numerical solution of a non-smooth eigenvalue problem
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Numerical Solution of a Non-Smooth Eigenvalue Problem
An Operator-Splitting Approach
A. Caboussat & R. Glowinski
1. Formulation. Motivation
Our main objective is the numerical solution of the following problem from Calculus of Variations
Compute
γ = inf v Σ ∫Ω|v|dx (NSEVP)
where: Ω is a bounded domain of R2 and
Σ = {v| v H01(Ω), ∫Ω|v|2dx = 1}.
Actually, γ = 2√ π , independently of the shape and size
of Ω (holds even for non-simply connected Ω and in
fact for unbounded Ω) (G. Talenti).
A natural question is then:
Why solve numerically a problem whose exact solution is known ?
(i) If I claim that it is a new method to compute π nobody will believe me.
(ii) (NSEVP) is a fun problem to test solution methods for non-smooth & non-convex optimization problems.
(iii) ∫Ω|v|dx arises in a variety of problems from Image
Processing and Plasticity.
Actually, our motivation for investigating (NSEVP) arises from the following problem from visco-plasticity :
u L2(0,T; H01(Ω)) C0([0,T ]; L2(Ω)); u(0) = u0,
(BFP) ρ∫Ω(∂u/∂t)(t)(v – u(t))dx + μ∫Ωu(t).(v – u(t))dx +
g[ ∫Ω|v|dx – ∫Ω|u(t)|dx ] ≥ C(t)∫Ω(v – u(t))dx,
v H01(Ω), a.e. t (0, T),
with ρ > 0, μ > 0, g > 0, Ω a bounded domain of R2 and u0
L2(Ω).
(BFP) models the flow of a Bingham visco-plastic fluid in an infinitely long cylinder of cross section Ω, C being the pressure drop per unit length. Suppose that C = 0 and that T = +∞; we can show that
(C-O.PR) u(t) = 0, t ≥ Tc,
with
Tc = (ρ/μλ0)ln[1 + (μλ0/γg)||u0||L2(Ω)],
λ0 being the smallest eigenvalue of – 2 in H01(Ω).
A similar cut-off property holds if after space discretization we usethe backward Euler scheme for the time discretization of (BFP),
with λ0 and γ replaced by their discrete analogues λ0h and γh.
Suppose that the space discretization is achieved via C0-piecewise
linear finite element approximations, we have then
|λ0h – λ0| = O(h2).
But what can we say about |γh – γ| ?
The main goal of this lecture is to look for answers to the
above question !
2. Some regularization procedures There are several ways to approximate (NSEVP) – at the
continuous level – by a better posed and/or smoother
variational problem. The most obvious candidate is clearly
γε = inf v Σ ∫Ω(|v|2 + ε2)½dx, (NSEVP.1)ε
a regularization quite popular in Image Processing.
Assuming that the above problem has a minimizer uε, this
minimizer verifies the following Euler-Lagrange equation
(reminiscent of the mean curvature equation):
First regularized problem:
.1d||
,on 0,in ||
2
22
xu
uuu
u
(RP.1)
.
(RP.1) is clearly a nonlinear eigenvalue problem for a
close variant of the mean curvature operator, the eigenva
lue being γε.
Another regularization, more sophisticated in some sense,
since this time the regularized problem has minimizers, is
provided (with ε > 0) by
γε = min v Σ [ ½ ε∫Ω|v|2dx + ∫Ω|v|dx ]. (NSEVP.2)ε
An associated Euler-Lagrange (multivalued) equation
reads as follows, also of the nonlinear (in fact, non-
smooth) eigenvalue type (as above the eigenvalue is γε):
– ε2uε + ∂j(uε) γεuε in Ω,
(RP.2) uε = 0 on ∂Ω,
∫Ω|uε|2dx = 1;
in (RP.2), ∂j(uε) is the subgradient at uε of the functional
j : H01(Ω) → R defined by
j(v) = ∫Ω|v|dx.
The solution of problems such as (RP.2) is discussed in
GKM (2007); the method used in the above referenceis of the operator-splitting/inverse power method type.
In order to avoid handling simultaneously two small
parameters, namely ε and h, we will address the solution
of
γ = inf v Σ ∫Ω|v|dx
without using any regularization (unless we consider the
space approximation as a kind of regularization, that it is
indeed).
