Numerical%20Solution%20of%20a%20Non-Smooth%20Eigenvalue%20Problem

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Numerical Solution of a Non-Smooth Eigenvalue
Problem

• An Operator-Splitting Approach
• A. Caboussat R. Glowinski

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1. Formulation. Motivation

• Our main objective is the numerical solution of
  the
• following problem from Calculus of Variations
• Compute
• ? inf v ?S ?O?vdx
  (NSEVP)
• where O is a bounded domain of R2 and
• S v v ? H01(O), ?Ov2dx
  1.

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Actually, ? 2v p , independently of the shape
and size

• of O (holds even for non-simply connected O and
  in
• fact for unbounded O) (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
  p nobody will believe me.
• (ii) (NSEVP) is a fun problem to test solution
  methods
• for non-smooth non-convex optimization
  problems.

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(iii) ?O?vdx 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(O))? C0(0,T
  L2(O)) u(0) u0,
• (BFP) ??O(?u/?t)(t)(v u(t))dx µ?O?u(t).?(v
  u(t))dx
• g ?O?vdx ?O?u(t)dx
  C(t)?O(v u(t))dx,
• ?? v ? H01(O), a.e. t ? (0, T),
• with ? gt 0, µ gt 0, g gt 0, O a bounded domain of
  R2 and u0
• ? L2(O).

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• (BFP) models the flow of a Bingham visco-plastic
  fluid in an
• infinitely long cylinder of cross section O, C
  being the pressure
• drop per unit length. Suppose that C 0 and that
  T 8 we can
• show that
• (C-O.PR) u(t) 0, ? t Tc,
• with
• Tc (?/µ?0)ln1
  (µ?0/?g)u0L2(O),
• ?0 being the smallest eigenvalue of ?2
  in H01(O).
• A similar cut-off property holds if after space
  discretization we use
• the backward Euler scheme for the time
  discretization of (BFP),
• with ?0 and ? replaced by their discrete
  analogues ?0h and ?h.

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• Suppose that the space discretization is achieved
  via C0-piecewise
• linear finite element approximations, we have
  then
• ?0h ?0
  O(h2).
• But what can we say about ?h
  ? ?
• The main goal of this lecture is to look for
  answers to the
• above question !

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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
• ?e inf v ?S ?O(?v2 e2)½dx,
  (NSEVP.1)e
• a regularization quite popular in Image
  Processing.
• Assuming that the above problem has a minimizer
  ue, this
• minimizer verifies the following Euler-Lagrange
  equation
• (reminiscent of the mean curvature equation)

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• First regularized problem

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• (RP.1) is clearly a nonlinear eigenvalue problem
  for a
• close variant of the mean curvature operator, the
  eigenva
• lue being ?e.
• Another regularization, more sophisticated in
  some sense,
• since this time the regularized problem has
  minimizers, is
• provided (with e gt 0) by
• ?e min v ?S ½ e?O?v2dx ?O?vdx .
  (NSEVP.2)e
• An associated Euler-Lagrange (multivalued)
  equation
• reads as follows, also of the nonlinear (in fact,
  non-
• smooth) eigenvalue type (as above the eigenvalue
  is ?e)

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• e?2ue ?j(ue) ? ?eue in O,
• (RP.2) ue 0 on ?O,
• ?Oue2dx 1
• in (RP.2), ?j(ue) is the subgradient at ue of the
  functional
• j H01(O) ? R defined by
• j(v) ?O?vdx.
• The solution of problems such as (RP.2) is
  discussed in
• GKM (2007) the method used in the above
  reference
• is of the operator-splitting/inverse power method
  type.

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• In order to avoid handling simultaneously two
  small
• parameters, namely e and h, we will address the
  solution
• of
• ? inf v ?S ?O?vdx
• without using any regularization (unless we
  consider the
• space approximation as a kind of regularization,
  that it is
• indeed).

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3. Finite Element Approximation

• (i) First, we introduce a family Ohh of
  polygonal approxi- mations of O, such that
• limh?0 Oh O.
• (ii) With each Oh 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

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• V0h v v ? C0(Oh??Oh), vT ? P1, ? T ? Th,
• v 0 on ?Oh.
• (iv) We approximate
• ? inf v ?S ?O?vdx
  (NSEVP)
• by
• ?h min v ?Sh ?Oh ?vdx
  (NSEVP)h

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• with
• Sh v v ? V0h, vL2(Oh)
  1.
• It is easy to prove that
• (i) Problem (NSEVP)h has a solution, that is
  there exists
• uh ? Sh such that
• ?Oh ?uhdx ?h.
• (ii) limh?0 ?h ? ( 2vp).
• We would like to investigate (computationally)
  the order
• of the convergence of ?h to ?. From the
  non-smoothness
• of the problem, we do not expect O(h2).

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4. An iterative method for the solutionof
(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-
• Rachfords and Peaceman-Rachfords. 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.

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• For formalism simplicity, we will use the
  continuous
• problem notation. We observe that there is
  equivalence
• between
• ? inf v ?S ?O?vdx
• and
• ? inf v, q, z ? E
  ?Oqdx,
• where
• E v, q, z v ? H01(O), q ? (L2(O))2, z ?
  L2(O),
• ?v q 0, v z 0, zL2(O) 1.

