Computing multiple solutions of partial differential equations
Patrick E. Farrell1
S. P. MacLachlan, T. J. Atherton, J. H. Adler, A. Majumdar, . . .
1University of Oxford
December 10, 2019
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Introduction
Section 1
Introduction
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Introduction
Can you conduct an experiment twice . . .
and get two different answers?
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Introduction
Can you conduct an experiment twice . . .
and get two different answers?
Axial displacement test of an Embraer aircraft stiffener.
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Introduction
Can you conduct an experiment twice . . .
and get two different answers?
Two different, stable configurations.P. E. Farrell (Oxford) Deflated continuation December 10, 2019 3 / 26
Introduction
Mathematical formulation
Compute the multiple solutions u of an equation
f(u, λ) = 0
f : V × R→ V ∗
as a function of a parameter λ.
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Introduction
Mathematical formulation
Compute the multiple solutions u of an equation
f(u, λ) = 0
f : V × R→ V ∗
as a function of a parameter λ.
Aircraft stiffener
u displacement, λ loading, f hyperelasticity
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Introduction
Mathematical formulation
Compute the multiple solutions u of an equation
f(u, λ) = 0
f : V × R→ V ∗
as a function of a parameter λ.
Aircraft stiffener
u displacement, λ loading, f hyperelasticity
Today
u director field or Q-tensor, f Oseen–Frank or Landau–de Gennes
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The classical algorithm
Section 2
The classical algorithm
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The classical algorithm
Branch switching
λ
u
Starting solution
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The classical algorithm
Branch switching
λ
u
Step I: continuation
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The classical algorithm
Branch switching
λ
u
Step II: continuation
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The classical algorithm
Branch switching
λ
u
Step III: detect bifurcation point
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The classical algorithm
Branch switching
λ
u
Step IV: compute eigenvectors and switch
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The classical algorithm
Branch switching
λ
u
Step V: continuation on branches
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The classical algorithm
Branch switching
λ
u
A disconnected diagram.
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The classical algorithm
Branch switching
Disconnected diagrams
The algorithm only computes branches connected to the initial datum.
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The classical algorithm
This work
Disconnected diagrams
An algorithm that can compute disconnected bifurcation diagrams.
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The classical algorithm
This work
Disconnected diagrams
An algorithm that can compute disconnected bifurcation diagrams.
Scaling
The computational kernel is exactly the same as Newton’s method.
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Deflation
Section 3
Deflation
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Deflation
The core idea
Deflation
Fix parameter λ. Given
I a Frechet differentiable residual F : V → V ∗
I a solution r ∈ V , F(r) = 0, F ′(r) nonsingular
construct a new nonlinear problem G : V → V ∗ such that:
I (Preservation of solutions) F(r) = 0 ⇐⇒ G(r) = 0 ∀ r 6= r;
I (Deflation property) Newton’s method applied to G will never convergeto r again, starting from any initial guess.
Find more solutions, starting from the same initial guess.
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Deflation
The core idea
Deflation
Fix parameter λ. Given
I a Frechet differentiable residual F : V → V ∗
I a solution r ∈ V , F(r) = 0, F ′(r) nonsingular
construct a new nonlinear problem G : V → V ∗ such that:
I (Preservation of solutions) F(r) = 0 ⇐⇒ G(r) = 0 ∀ r 6= r;
I (Deflation property) Newton’s method applied to G will never convergeto r again, starting from any initial guess.
Find more solutions, starting from the same initial guess.
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Deflation
The core idea
Deflation
Fix parameter λ. Given
I a Frechet differentiable residual F : V → V ∗
I a solution r ∈ V , F(r) = 0, F ′(r) nonsingular
construct a new nonlinear problem G : V → V ∗ such that:
I (Preservation of solutions) F(r) = 0 ⇐⇒ G(r) = 0 ∀ r 6= r;
I (Deflation property) Newton’s method applied to G will never convergeto r again, starting from any initial guess.
Find more solutions, starting from the same initial guess.
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Deflation
The core idea
Deflation
Fix parameter λ. Given
I a Frechet differentiable residual F : V → V ∗
I a solution r ∈ V , F(r) = 0, F ′(r) nonsingular
construct a new nonlinear problem G : V → V ∗ such that:
I (Preservation of solutions) F(r) = 0 ⇐⇒ G(r) = 0 ∀ r 6= r;
I (Deflation property) Newton’s method applied to G will never convergeto r again, starting from any initial guess.
