ( µ 1 ) , ..., u N ( µ N ) } , ∀ µ 1 , ..., µ N ∈ D . (5) Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 8/24

Essentials of Reduced Basis Method (RBM) Greedy Algorithm [ Hesthaven et al 2014 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithm 1 Greedy algorithm Input: Training set Ξ ⊂ D including all of µ , i.e., Ξ := { µ 1 , . . . , µ l } . Output: RB basis represented by the projection matrix V . 1: Choose µ ∗ ∈ Ξ arbitrarily 2: Solve FOM for u N ( µ ∗ ) 3: S 1 = { µ ∗ } , V 1 = [ u N ( µ ∗ )], N = 1 4: while max µ ∈ Ξ ∆ N ( µ ) ≥ ǫ do µ ∗ = arg max µ ∈ Ξ ∆ N ( µ ) 5: Solve FOM for u N ( µ ∗ ) 6: S N +1 = S N ∪ µ ∗ , V N +1 = [ V N u N ( µ ∗ )] 7: Orthonormalize the columns of V N +1 8: N = N + 1 9: 10: end while Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 9/24

Essentials of Reduced Basis Method (RBM) Computational complexity of the Reduced order Model (ROM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonaffine parameter dependence ( A 1 + µ A 2 ) u N ( µ ) = ρ + b ( µ ) , µ ∈ D . (6) Consider the reduced order model (ROM); ( ˆ ˆ ) u N ( µ ) + V T A 1 + µ A 2 = ρ ˆ b ( µ ) , (7) ���� ���� ���� ���� � �� � ���� N ×N N × N N × N N × 1 N × 1 N× 1 where ˆ A 1 = V T A 1 V , ˆ A 2 = V T A 2 V , ˆ ρ = V T ρ , and N ≪ N . matrix-vector products require 2 N N flops, full evaluation of b ( µ ). Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 10/24

Essentials of Reduced Basis Method (RBM) Discrete Empirical Interpolation Method (DEIM) [ Chaturantabut 2010 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5 Compute snapshot matrix F = [ b ( µ 1 ) , . . . , b ( µ l )] ∈ R N× l , Singular values apply SVD to F : F = U F Σ W T , 10 − 5 U F ∈ R N× l , Σ ∈ R l × l , and W ∈ R l × l , Σ = diag ( σ 1 , . . . , σ l ) s.t, σ 1 > . . . > σ l ≥ 0, 10 − 15 0 5 10 15 20 l � σ i Number of singular values ǫ svd = 10 − 13 . i = r +1 < ǫ svd , l � σ i Figure: Decay of singular values 1=1 Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 11/24

Essentials of Reduced Basis Method (RBM) Discrete Empirical Interpolation Method (DEIM) [ Volkwein 2010 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Select basis set { u F i } r i =1 of rank r from U F which solves, � l j =1 � x j − � r i =1 � x j , u F i � u F i � 2 s . t . � u i , u j � = δ ij , arg min 2 , { u F i } r i =1 DEIM determines U F c ( µ ) s.t, b ( µ ) ≈ U F c ( µ ) , c ( µ ) ∈ R r , determine c ( µ ) by selecting r rows from b ( µ ) = U F c ( µ ), suppose P T U is nonsingular, for P = [ e ℘ 1 , . . . , e ℘ r ] ∈ R N× r , then, P T b ( µ ) = P T U F c ( µ ) = ⇒ c ( µ ) = ( P T U F ) − 1 P T b ( µ ) , (8) ∴ b ( µ ) ≈ U F ( P T U F ) − 1 P T b ( µ ) . (9) ROM with DEIM approximation becomes, ( ˆ ˆ ) u N ( µ ) + V T U F ( P T U F ) − 1 P T b ( µ ) A 1 + µ A 2 = ρ ˆ . (10) ���� ���� ���� � �� � � �� � � �� � N × N N × N N × 1 N × 1 N × r r × 1 Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 12/24

Essentials of Reduced Basis Method (RBM) Discrete Empirical Interpolation Method (DEIM) [ Chaturantabut 2010 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithm 2 DEIM algorithm Input: Basis { u F i } r i =1 for F . ℘ = [ ℘ 1 , . . . , ℘ r ] T ∈ R r . Output: DEIM basis U F and indices � 1: ℘ 1 = arg max {| u F 1 |} , 2: U F = [ u F 1 ], P = [ e ℘ 1 ] , � ℘ = [ ℘ 1 ]. 3: for i = 2 to r do Solve ( P T U F ) α = P T u F i for α , where α = ( α 1 , . . . , α i − 1 ) T , 4: r = u F i − U F α , 5: ℘ i = arg max {| r |} , 6: � � � ℘ U F ← [ U F u F i ], P ← [ P e ℘ i ], � ℘ ← . 7: ℘ i 8: end for Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 13/24