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A Fast Continuation Method for theOrnstein-Zernike Equations
C. T. Kelley
Joint work with
Monte Pettitt, Yannis Kevrekidis, Liang Qiao
Department of Mathematics
Center for Research in Scientific Computation
North Carolina State University
Raleigh, North Carolina, USA
NCSU
November 2, 2004
Supported by NSF, ARO.
C. T. Kelley – p.1
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Outline
• The Ornstein-Zernike (OZ) Equations
• Fast solvers for compact fixed point problemsApplication to OZ + uniqueness problems
• Path following: introductionNonlinear solversPseudo-arclength continuation
• Multilevel method.
• Results
C. T. Kelley – p.2
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OZ Equations: O-Z, 1914
Used to calculate probability distributions of atoms in fluidstates. Unknowns are h,c ∈C[0,L].
• h: radial pair correlation function, observable
• c: direct correlation function, defined by IE
Integral Equation:
h(r)− c(r)−ρ(h∗ c)(r)
where
(h∗ c)(r) =
∫
R3c(‖r− r′‖)h(‖r′‖)dr′.
C. T. Kelley – p.3
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Algebraic Closure Constraint
exp(−βU(r)+h(r)− c(r))−h(r)−1 = 0.
where u is the Lennard-Jones potential.
U(r) = 4ε(
(σr
)12−(σ
r
)6)
.
C. T. Kelley – p.4
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Parameters
Data are parameters
• ρ: number density, sometimes unknown
• β = 1/(absolute temperature×Boltzmann’s constant)
• ε : well depth of the potential
• σ : determines size of the particles
C. T. Kelley – p.5
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Discretization
• Uniform grid on [0,L]
• Trapezoid rule for integration
• Discrete Hankel transform for evaluation of integrals
H (h)(k) = 4π∫ ∞
0
sin(kr)kr
h(r)r2dr
andh∗ c = H
−1(hc).
• Fast evaluation via FFT
C. T. Kelley – p.6
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Solution: ρ = .2,σ = 2;ε = .1;β = 10;L = 9
0 1 2 3 4 5 6 7 8 9−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
r
hc
C. T. Kelley – p.7
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Reduction to single equation
Let g = h− c, then the closure constraint expresses c as afunction of g.
c(r) = c(g(r)) = exp(−βU(r)+g(r))−g(r)−1.
The integral equation is
h−ρc∗h = c.
Take Hankel transforms
h−ρ hc = c,
and obtain h = c/(1−ρ c).
C. T. Kelley – p.8
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g→ c→ h leads to . . .
h = h(c(g)) = c(g)+K (g).
Subtract c and obtain a fixed point problem for g.
g = h(c(g))− c(g) = K (g).
K is a nonlinear integral operator with compact Fréchetderivative.
C. T. Kelley – p.9
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Alternative: reduce to single equation in c
• c→ h(c) via solution of integral equation
• h(c)− c = G (c), G compact
• K (c) = exp(−βU −G (c))−G (c)−1
Compact fixed point problem:
c = K (c)
C. T. Kelley – p.10
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More General OZ Equations
Unknowns h,c,ρ,∈C[0,L]
h(r) = exp(−βU(r)+h(r)− c(r))−1
h(r) = c(r)+∫ r
0c(r− r′)ρ(r′)h(r′)dr′
ρ(r) = A1exp
(
−βU(r)+
∫ r
0ρ(r− r′)c(r′)dr′
)
.
Also matrix-valued unknowns.
C. T. Kelley – p.11
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Compact Fixed Point Problems
We’re worried about problems like
F(u) = u−K (u) = 0, on a Banach space X ,
where
• K ∈C1LIP(X).
• K ′ ∈Com(X).
• Compactness will lead to fast solvers.
C. T. Kelley – p.12
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How to exploit compactness
• Discretization
C. T. Kelley – p.13
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How to exploit compactness
• Discretization• Almost every reasonable scheme works, but
C. T. Kelley – p.13
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How to exploit compactness
• Discretization• Almost every reasonable scheme works, but• some approximations to K ′ converge in norm.
C. T. Kelley – p.13
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How to exploit compactness
• Discretization• Almost every reasonable scheme works, but• some approximations to K ′ converge in norm.
