Linear Variational Problems Part I

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Lecture 1

• Linear Variational Problems (Part I)

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1. Motivation For those participants wondering
why we start a course dedicated to nonlinear
problems by discussing a particular family of
linear problems let us say

• that
• (i) The Newtons method provides a systematic way
  to reduce the solution of some nonlinear problems
  to the solution of a sequence of linear problems.

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1. Motivation For those participants wondering
why we start a course dedicated to nonlinear
problems by discussing a particular family of
linear problems let us say

• that
• (i) The Newtons method provides a systematic way
  to reduce the solution of some nonlinear problems
  to the solution of a sequence of linear problems.
• (ii) During this course, we will show that there
  are other methods than Newtons which rely on
  linear solvers to achieve the solution of
  nonlinear problems. An example of such methods
  will be provided by the solution of nonlinear
  problems in Hilbert spaces by conjugate gradient
  algorithms.

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1. Motivation For those participants wondering
why we start a course dedicated to nonlinear
problems by discussing a particular family of
linear problems let us say

• that
• (i) The Newtons method provides a systematic way
  to reduce the solution of some nonlinear problems
  to the solution of a sequence of linear problems.
• (ii) During this course, we will show that there
  are other methods than Newtons which rely on
  linear solvers to achieve the solution of
  nonlinear problems. An example of such methods
  will be provided by the solution of nonlinear
  problems in Hilbert spaces by conjugate gradient
  algorithms.
• We will start our discussion with the
  Newtons method.

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2. The Newtons method in Hilbert spaces

• Let V be a Hilbert space (real for simplicity)
  we denote by
• V the dual space of V and by lt. , .gt the duality
  pairing
• between V and V. Consider now an operator A
  (possibly
• nonlinear) mapping V into V and the following
  equation
• (E) A(u) 0.
• Suppose that operator A is F-differentiable in
  the
• neighborhood of a solution u of (E) if v is
  sufficiently close
• to u we have then

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0 A(u) A(v u v) A(v) A(v)(u v)
o(v u),

• with A(v) (? L( V, V)) the differential of A at
  v. The abo-
• ve relation suggest the following algorithm,
  classically
• known as the Newton-Raphson method
• (1) u0 is given in V
• then, for n 0, un ? un1 by
• (2) un1 un A(un)1A (un).

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Suppose that A(u) ? Isom(V, V) and that A is
locally Lipschitz continuous, then if u0 is
sufficiently close

• to u we have
• limn?8 un u,
• the convergence being super-linear. From a
  practical
• point of view it may be more appropriate to write
  (2) as
• (3) A(un)(un1 un)
  A(un),
• or even better (and equivalently) as
• un1 un ? V,
• (VF)
• ltA(un)(un1 un),vgt
  ltA(un),vgt, ? v ? V.

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Many remarks are in order, but the only one we
will make is that (VF) is a linear variational
problem (in the sense of,

• e.g., J.L. LIONS) since the bilinear functional
• v, w ? lt A(un)v,wgt VV?R
• and the linear functional
• v? lt A(un),vgt V?R
• are both continuous.
• 3. On a family of linear variational problems in
  Hilbert spaces The Lax-Milgram Theorem
• We are going to focus our discussion on a
  particular family
• of linear problems, namely to those problems
  defined by

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considering a triple V, a, L verifying the
followingconditions

• (i) V is a real Hilbert space for the scalar
  (inner) product (.,.) and the associated norm
  . .
• (ii) a VV ? R is bilinear, continuous and
  V-elliptic (this last property meaning that there
  exists ? gt 0 such that
• a(v,v) ?
  v2, ? v ?V ).
• (iii) L V ? R is linear and continuous.
• Remark 1 We do not assume that the bilinear
  functional
• a is symmetric.

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Remark 2 If V is a finite dimensional space, the
V-ellipticity of a?? the positive definiteness of
a (not necessarily true if dim V is infinite).

• To V, a, L we associate the following family of
  linear
• (variational) problems
• Find u ? V such that
• (LVP)
• a(u,v) L(v), ? v ? V.

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Why do we consider such a family of linear
problems ?Among the many reasons, let us mention
the main ones
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Why do we consider such a family of linear
problems ?Among the many reasons, let us mention
the main ones

• 1) Many applications in Mechanics, Physics,
  Control, Image Processing,.. lead to such
  problems once the convenient Hilbert space has
  been identified (exemples of such situations will
  be encountered during this course).