Find more solutions, starting from the same initial guess.
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Deflation
The core idea
Deflation
Fix parameter λ. Given
I a Frechet differentiable residual F : V → V ∗
I a solution r ∈ V , F(r) = 0, F ′(r) nonsingular
construct a new nonlinear problem G : V → V ∗ such that:
I (Preservation of solutions) F(r) = 0 ⇐⇒ G(r) = 0 ∀ r 6= r;
I (Deflation property) Newton’s method applied to G will never convergeto r again, starting from any initial guess.
Find more solutions, starting from the same initial guess.
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Deflation
Finding many solutions from the same guess
F
F
F
Starting setup
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Deflation
Finding many solutions from the same guess
F
F
F
F
Step I: Newton from initial guess
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Deflation
Finding many solutions from the same guess
F
F
F
F
Step II: deflate solution found
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Deflation
Finding many solutions from the same guess
F
F
F
F
F
Step I: Newton from initial guess
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Deflation
Finding many solutions from the same guess
F
F
F
F
F
Step II: deflate solution found
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Deflation
Finding many solutions from the same guess
F
F
F
F
F
F
Step I: Newton from initial guess
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Deflation
Finding many solutions from the same guess
F
F
F
F
F
F
Step II: deflate solution found
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Deflation
Finding many solutions from the same guess
F
F
F
F
F
F
Step III: termination on nonconvergence
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Deflation
Finding many solutions from the same guess
F
F
F
F
F
FF
Step III: termination on nonconvergence
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Deflation
Construction of deflated problems
A nonlinear transformation
G(u) =M(u; r)F(u)
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Deflation
Construction of deflated problems
A nonlinear transformation
G(u) =M(u; r)F(u)
A deflation operator
We say M(u; r) is a deflation operator if for any sequence u→ r
lim infu→r
‖G(u)‖V ∗ = lim infu→r
‖M(u; r)F(u)‖V ∗ > 0
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Deflation
Construction of deflated problems
A nonlinear transformation
G(u) =M(u; r)F(u)
A deflation operator
We say M(u; r) is a deflation operator if for any sequence u→ r
lim infu→r
‖G(u)‖V ∗ = lim infu→r
‖M(u; r)F(u)‖V ∗ > 0
Theorem (F., Birkisson, Funke, 2014)
This is a deflation operator for p ≥ 1:
M(u; r) =
(1
‖u− r‖p + 1
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Deflation
Deflated continuation
λ
u
Starting solution
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Deflation
Deflated continuation
λ
u
Step I: continuation
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Deflation
Deflated continuation
λ
u
Step II: continuation
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Deflation
Deflated continuation
λ
u
Step III: deflate
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Deflation
Deflated continuation
λ
u
Step III+: solve deflated problem
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Deflation
Deflated continuation
λ
u
Step III: deflate
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Deflation
Deflated continuation
λ
u
Step III+: solve deflated problem
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Deflation
Deflated continuation
λ
u
Step IV: continuation on branches
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Deflation
Deflated continuation
λ
u
A disconnected diagram.
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Computations
Section 4
Computations
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Computations
Newton–Krylov
A question
How do we solve the deflated problem?
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Computations
Newton–Krylov
A Newton step
JF (u)∆uF = −F (u)
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Computations
Newton–Krylov
A Newton step
JF (u)∆uF = −F (u)
A deflated Newton step
JG(u)∆uG = −G(u)
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Computations
Newton–Krylov
A Newton step
JF (u)∆uF = −F (u)
A deflated Newton step
JG(u)∆uG = −G(u)
Deflated residual
G(u) = M(u; r)F (u)
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Computations
Newton–Krylov
A Newton step
JF (u)∆uF = −F (u)
A deflated Newton step
JG(u)∆uG = −G(u)
Deflated Jacobian
JG(u) = M(u; r)JF (u) + F (u)M ′(u; r)T
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Computations
Newton–Krylov
A Newton step
JF (u)∆uF = −F (u)
A deflated Newton step
JG(u)∆uG = −M(u)F (u)
Deflated Jacobian
JG(u) = M(u; r)JF (u) + F (u)M ′(u; r)T
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Computations
Newton–Krylov
A Newton step
JF (u)∆uF = −F (u)
A deflated Newton step
JG(u)∆uG = −G(u)
Sherman–Morrison–Woodbury
∆uG = τ∆uF
where τ ∈ R is a simple function of J−1F F,M, and M ′.