• Solvers
C. T. Kelley – p.13
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How to exploit compactness
• Discretization• Almost every reasonable scheme works, but• some approximations to K ′ converge in norm.
• Solvers• Krylov solvers need no preconditioning (in theory).
C. T. Kelley – p.13
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How to exploit compactness
• Discretization• Almost every reasonable scheme works, but• some approximations to K ′ converge in norm.
• Solvers• Krylov solvers need no preconditioning (in theory).• Multilevel methods are easy to design.
C. T. Kelley – p.13
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How to exploit compactness
• Discretization• Almost every reasonable scheme works, but• some approximations to K ′ converge in norm.
• Solvers• Krylov solvers need no preconditioning (in theory).• Multilevel methods are easy to design.• No smoothers are needed.
C. T. Kelley – p.13
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How to exploit compactness
• Discretization• Almost every reasonable scheme works, but• some approximations to K ′ converge in norm.
• Solvers• Krylov solvers need no preconditioning (in theory).• Multilevel methods are easy to design.• No smoothers are needed.
• Fast evaluation (O(N log(N)) is common.
C. T. Kelley – p.13
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How to exploit compactness
• Discretization• Almost every reasonable scheme works, but• some approximations to K ′ converge in norm.
• Solvers• Krylov solvers need no preconditioning (in theory).• Multilevel methods are easy to design.• No smoothers are needed.
• Fast evaluation (O(N log(N)) is common.
• Newton-Krylov, Newton-MG nonlinear solvers workwith no surprises (most of the time).
C. T. Kelley – p.13
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World’s Easiest Example
Linear Fredholm equation:
(I−K)u(x) = u(x)−∫ 1
0k(x,y)u(y)dy = f (x),
f ∈ X = C[0,1], k ∈C([0,1]× [0,1])Approximating space: Vh = span {φi}
Ph is a projection onto Vh, and we seek uh ∈Vh.
uh(x)−Khuh(x) = uh(x)−∫ 1
0kh(x,y)u
h(y)dy = Ph f (x)
where, kh(x,y) = ∑Nhi, j=1k(xi,x j)φi(x)φ j(y)
C. T. Kelley – p.14
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Properties of Discretization
• Kh operates on the function space
C. T. Kelley – p.15
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Properties of Discretization
• Kh operates on the function space
• Kh→ K in the operator norm
C. T. Kelley – p.15
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Properties of Discretization
• Kh operates on the function space
• Kh→ K in the operator norm
• Lots of flexibility in PhStrong convergence to I is all you need.
C. T. Kelley – p.15
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Properties of Discretization
• Kh operates on the function space
• Kh→ K in the operator norm
• Lots of flexibility in PhStrong convergence to I is all you need.
• If I−K is nonsingular, then
uh = (I−Kh)−1Ph f → (I−K)−1 f
Solve finite dimensional system for nodal values.
C. T. Kelley – p.15
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Properties of Discretization
• Kh operates on the function space
• Kh→ K in the operator norm
• Lots of flexibility in PhStrong convergence to I is all you need.
• If I−K is nonsingular, then
uh = (I−Kh)−1Ph f → (I−K)−1 f
Solve finite dimensional system for nodal values.
• Other choices of Kh are possibleStandard quadrature rule + fine-to-coarse by averaging
C. T. Kelley – p.15
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Nystrom interpolation
• Solve uh−Khuh = f rather than uh−Khuh = Ph f .
• Multiply by Ph and use Kh = KhPh = PhKh to get
(Phu)−PhKh(Phu) = Ph f .
Finite dimensional system.Solve for uh = Phuh.
• uh = f +Khuh
C. T. Kelley – p.16
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Performance of GMRES
Avoid the O(N3h ) cost of a direct solver, and compute
uh = (I−Kh)−1Ph f =
Nh
∑i=1
uhi φi ∈Vh.
with GMRES.
• Continuous problem: superlinear convergence
• Discrete problem: mesh independent performance
• Cost: One Khv evaluation/linear iterationThink Nh logNh work if done slickly.
Nested iteration (aka grid sequencing) is a good idea.