3. Finite Element Approximation(i) First, we introduce a family {Ωh}h of polygonal approxi-
mations of Ω, such that
limh→0 Ωh = Ω.
(ii) With each Ωh we associate a triangulation Th verifying
the usual assumptions of: (a) compatibility between triangles, and (b) regularity.
(iii) With each Th we associate the finite dimensional space
V0h defined (classically) as follows:
V0h = {v| v C0(Ωh∂Ωh), v|T P1, T Th,
v = 0 on ∂Ωh}.
(iv) We approximate
γ = inf v Σ ∫Ω|v|dx (NSEVP)
by
γh = min v Σh ∫Ωh |v|dx (NSEVP)h
with
Σh = {v| v V0h, ||v||L2(Ωh) = 1}. It is easy to prove that:
(i) Problem (NSEVP)h has a solution, that is there exists
uh Σh such that
∫Ωh |uh|dx = γh.
(ii) limh→0 γh = γ ( = 2√π).
We would like to investigate (computationally) the order
of the convergence of γh to γ. From the non-smoothness
of the problem, we do not expect O(h2).
4. An iterative method for the solution
of (NSEVP)h We are going to look for robustness, modularity and simplicity of programming instead of performance measured in number of elementary operations (this is not image processing and/or real time). At ADI 50 ( December
2005, at Rice University), we showed that the inverse power method for eigenvalue computations has an operator-splitting interpretation; we also showed the equivalence between some augmented Lagrangian algorithms and ADI methods such as Douglas-Rachford’s and Peaceman-Rachford’s. For problem
(NSEVP)h we think that it is simpler to take the AL approa-ch, keeping in mind that it will lead to a ‘disguised’ ADI method.
For formalism simplicity, we will use the continuous
problem notation. We observe that there is equivalence
between
γ = inf v Σ ∫Ω|v|dxand
γ = inf {v, q, z} E ∫Ω|q|dx,
where
E = {{v, q, z}| v H01(Ω), q (L2(Ω))2, z L2(Ω),
v – q = 0, v – z = 0, ||z||L2(Ω) = 1}.
The above equivalence suggests introducing the followingaugmented Lagrangian functional
Lr : (H01(Ω)×Q×L2(Ω))×(Q×L2(Ω)) → R
defined as follows, with Q = (L2(Ω))2 and r = {r1, r2}, ri > 0,
Lr(v, q, z; μ1, μ2) = ∫Ω|q|dx + ½ r1 ∫Ω|v – q|2dx
+ ½ r2 ∫Ω|v – z|2dx + ∫Ω(v – q).μ1dx
+ ∫Ω(v – z)μ2dx
We consider then, the following saddle-point problem
Find {{u, p, y}, {λ1, λ2}} (H01(Ω)×Q×S)×(Q×L2(Ω))
such that
Lr(u, p, y; μ1, μ2) ≤ Lr(u, p, y; λ1, λ 2) ≤ Lr(v, q, z; λ1, λ 2), (SDP-P) {{v, q, z}, {μ1, μ2}} (H0
1(Ω)×Q×S)×(Q×L2(Ω)),
with S = {z| z L2(Ω), ||z||L2(Ω) = 1}.
Suppose that the above saddle-point problem has a solution. We have then p = u, y = u, u being a minimizer for the original mimimization problem (the primal one).
To solve the above saddle-point problem, we advocate
the algorithm below (called ALG 2 by some practitioners
(BB)):
(1) {u –1, {λ10, λ2
0}} is given in H01(Ω)×(Q×L2(Ω));
for n ≥ 0, assuming that {un – 1, {λ1n, λ2
0}} is known,
solve:
(2) {pn, yn} = arg min{q, z} Q×S Lr(un – 1, q, z; λ1n, λ 2
n),
then
(3) un = arg minv Lr(v, pn, yn; λ1n, λ 2
n), v H01(Ω),
(4) λ1n+1 = λ1
n + r1(un – pn), λ2n+1 = λ2
n + r2(un – yn).
