reduced basis method for poisson-boltzmann equation · reduced basis method for poisson-boltzmann...

Reduced Basis Method forPoisson-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

MotivationElectrostatic Interactions [Holst ’94]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Complexity of a charged particle in solution surrounded by other chargedparticles.

Figure: 2-D view of the 3-D Debye-Huckel model.

IntroductionPoisson-Boltzmann Equation (PBE) [Holst ’94]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

−~∇.(ε(x)~∇u(x)) + k2(x) sinh(u(x)) = (4πe2

Nm∑i=1

ziδ(x − xi ), in Ω ∈ R3,

u(x) = (e2c

kBT)Nm∑i=1

zie−k(d−ai )

εw (1 + kai )don ∂Ω, d = |x − xi |, (1)

u(∞) = 0.

k2 = 8πe2c I

1000εkBT, (I = µ) = 1

∑Ni=1 ciz

u(x) = ecψ(x)kBT

IntroductionPoisson-Boltzmann Coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ε(x) =

ε1 ≈ 2 if x ∈ Ω1

ε2(= ε3) ≈ 80 if x ∈ Ω2or Ω3

, k(x) =

0 if x ∈ Ω1or Ω2√ε3k if x ∈ Ω3

Figure: PBE coefficients

Source: Introduction to Molecular Electrostatics with APBS, Robert Konecny

IntroductionLinearized PBE (LPBE) [Fogolari et al ’99,Holst ’94]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Assumption: ψ(x) 1.

− ~∇.(ε(x)~∇u(x)) + k2(x)u(x) = (4πe2

Nm∑i=1

ziδ(x − xi ), (2)

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.

Finite Difference Discretization

Centered finite differences of LPBE [Simakov2013]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

dx2εi+ 1

2,j ,k(ui+1,j ,k − ui ,j ,k) +

dx2εi− 1

2,j ,k(ui ,j ,k − ui−1,j ,k)− 1

dy2εi ,j+ 1

2,k(ui ,j+1,k − ui ,j ,k)

dy2εi ,j− 1

2,k(ui ,j ,k − ui ,j−1,k)− 1

dz2εi ,j ,k+ 1

2(ui ,j ,k+1 − ui ,j ,k) +

dz2εi ,j ,k− 1

2(ui ,j ,k − ui ,j ,k−1)

+ k2i ,j ,kui ,j ,k = Cqi ,j ,k . (3)

(a) Discretization of continuous variables (b) Molecular surfaces and volumes

Essentials of Reduced BasisMethod (RBM)

Introduction [Benner et al ’2015, Eftang ’2011]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Model Reduction: FOM to ROM

Replace FOM AuN (µ) = f N (µ), µ ∈ D,

with ROM AuN(µ) = fN(µ), uN(µ) ≈ uN (µ), N N .

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,

MN = uN (µ) : µ ∈ D. (4)

RB space V is built upon 4 - generated by greedy algorithm,

range(V ) = spanuN (µ1), ..., uN (µN), ∀µ1, ..., µN ∈ D. (5)

Greedy Algorithm [Hesthaven et al 2014]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Algorithm 1 Greedy algorithm

Input: Training set Ξ ⊂ D including all of µ, i.e., Ξ := µ1, . . . , µl.Output: RB basis represented by the projection matrix V .1: Choose µ∗ ∈ Ξ arbitrarily2: Solve FOM for uN (µ∗)3: S1 = µ∗, V1 = [uN (µ∗)], N = 14: while max

µ∈Ξ∆N(µ) ≥ ε do

5: µ∗ = arg maxµ∈Ξ

∆N(µ)

6: Solve FOM for uN (µ∗)7: SN+1 = SN ∪ µ∗, VN+1 = [VN uN (µ∗)]8: Orthonormalize the columns of VN+1

9: N = N + 1

10: end while

Computational complexity of the Reduced order Model (ROM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Nonaffine parameter dependence

(A1 + µA2)uN (µ) = ρ+ b(µ), µ ∈ D. (6)

Consider the reduced order model (ROM);

( A1︸︷︷︸N×N

+µ A2︸︷︷︸N×N

) uN(µ)︸︷︷︸N×1

= ρ︸︷︷︸N×1

+ V T︸︷︷︸N×N

b(µ)︸︷︷︸N×1

where A1 = V TA1V , A2 = V TA2V , ρ = V Tρ, and N N .

matrix-vector products require 2NN flops,

full evaluation of b(µ).

Discrete Empirical Interpolation Method (DEIM) [Chaturantabut 2010]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 5 10 15 2010−15

10−5

Number of singular values

Figure: Decay of singular values

Compute snapshot matrixF = [b(µ1), . . . , b(µl)] ∈ RN×l ,apply SVD to F : F = UFΣW T ,

UF ∈ RN×l , Σ ∈ Rl×l , andW ∈ Rl×l ,

Σ = diag(σ1, . . . , σl) s.t,σ1 > . . . > σl ≥ 0,

l∑i=r+1

l∑1=1

< εsvd , εsvd = 10−13.