C. T. Kelley – p.17
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Multilevel Method; K 95
Since Kh→ K in the operator norm,
• (I−KH) (h << H) might be a good preconditioner forGMRES
C. T. Kelley – p.18
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Multilevel Method; K 95
Since Kh→ K in the operator norm,
• (I−KH) (h << H) might be a good preconditioner forGMRES
• Richardson iteration is a better idea thanks to LOWSTORAGE.
u← u− (I−KH)−1((I−Kh)u−Ph f )
C. T. Kelley – p.18
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Multilevel Method; K 95
Since Kh→ K in the operator norm,
• (I−KH) (h << H) might be a good preconditioner forGMRES
• Richardson iteration is a better idea thanks to LOWSTORAGE.
u← u− (I−KH)−1((I−Kh)u−Ph f )
• H suff small implies
C. T. Kelley – p.18
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Multilevel Method; K 95
Since Kh→ K in the operator norm,
• (I−KH) (h << H) might be a good preconditioner forGMRES
• Richardson iteration is a better idea thanks to LOWSTORAGE.
u← u− (I−KH)−1((I−Kh)u−Ph f )
• H suff small implies• Krylovs independent of H.
C. T. Kelley – p.18
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Multilevel Method; K 95
Since Kh→ K in the operator norm,
• (I−KH) (h << H) might be a good preconditioner forGMRES
• Richardson iteration is a better idea thanks to LOWSTORAGE.
u← u− (I−KH)−1((I−Kh)u−Ph f )
• H suff small implies• Krylovs independent of H.• One iteration/level suffices.
C. T. Kelley – p.18
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Nonlinear Problems
Generalization to the nonlinear case is easy,
u← u− (I−K′
H(uH))−1Fh(u)
if you’re careful about the fine-to-coarse transfer.If coarse mesh suff fine,
• Krylovs/Newton independent of H
• one Newton/level suffices.
C. T. Kelley – p.19
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Nested Iteration: Bottom up; K 95
h = H, i = 0Solve FH(uH) = 0 to high accuracy.u← uH
for i = 1, . . .m doh← h/2u← u− (I−K ′
H(uH))−1Fh(u)end for
• All the linear solver work is on the coarse mesh.
• Only two grids H and h active at any time.
• Cost of solve to truncation error:< 3 fine mesh evals, depending on cost of Kh
C. T. Kelley – p.20
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Works great for OZ! K., Pettit 2004
Iteration statistics for three nested iterations
• Multilevel, Newton-GMRES, Picard
• Formulation in c:c→ h(c) via integral equationc = K (c) via constraint
• Tabulate:i fG = fine mesh GMRES/Newton (average)
icG = coarse GMRES/Newton (average)incoming nonlinear residual Rh (R2h ≈ 4Rh)
C. T. Kelley – p.21
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Iteration Statistics: h = 1/(N-1)
Picard Newton-GMRES Multilevel
N Rδ i fG Rδ i f
G Rδ icG
65 3.5900e+00 650 3.5900e+00 85 3.5900e+00 85
129 1.3696e-01 11 1.3696e-01 4 1.3696e-01 8
257 2.0031e-02 3 2.9413e-02 5 4.1900e-02 7
513 4.8144e-03 9 6.9937e-03 5 9.4120e-03 7
1025 2.3568e-03 14 1.5400e-03 5 2.0205e-03 7
2049 3.6543e-04 15 3.5596e-04 7 4.6015e-04 8
4097 8.2396e-05 22 8.4570e-05 5 1.0831e-04 8
8193 2.2253e-05 38 2.0784e-05 7 2.6411e-05 8
16385 4.0075e-06 48 5.2729e-06 8 6.5042e-06 8
32769 9.7738e-07 32 1.2263e-06 5 1.6132e-06 8
65537 2.3869e-07 44 3.0647e-07 7 4.0169e-07 8
C. T. Kelley – p.22
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Mission Accomplished?
• We found two solutions;
C. T. Kelley – p.23
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Mission Accomplished?
• We found two solutions; one was wrong.
C. T. Kelley – p.23
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Mission Accomplished?
• We found two solutions; one was wrong.
• Monte knew which one was correct.
