Linear Variational Problems Part II

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Lecture 2

• Linear Variational Problems (Part II)

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Conjugate Gradient Algorithms for Linear
Variational Problems in Hilbert Spaces

• 1.Introduction. Synopsis. Conjugate gradient
  algorithms
• are among the most popular methods of Scientific
• Computing. Introduced initially for the solution
  of linear
• finite dimensional problems they have found
  applications
• for the solution of nonlinear and/or infinite
  dimensional
• problems and, combined with least-squares can be
• applied to the solution of critical point
  problems which
• are not minimization ones. We will begin our
  discussion
• with the solution of linear variational problems
  (LVP)
• in Hilbert spaces when the bilinear functional
  a(,.,) is
• symmetric.

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2. Formulation of the basic problem. The basic
problemto be considered was discussed in Lecture
1 it reads as follows (with a(.,.) symmetric,
here)

• Find u ? V such that
• (LVP)
• a(u,v) L(v), ? v ? V,
• with V, a, L as in Lecture
  I.

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3. Description of the Conjugate Gradient
Algorithm. The algorithm reads as follows

• Step 0 Initialization
• (1) u0 is given in V
• Solve
• g0 ? V,
• (2)
• (g0,v) a(u0,v) L(v), ? v ? V,
• Set (if g0 ? 0)
• (3) w0 g0.

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For n 0, assuming that un, gn and wn are known,
the last two different from 0, we update them as
follows

• Step 1 Descent
• (4) ?n gn2/a(wn, wn),
• (5) un1 un ?nwn.
• Step 2 Testing Convergence Updating wn
• Solve gn1 ? V,
• (6)
• (gn1, v) (gn, v) ?na(wn,v),
  ? v ? V.

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If gn 1/g0 ? tol. take u un 1 else

• (7) ?n gn 12/gn 2,
• (8) wn 1 gn 1 ?nwn .
• Do n n 1 and return to (4).
• We observe that the CG algorithm requires the
  solution of
• one linear variational problem per iteration,
  implying that
• the choice of the inner product is critical (in
  finite dimension
• this is related to the choice of the
  preconditioning matrix)

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4. Convergence of the CG algorithm

• Theorem Suppose that tol. 0 in algorithm
  (1)-(8) then,
• ? u0 ? V, we have
• limn ?8 un u,
• with u the solution of (LVP). Moreover if dim V
  d lt 8,
• there exists N ? d, such that uN u (finite
  termination
• property) .
• Proof See, e.g., RG, HNA, Vol. IX, Chapter 3
  (2003).

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From a practical point of view the most important
result is (Meinardius-Daniel)

• un u ? C u0 u (v?a
  1)/(v?a 1)n
• with
• ?a supv ??a(v,v) / infv
  ??a(v,v)
• ? being the unit sphere of V (? v ? V, v
  1).

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The Finite Dimensional Case(1)

• Consider
• (LE) Ax b,
• with A a SPD d d real matrix and b ? Rd. Since
  (LE) is
• equivalent to
• x ? Rd,
• (LEV)
• Ax.y b.y, ? y
  ? Rd,
• we can apply CG algorithms to the solution of
  (LE).

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The Finite Dimensional Case(2)

• Suppose now that the inner product of Rd is
  defined by
• (y,z) Sy.z
• with S another SPD matrix, then
• ?a ?( S 1A),
• a well-known result (!)

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5. An application to the Control of Elliptic PDEs

• Let us consider the following control problem for
  an
• elliptic PDE
• u ? L2(?),
• 
  (ECP)
• J(u) ? J(v), ? v ?
  L2(?),
• with
• J(v) ½??v2dx ½ k ?O
  y yd2dx
• and
• ?y f v ?? in O, y
  0 on ?O. (SE)
• Above ? ? O,O ? O and k gt 0.

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• The above control problem has a unique solution
  characterized
• by
• DJ(u) 0
  (OC)
• where, ? v ? L2(?),
• DJ(v) v
  p?
• p being the unique solution of the following
  adjoint equation
• ?p k(y yd)?O, p 0 on
  ?O. (ASE)
• To solve (OC) we advocate the following conjugate
  gradient
• algorithm

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• (1) u0 is given in
  L2(?).
• Solve
• (2) ?y0 f u0?? in O, y0 0
  on ?O
• and
• (3) ?p0 k(y0 yd)?O, p0
  0 on ?O.
• Set
• (4) g0 u0
  p0?,
• (5) w0 g0.
  

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• For n 0, un, gn and wn known, the last two ? 0,
  we compute un1,
• gn1 and, if necessary, wn1 as follows
• Solve
• (6) ??y0 w0?? in O, ?y0 0 on
  ?O
• and
• (7) ??p0 k?y0?O, ?p0 0
  on ?O.
• Set
• (8) ?g0 w0
  ?p0?,
• and compute

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• (9) ?n ?? gn2dx /
  ???gnwndx,
• (10) un1 un ?n wn,
• (11) gn1 gn ?n ?gn.
• If ?? gn12dx / ?? g02dx ? tol. take u
  un1 otherwise, compute
• (12) ?n ?? gn12dx / ??
  gn2dx,
• (13) wn1 gn1 ?nwn.
• Do n n 1 and return to (6).

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• It can be shown that, generically speaking, the
  number of
• iterations necessary to obtain convergence varies
  like
• k½
  log(1/tol.)
• For more information and examples on the
  application of
• conjugate gradient algorithms to the solution of
  control
• problems, see, e.g.,
• R.GLOWINSKI, J.L. LIONS, J. HE, Exact and
• Approximate Controllability for Distributed
  Parameters
• Systems A Numerical Approach, Cambridge
• University Press, 2008.

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