Discrete Empirical Interpolation Method (DEIM) [Volkwein 2010]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Select basis set uFi ri=1 of rank r from UF which solves,

arg minuFi

∑lj=1 ‖xj −

∑ri=1〈xj , uFi 〉uFi ‖2

2, s.t.〈ui , uj〉 = δij ,

DEIM determines UF c(µ) s.t, b(µ) ≈ UF c(µ), c(µ) ∈ Rr ,

determine c(µ) by selecting r rows from b(µ) = UF c(µ),

suppose PTU is nonsingular, for P = [e℘1 , . . . , e℘r ] ∈ RN×r , then,

PTb(µ) = PTUF c(µ) =⇒ c(µ) = (PTUF )−1PTb(µ), (8)

∴ b(µ) ≈ UF (PTUF )−1PTb(µ). (9)

ROM with DEIM approximation becomes,

( A1︸︷︷︸N×N

+µ A2︸︷︷︸N×N

) uN(µ)︸︷︷︸N×1

= ρ︸︷︷︸N×1

+V TUF (PTUF )−1︸︷︷︸N×r

PTb(µ)︸︷︷︸r×1

. (10)

Discrete Empirical Interpolation Method (DEIM) [Chaturantabut 2010]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Algorithm 2 DEIM algorithm

Input: Basis uFi ri=1 for F .Output: DEIM basis UF and indices ~℘ = [℘1, . . . , ℘r ]T ∈ Rr .1: ℘1 = arg max|uF1 |,2: UF = [uF1 ], P = [e℘1 ], ~℘ = [℘1].3: for i = 2 to r do4: Solve (PTUF )α = PTuFi for α, where α = (α1, . . . , αi−1)T ,5: r = uFi − UFα,6: ℘i = arg max|r |,

7: UF ← [UF uFi ], P ← [P e℘i ], ~℘←[~℘℘i

8: end for

DEIM Approximation Error [Feng et al 2016, Wirtz et al 2014]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

DEIM error is given by,

eDEIM = b(µ)− b(µ) = Π2(I − Π)b(µ), (11)

Π and Π2 are oblique projectors defined as,

Π = UF (PTUF )−1PT , (12)

Π2 = (I − Π)UF (PT (I − Π)UF )−1PT , (13)

UF = U∗F (:, r + 1 : r∗) and P = P∗(:, r + 1 : r∗) such that

U∗F = [UF , UF ] and P∗ = [P, P],

b(µ) = U∗F ((P∗)TU∗F )−1(P∗)Tb(µ). (14)

A Posteriori Error Estimation [Quarteroni et al 2016]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Compute residual due to DEIM,

rDEIMN (uN ;µ) = (ρ+ b(µ))− AN (µ)uN(µ), (15)

general residual becomes,

rN(uN ;µ) = (ρ+ b(µ))− AN (µ)uN(µ)

= (ρ+ b(µ))− AN (µ)uN(µ) + b(µ)− b(µ)

= rDEIMN (uN ;µ) + b(µ)− b(µ)︸︷︷︸

:=eDEIM

a posteriori error can be derived from 16 by,

rN(uN ;µ) = AN (µ)uN (µ)− AN (µ)uN(µ)

= AN (µ)e(µ)(17)

A Posteriori Error Estimation [Quarteroni ’2015]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The error e(µ) := uN (µ)− uN(µ) is given by

e(µ) = (AN (µ))−1rN(uN ;µ), (18)

obtain an upper bound for the 2-norm of the error,

‖e(µ)‖2 ≤ ‖(AN )−1(µ)‖2‖rN(uN ;µ)‖2 =‖rN(uN ;µ)‖2

σmin(AN (µ))=: ∆N(µ),

where σmin(AN (µ)) is the smallest singular value of AN (µ),

in our case the a posteriori error is,

‖e(µ)‖2 ≈ ‖rN(uN ;µ)‖2 = ∆N(µ). (20)

Numerical Results

Finite Difference Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0 2 4 6 8 1010−12

10−8

10−4

Iteration number

Figure: Relative residual for PCG

Consider LPBE (1),

parameter domainµ ∈ D = [0.05, 0.15],

physical domainΩ = 60A× 60A× 60A,

dimension N = 2, 146, 689,

PQR file,

Cubic B-spline interpolation(basis spline),

PCG with algebraic multigridv-cycle preconditioner.

Numerical Results

Computational time to solve uN (µ) is ≈ 50 seconds on average.

Figure: uN (µ) at µ = 0 Figure: uN (µ) at µ = 0.05

Numerical Results

Figure: uN (µ) at µ = 0.15 Figure: uN (µ) at µ = 0.5

Numerical Results

Reduced Basis Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Computational time to solve uN(µ) is ≈ 0.065 seconds on average.

True error = ‖uN (µ)− uN(µ)‖2, ∆maxN (µ) = max

µ∈Ξ‖rN(uN ;µ)‖2, Relative ∆max

N (µ) =∆max

N (µ)‖uN(µ∗)‖2

. µ∗ = arg maxµ∈Ξ‖rN(uN ;µ)‖2.

True error Maximal error

1 2 3 4 5 6

10−5

10−3

10−1

Reduced Dimension N

(a) Maximal versus true error

1 2 3 4 5 6

10−9

10−7

10−5

10−3

10−1

Reduced Dimension N

(b) Relative ∆maxN (µ) vs true error

Figure: Comparison between true error and maximal error

Numerical Results

Reduced Basis Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

True error Error estimate

5 10 15 20

10−8

10−6

10−4

Parameter (µ) sample size

Figure: Error estimate versus true error

Numerical Results

Error analysis between FDM and RBM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure: Absolute error at µ = 0.05101

Conclusions and Outlook

Conclusions and Outlook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Conclusions

Applied RBM to LPBE with ionic strength as meaningful parameter.

RBM reduces the high dimensional FOM by a factor of ≈ 360, 000and computational time by a factor of approximately over 800,

DEIM error costly in online stage,

error estimator provided fast convergence to the RB approximation.

Outlook

Develop an efficient error estimator,

reduce DEIM error costs.

Thank you for your attention!Cleophas Kweyu, kweyu@mpi-magdeburg.mpg.de WIAM16, August 31- September 2, 2016 23/24

RBM Summary

Conclusions and Outlook [Quarteroni et al 2016]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure: RB workflow

reduced basis method for poisson-boltzmann equation · reduced basis method for poisson-boltzmann...

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