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)