C. T. Kelley – p.23
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Mission Accomplished?
• We found two solutions; one was wrong.
• Monte knew which one was correct. Tim did not.
C. T. Kelley – p.23
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Mission Accomplished?
• We found two solutions; one was wrong.
• Monte knew which one was correct. Tim did not.
• One can get one or the other by• varying the initial iterate,• varying the initial grid, or• varying the details of the algorithm,
C. T. Kelley – p.23
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Mission Accomplished?
• We found two solutions; one was wrong.
• Monte knew which one was correct. Tim did not.
• One can get one or the other by• varying the initial iterate,• varying the initial grid, or• varying the details of the algorithm,
• which motivates a parametric (σ ,ε ,ρ . . .)study of the OZ equations.
C. T. Kelley – p.23
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Path Following
F : X× [a,b], F smooth, X a Banach space.Objective: Solve F(u,λ ) = 0 for λ ∈ [a,b]Obvious approach:
Set λ = a, solve F(u,λ ) = 0 withNewton-(MG, GMRES, . . . ) to obtain u0 = u(λ ).while λ < b do
Set λ = λ + dλ .Solve F(u,λ ) = 0 with u0 as the initial iterate.u0← u(λ )
end whileThe implicit function theorem says: You will not find twosolutions with identical parameter values this way.
C. T. Kelley – p.24
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What’s the problem?
• Multiple solutions, hysteresis
• No solutions
C. T. Kelley – p.25
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What’s the problem?
• Multiple solutions, hysteresis
• No solutions
A fix: Pseudo-arclength continuation.Set x = (u,λ ) and solve G(x,s) = 0, where, for example
G(x,s) =
(
FN
)
=
(
F(u(s),λ (s))uT (u−u0)+ λ T (λ −λ0)− (s− s0)
)
.
C. T. Kelley – p.25
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What’s the problem?
• Multiple solutions, hysteresis
• No solutions
A fix: Pseudo-arclength continuation.Set x = (u,λ ) and solve G(x,s) = 0, where, for example
G(x,s) =
(
FN
)
=
(
F(u(s),λ (s))uT (u−u0)+ λ T (λ −λ0)− (s− s0)
)
.
s is an artificial “arclength” parameter.u0 and λ0 are from the previous step.u≈ du/ds and λ ≈ dλ/ds,
(say by differences using s0 and s−1).
C. T. Kelley – p.25
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Simple Folds
We follow solution paths {x(s)}.Assume that F is smooth and
• Gx is nonsingular (not always true)So implicit function theorem holds in s.
We are assuming that there is no true bifurcation and thatthe singularity in λ is at worst simple fold.
dim(Null(Fu)) = 1,Fλ 6= Ran(Fu)
C. T. Kelley – p.26
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Arclength Continuation Algorithm
Set λ = a, s = 0 solve F(u,λ ) = 0 withNewton-(MG, GMRES, . . . ) to obtain u0.Estimate ds, u, λ .while s < smax do
s← s+ds.Solve G(x,s) = 0 with u0 as the initial iterate.x0← xUpdate ds, u, λ .
end while
C. T. Kelley – p.27
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How should compactness help?
• Newton-Krylov solvers: Ferng-K(00),K, Kevrekidis, Qiao (04)
C. T. Kelley – p.28
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How should compactness help?
• Newton-Krylov solvers: Ferng-K(00),K, Kevrekidis, Qiao (04)• Mesh-independent performance for compact
ranges of s,
C. T. Kelley – p.28
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How should compactness help?
• Newton-Krylov solvers: Ferng-K(00),K, Kevrekidis, Qiao (04)• Mesh-independent performance for compact
ranges of s,• Preconditioning easy or unnecessary(?).
C. T. Kelley – p.28
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How should compactness help?
• Newton-Krylov solvers: Ferng-K(00),K, Kevrekidis, Qiao (04)• Mesh-independent performance for compact
ranges of s,• Preconditioning easy or unnecessary(?).
• Multilevel solvers
C. T. Kelley – p.28
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How should compactness help?
• Newton-Krylov solvers: Ferng-K(00),K, Kevrekidis, Qiao (04)• Mesh-independent performance for compact
ranges of s,• Preconditioning easy or unnecessary(?).