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Computations
Newton–Krylov
Scaling of deflated continuation
With a good preconditioner, you can do bifurcation analysis at scale.
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Applications
Section 5
Applications
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Applications Nonlinear PDEs
Application: Carrier’s problem
Carrier’s problem (Carrier 1970, Bender & Orszag 1999)
ε2y′′ + 2(1− x2)y + y2 − 1 = 0, y(−1) = 0 = y(1).
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Applications Nonlinear PDEs
Application: Carrier’s problem
0.05 0.1 0.25 0.7−200
−150
−100
−50
0
50
100
150
200
ε
y′(−
1)‖y‖2
Solutions of ε2y′′+2(1− x2)y+ y2 −1 = 0
Pitchfork bifurcation
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Applications Nonlinear PDEs
Application: Carrier’s problem
0.05 0.1 0.25 0.7−200
−150
−100
−50
0
50
100
150
200
ε
y′(−
1)‖y‖2
Solutions of ε2y′′+2(1− x2)y+ y2 −1 = 0
Pitchfork bifurcationFold bifurcation
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Applications Nonlinear PDEs
Application: Carrier’s problem
0.05 0.1 0.25 0.7−200
−150
−100
−50
0
50
100
150
200
ε
y′(−
1)‖y‖2
Solutions of ε2y′′+2(1− x2)y+ y2 −1 = 0
Pitchfork bifurcationFold bifurcation
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Applications Nonlinear PDEs
Application: Carrier’s problem
0.05 0.1 0.25 0.7−200
−150
−100
−50
0
50
100
150
200
ε
y′(−
1)‖y‖2
Solutions of ε2y′′+2(1− x2)y+ y2 −1 = 0
Pitchfork bifurcationFold bifurcation
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Applications Nonlinear PDEs
Application: Carrier’s problem
Pitchfork bifurcations
ε ≈ 0.472537
n
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Applications Nonlinear PDEs
Application: Carrier’s problem
Pitchfork bifurcations
ε ≈ 0.472537
n
Connected Computed Asymptotic Relativecomponent ε estimate error
1 0.46886251 0.472537 0.7837%2 0.23472529 0.236269 0.6574%3 0.15703946 0.157512 0.3012%4 0.11798359 0.118134 0.1278%
Computed and estimated parameter values for the first four pitchfork bifurcations.
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Applications Nonlinear PDEs
Application: Carrier’s problem
Fold bifurcations
ε ≈ 0.472537
n− 0.8344n
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Applications Nonlinear PDEs
Application: Carrier’s problem
Fold bifurcations
ε ≈ 0.472537
n− 0.8344n
Connected Computed Asymptotic Relativecomponent ε estimate error
2 0.28522538 0.298545 4.670%3 0.17186970 0.173608 1.011%4 0.12421206 0.124634 0.3397%5 0.09762446 0.0977706 0.1497%
Computed and estimated parameter values for the first four fold bifurcations.
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Applications Nonlinear PDEs
Application: Freedericksz transition
Description
A classical pitchfork bifurcation. Below a certain electric field threshold,the liquid crystal remains undistorted. Beyond a critical strength V , thedirector twists to align with the field.
Minimise Oseen–Frank energy on a unit square subject to
I n periodic in x and parallel to x-axis along y = 0, y = 1
I Frank constants (K1,K2,K3) = (1, 0.62903, 1.32258) (5CB)
I electric potential φ(x, 0) = 0, φ(x, 1) = V
I permittivity of free space ε0 = 1.42809
I perpendicular dielectric permittivity ε⊥ = 7
I dielectric anisotropy εa = 11.5
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Applications Nonlinear PDEs
Application: Freedericksz transition
Bifurcation diagrams for maximum angular tilt and free energy as a function ofV . The critical voltage is V ∗ ≈ 0.775.
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Applications Nonlinear PDEs
Application: Freedericksz transition
Three solutions for V = 1.1.