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• The above equivalence suggests introducing the
  following
• augmented Lagrangian functional
• Lr (H01(O)QL2(O))(QL2(O)) ? R
• defined as follows, with Q (L2(O))2 and r
  r1, r2, ri gt 0,
• Lr(v, q, z µ1, µ2) ?Oqdx ½ r1 ?O?v
  q2dx
• ½ r2 ?Ov z2dx ?O(?v
  q).µ1dx
• ?O(v z)µ2dx

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• We consider then, the following saddle-point
  problem
• Find u, p, y, ?1, ?2 ? (H01(O)QS)(QL2(O)
  )
• such that
• Lr(u, p, y µ1, µ2) Lr(u, p, y ?1, ? 2)
  Lr(v, q, z ?1, ? 2),
• (SDP-P)
  
• ? v, q, z, µ1, µ2 ? (H01(O)QS)(QL2(
  O)),
• with
• S z z ? L2(O), zL2(O)
  1.
• Suppose that the above saddle-point problem has a
  solut
• ion. We have then p ?u, y u, u being a
  minimizer for
• the original mimimization problem (the primal
  one).

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• To solve the above saddle-point problem, we
  advocate
• the algorithm below (called ALG 2 by some
  practitioners
• (BB))
• (1) u 1, ?10, ?20 is given in
  H01(O)(QL2(O))
• for n 0, assuming that un 1, ?1n, ?20 is
  known,
• solve
• (2) pn, yn arg minq, z ?QS Lr(un 1, q,
  z ?1n, ? 2n),
• then
• (3) un arg minv Lr(v, pn, yn ?1n, ? 2n), v ?
  H01(O),
• (4) ?1n1 ?1n r1(?un pn), ?2n1 ?2n
  r2(un yn).

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• The above algorithm is easy to implement since
• (i) Problem (3) is equivalent to the following
  linear variational problem in H01(O)
• un ? H01(O),
• r1?O?un.?v dx r2 ?Ounv dx ?O(r1pn ?1n ).?v
  dx
• ?O(r2yn ?2n )v dx, ? v ? H01(O).
• The solution of the discrete analogue of the
  above
• problem is a simple task nowadays.

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• (ii) Problem (2) decouples as
• (a) pn arg min q ? Q ½ r1 ?O q2dx ?Oqdx
• ?O(r1?un
  ?1n).qdx .
• (b) yn arg min z ? S ½ r2 ?O z2dx ?O(r2un
  ?2n)zdx .
• Both problems have closed form solutions indeed,
  since
• zL2(O) 1, ? z ? S, one has
• yn (r2un ?2n) / r2un
  ?2n L2(O).

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• 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 O,
• pn(x) (1/r1) (1 1/Xn(x))
  Xn(x),
• where
• Xn(x) r1?un(x)
  ?1n(x).

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5. Numerical experiments

• First Test Problem O is the unit disk

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Unit Disk Test Problem

• Variation of ?h versus h

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Unit Disk Test Problem

• Variation of ?h ? versus h

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Unit Disk Test Problem

• Visualisation of the coarse mesh solution

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Unit Disk Test Problem

• Visualisation of the fine mesh solution

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Unit Disk Test Problem

• Coarse mesh solution contours

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Unit Disk Test Problem

• Fine mesh solution contours

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Unit Disk Test Problem

• Fine mesh solution contours (details)

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Second Test Problem O is the unit square

• Coarse mesh

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Unit Square Test Problem

• Fine mesh
• 
  

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Unit Square Test Problem

• Variation of ?h versus h

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Unit Square Test Problem

• Variation of ?h ? versus h

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Unit Square Test Problem

• Visualisation of the coarse mesh solution

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Unit Square Test Problem

• Visualisation of the fine mesh solution

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Unit Square Test Problem

• Contours of the coarse mesh solution
• 
  

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Unit Square Test Problem

• Contours of the fine mesh solution

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Unit Square Test Problem
Contours of the fine mesh solution (details)
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• Circular Ring Test Problem (coarse mesh)

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• Circular Ring Test Problem (fine mesh)

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• A GENERALIZATION
• Compute for O ? R2
• ? infv?? ?O ?vdx
  
• with
• ? v v ? (H10(O))2,
  v(L2(O))2 1.

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• Conjecture (unless it is a classical result)

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• Square (coarse mesh)

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Square (fine mesh)
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Disk (coarse mesh)
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Disk (fine mesh)
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• The results of our numerical computations
• suggest very strongly that the value we conjectu-
• red for ? is the good one.

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APPLICATION to a SEDIMENTATION PROBLEM

• The following problem has been considered by C.
  Evans
• L. Prigozhin
• ?u/?t ?IK(u) ? f in O
  (0, T),
• 
  (SP)
• u(0) u0,
• with O ? R2 and
• K v v ? H1(O), ?v ? C, v g on G0 ( ?
  ?O).

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• 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) ? ?t f n.
• Equation (2) is the Euler-Lagrange equation of
  the following
• problem from Calculus of Variations
• (MP) un arg minv ?K ½ ?Ov2 dx ?O(un 1
  ?t f n)vdx.

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• The minimization problem (MP) is equivalent to
• un, pn
• arg minv, q ?K ½ ?Ov2 dx ?O(un
  1 ?t f n)vdx,
• with
• K v, q v ? H1(O), v g on G0, q ? C, ?v
  q 0.
• We can compute un, pn via the following
  augmented
• Lagrangian
• Lr(v, q µ) ½ r ?O ?v q2 dx ?O µ.(?v q)
  dx
• ½ ?Ov2 dx ?O(un 1 ?t f n)vdx.

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• River sand pile FE mesh

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River sand pile (2)
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River sand pile (3)
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• River sand pile (4)

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Rectangular pond sand pile (1)
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Rectangular pond sand pile (2)
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