• Multilevel solvers• Easy to build. Compactness smooths for you.
C. T. Kelley – p.28
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How should compactness help?
• Newton-Krylov solvers: Ferng-K(00),K, Kevrekidis, Qiao (04)• Mesh-independent performance for compact
ranges of s,• Preconditioning easy or unnecessary(?).
• Multilevel solvers• Easy to build. Compactness smooths for you.• Appropriate coarse grid data depend on s.
C. T. Kelley – p.28
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Multilevel Approach
Pathfollowing on coarse mesh + nested iteration fails.
• F(u,λ ) = u−K (u,λ )
• λ (s) is sensitive to the mesh.
• Track path on fine mesh.
• Use coarse mesh problem to approximate K u
Apply GMRES to new problem.
C. T. Kelley – p.29
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Coarse mesh problem construction
For continuation in λ• xh = xh +dx, Euler predictor on fine mesh.
• uH = IHh (uh), λ = λ H = λ h.
• Build KH = IhHK H
u (uH ,λ )IHh
• Norm convergent (K, 1995) if IHh is done right
degenerate kernel approximation
• Approximate Newton step by solvings−KHs =−Fh(uH ,λ ).Fine mesh residual and coarse mesh solve.
C. T. Kelley – p.30
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Continuation in s
Approximate Gx by
GH,hu,λ (u,λ )≡
I−∂K H(IHh u,λ )/∂u −∂K H(IH
h u,λ )/∂λ
(IHh u)T λ
.
and apply GMRES.
C. T. Kelley – p.31
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Continuation in s
Approximate Gx by
GH,hu,λ (u,λ )≡
I−∂K H(IHh u,λ )/∂u −∂K H(IH
h u,λ )/∂λ
(IHh u)T λ
.
and apply GMRES.
• Operator-function product is now on coarse mesh.
• Works for “black-box” functions. Flexible choice of K H .
• Theory follows from older work,if you coarsen only in K , not in G.
C. T. Kelley – p.31
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Details
• F and N may require scaling to make Newtonterminate properly
C. T. Kelley – p.32
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Details
• F and N may require scaling to make Newtonterminate properly
• ds must be controlled by watching for
C. T. Kelley – p.32
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Details
• F and N may require scaling to make Newtonterminate properly
• ds must be controlled by watching for• deviation of Newton’s/(step in s) from target
C. T. Kelley – p.32
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Details
• F and N may require scaling to make Newtonterminate properly
• ds must be controlled by watching for• deviation of Newton’s/(step in s) from target• curvature estimation
C. T. Kelley – p.32
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Details
• F and N may require scaling to make Newtonterminate properly
• ds must be controlled by watching for• deviation of Newton’s/(step in s) from target• curvature estimation• true bifurcation
C. T. Kelley – p.32
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Details
• F and N may require scaling to make Newtonterminate properly
• ds must be controlled by watching for• deviation of Newton’s/(step in s) from target• curvature estimation• true bifurcation
• occasional testing for bifurcation
C. T. Kelley – p.32
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Numerical Results: Three Solution Paths
For each solution we continue in ρ, and plot three scalars:
• Excess number∫
r2h(r)dr
• Pressure∫
r3U ′(r)(h(r)+1)dr
• Compressibility∫
r2c(r)dr
as functions of ρ.
C. T. Kelley – p.33
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Path through physical solution
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−150
−100
−50
0
50
100
150
200
250
ρ
compessibilityexcess numberpressure
C. T. Kelley – p.34
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Path through non-physical solution
10−3
10−2
10−1
100
101
100
101
102
103
104
ρ
Solution B
compessibilityexcess numberpressure/20
C. T. Kelley – p.35
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Path through new solution
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018−2500
−2000
−1500
−1000
−500
0
500
1000
ρ
compessibility
excess number
pressure
C. T. Kelley – p.36
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Conclusions
• OZ integro-algebraic equationsElimination leads to compact fixed point problem
• Multilevel method for integral equations
• Solves OZ, but finds too many solutions
• Bottom-up nesting goes the wrong way for continuation
• Top down works; currently 30% faster than GMRES
C. T. Kelley – p.37