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Applications Nonlinear PDEs
Application: escape and disclination solutions
Minimise Oseen–Frank energy on a unit square subject to
I n radial from the centre
I Frank constants (K1,K2,K3) = (1, 3, 1.2)
I no electric field present
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Applications Nonlinear PDEs
Application: escape and disclination solutions
Two escape and one disclination solution, with energies (9.971, 24.042, 9.971).The energy of the middle solution diverges with mesh refinement.
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Applications Nonlinear PDEs
Application: square well filled with nematic LCs
We consider the square wells filled with nematic liquid crystals consideredby Tsakonas et al. (Appl. Phys. Lett, 2007) and Majumdar et al.
Minimise Landau–de Gennes energy on a square subject to
I Q11 ≥ 0 on horizontal edges,
I Q11 ≤ 0 on vertical edges,
I Q12 = 0 on ∂Ω.
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Applications Nonlinear PDEs
Application: square well filled with nematic LCs
Bifurcation diagram showing stable states as a function of square edge length D.
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Applications Nonlinear PDEs
Application: square well filled with nematic LCs
21 different stationary points, coloured by the order parameter, for D = 1.5µm.
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Applications Nonlinear PDEs
Application: geometrically frustrated cholesteric
Description
In the absence of boundaries, the cholesteric adopts a helical structurewith a preferred pitch q0. On an ellipse, the boundary conditions precludethe energetically preferred uniformly twisted state. This frustration isresolved by deformation of the cholesteric layers or the introduction ofdefects, in multiple ways.
Minimise Oseen–Frank energy with cholesteric term in an ellipse subject to
I n = (0, 0, 1) on the boundary
I Frank constants (K1,K2,K3) = (1, 3.2, 1.1)
I no electric field
as a function of cholesteric pitch q0.
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Applications Nonlinear PDEs
Application: geometrically frustrated cholesteric
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Applications Nonlinear PDEs
Application: geometrically frustrated cholesteric
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Applications Nonlinear PDEs
Application: geometrically frustrated cholesteric
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Applications Nonlinear PDEs
Application: smectic-A
Pevnyi–Selinger–Sluckin (2014) model
Minimise
J(n) =
∫Ω
a
2δρ2 +
c
4δρ4 +B
∣∣∇∇δρ+ q2n⊗ nδρ∣∣2 +
K
2|∇n|2 dx
subject to n · n = 1, periodic in x, Dirichlet on n for y ∈ 0, 1.
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Applications Nonlinear PDEs
Application: smectic-A
Pevnyi–Selinger–Sluckin (2014) model
Minimise
J(n) =
∫Ω
a
2δρ2 +
c
4δρ4 +B
∣∣∇∇δρ+ q2n⊗ nδρ∣∣2 +
K
2|∇n|2 dx
subject to n · n = 1, periodic in x, Dirichlet on n for y ∈ 0, 1.
Difficult discretisation
Finite element discretisation is difficult because δρ ∈ H2(Ω).
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Applications Nonlinear PDEs
Application: smectic-A
Pevnyi–Selinger–Sluckin (2014) model
Minimise
J(n) =
∫Ω
a
2δρ2 +
c
4δρ4 +B
∣∣∇∇δρ+ q2n⊗ nδρ∣∣2 +
K
2|∇n|2 dx
subject to n · n = 1, periodic in x, Dirichlet on n for y ∈ 0, 1.
Difficult discretisation
Finite element discretisation is difficult because δρ ∈ H2(Ω).
Solution: nonconforming Morley element
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Applications Nonlinear PDEs
Application: smectic-A
δρ for some of 73 solutions found for q = 30, a = −10, c = 10, B = 10−5,K = 0.3.
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Conclusion
Conclusions
I Multiple solutions are ubiquitous and important in liquid crystals.
I Deflation is a useful technique for finding them.
I Deflated problems can be solved efficiently.
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Conclusion
Conclusions
I Multiple solutions are ubiquitous and important in liquid crystals.
I Deflation is a useful technique for finding them.
I Deflated problems can be solved efficiently.
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Conclusion
Conclusions
I Multiple solutions are ubiquitous and important in liquid crystals.
I Deflation is a useful technique for finding them.
I Deflated problems can be solved efficiently.
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