The above algorithm is easy to implement since:
(i) Problem (3) is equivalent to the following linear variational problem in H0
1(Ω)
un H01(Ω),
r1∫Ωun.v dx + r2 ∫Ωunv dx = ∫Ω(r1pn – λ1n ).v dx
+ ∫Ω(r2yn – λ2n )v dx, v H0
1(Ω).
The solution of the discrete analogue of the above
problem is a simple task nowadays.
(ii) Problem (2) decouples as
(a) pn = arg min q Q [½ r1 ∫Ω |q|2dx + ∫Ω|q|dx
– ∫Ω(r1un + λ1n).qdx ].
(b) yn = arg min z S [½ r2 ∫Ω |z|2dx – ∫Ω(r2un + λ2n)zdx ].
Both problems have closed form solutions; indeed, since
||z||L2(Ω) = 1, z S, one has
yn = (r2un + λ2n) / ||r2un + λ2
n ||L2(Ω).
Similarly, the minimization problem in (a) can be solved
point-wise (one such elementary problem for each triangle
of Th, in practice). We obtain then, a.e. on Ω,
pn(x) = (1/r1) (1 – 1/|Xn(x)|)+ Xn(x),
where
Xn(x) = r1un(x) + λ1n(x).
5. Numerical experimentsFirst Test Problem: Ω is the unit disk
Unit Disk Test Problem
Variation of γh versus h
Unit Disk Test Problem
Variation of γh – γ versus h
Unit Disk Test Problem
Visualisation of the coarse mesh solution
Unit Disk Test Problem
Visualisation of the fine mesh solution
Unit Disk Test Problem
Coarse mesh solution contours
Unit Disk Test Problem
Fine mesh solution contours
Unit Disk Test Problem
Fine mesh solution contours (details)
Second Test Problem: Ω is the unit square
Coarse mesh
Unit Square Test Problem
Fine mesh
Unit Square Test Problem
Variation of γh versus h
Unit Square Test Problem
Variation of γh – γ versus h
Unit Square Test Problem Visualisation of the coarse mesh solution
Unit Square Test Problem
Visualisation of the fine mesh solution
Unit Square Test Problem
Contours of the coarse mesh solution
Unit Square Test Problem
Contours of the fine mesh solution
Unit Square Test Problem
Contours of the fine mesh solution (details)
Circular Ring Test Problem (coarse mesh)
Circular Ring Test Problem (fine mesh)
A GENERALIZATION
Compute for Ω R2
γ* = infv ∫Ω |v|dx
with
= {v| v (H10(Ω))2, ||v||(L2(Ω))2 = 1}.
Conjecture (unless it is a classical result):
...04805.33
2sin1
2
13
2
0
d*γ
Square (coarse mesh)
Square (fine mesh)
Disk (coarse mesh)
Disk (fine mesh)
The results of our numerical computations
suggest very strongly that the value we conjectu-
red for γ* is the good one.
APPLICATION to a SEDIMENTATION PROBLEM
The following problem has been considered by C. Evans &
L. Prigozhin
u/t + IK(u) f in Ω × (0, T),
(SP)
u(0) = u0,
with Ω R2 and
K = {v | v H1(Ω), |v| C, v = g on Γ0 ( Ω)}.
After time-discretization by the backward Euler scheme, we
obtain
(1) u0 = u0 ;
n ≥ 1, un – 1 → un as follows
(2) un – un – 1 + IK(un) Δt f n.
“Equation” (2) is the Euler-Lagrange equation of the following
problem from Calculus of Variations:
(MP) un = arg minv K [ ½ Ωv2 dx – Ω(un – 1 + Δt f n)vdx].
The minimization problem (MP) is equivalent to:
{un, pn } =
arg min{v, q} K [½ Ωv2 dx – Ω(un – 1 + Δt f n)vdx],
with
K = {{v, q}| v H1(Ω), v = g on Γ0, |q| C, v – q = 0}.
We can compute {un, pn } via the following augmented
Lagrangian
Lr(v, q; μ) = ½ r Ω |v – q|2 dx + Ω μ.(v – q) dx
+ ½ Ωv2 dx – Ω(un – 1 + Δt f n)vdx.
River sand pile: FE mesh
River sand pile (2)
River sand pile (3)
River sand pile (4)
Rectangular pond sand pile (1)
Rectangular pond sand pile (2)
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