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Why do we consider such a family of linear
problems ?Among the many reasons, let us mention
the main ones

• 1) Many applications in Mechanics, Physics,
  Control, Image Processing,.. lead to such
  problems once the convenient Hilbert space has
  been identified (exemples of such situations will
  be encountered during this course).
• 2) Let us return to the solution of A(u) 0 by
  the Newtons method and suppose that A J where
  J is a strongly convex twice differentiable
  functional (i.e., ? ? gt 0, such that ltJ(w)
  J(v), w vgt ? w v2, ? v, w),
• then the linear problem that we have to solve
  at each
• iteration to obtain un 1 belongs to the
  (LVP) family.

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Strictly speaking, a problem is (was) variational
if it can be viewed as the Euler-Lagrange
equation of some problem

• from the Calculus of Variations. In the
  particular case of
• (LVP) this supposes that a is symmetric indeed,
  if a is
• symmetric, then (LVP) is equivalent to
• Find u ? V such that
• (MP)
• J(u) ? J(v), ? ? V,
• with J(v) ½ a(v,v) L(v). Following the
  influence of J.L.
• Lions and G. Stampacchia (mid 1960s) many
  authors
• use the terminology variational equations (and of
  course
• inequalities) even when a(.,.) is not symmetric.

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The Lax-Milgram Theorem

• Suppose that the triple V,a,L verifies
  (i)-(iii)
• then (LVP) has a unique solution.
• For a proof, see, e.g., Atkinson Han
  (Springer). There
• are several proofs of the LM Theorem, our
  favorite one
• relies on the Banach fixed point theorem and can
  be
• generalized to variational inequalities.
• Actually, the LM Theorem can be generalized to
  complex
• Hilbert spaces by assuming that a(.,.) is
  sesquilinear,
• continuous and verifies Re a(v,v) ? v2, ? v
  ? V.

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Among the many comments which can be associated
with the Lax-Milgram Theorem, we will focus on
the following one, considering its computational
implications

• The fixed point based proof of the LM Theorem is
  based
• is based on the following observations

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Among the many comments which can be associated
with the Lax-Milgram Theorem, we will focus on
the following one, considering its computational
implications

• The fixed point based proof of the LM Theorem is
  based
• on the following observations
• (1) From the Riescz representation Theorem, there
  exists a unique pair A,l such that
• A ? L(V,V), (Av,w) a(v,w), ? v, w ? V, ? ?
  A,
• (l,v) L(v), ? v ? V.

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Among the many comments which can be associated
with the Lax-Milgram Theorem, we will focus on
the following one, considering its computational
implications

• The fixed point based proof of the LM Theorem is
  based
• on the following observations
• (1) From the Riescz representation Theorem, there
  exists a unique pair A,l such that
• A ? L(V,V), (Av,w) a(v,w), ? v, w ? V, ? ?
  A,
• (l,v) L(v), ? v ? V.
• (2) If 0 lt ? lt 2? A 2 the mapping v ? v
  ?(Av l) is a
• uniformly strict contraction of V.

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Among the many comments which can be associated
with the Lax-Milgram Theorem, we will focus on
the following one, considering its computational
implications

• The fixed point based proof of the LM Theorem is
  based
• on the following observations
• (1) From the Riescz representation Theorem, there
  exists a unique pair A,l such that
• A ? L(V,V), (Av,w) a(v,w), ? v, w ? V, ? ?
  A,
• (l,v) L(v), ? v ? V.
• (2) If 0 lt ? lt 2? A 2 the mapping v ? v
  ?(Av l) is a
• uniformly strict contraction of V.
• (3) (LVP) and Au l are equivalent.

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It follows from observations (1)-(3) that if ?
verifies0 lt ? lt 2? A 2

• we have geometric convergence of the following
• algorithm, ? u0 ? V
• (1) u0 is given in V,
• n 0, un ? un1 by
• (2) un1 un ?(Aun l).
• The practical interest of (1)-(2) as written
  above is limited
• by the fact that in general A and l are unknown.

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A more practical form of (1)-(2) is obtained by
replacing relation (2) by the following
equivalent one

• un1 ? V,
• (2)
• (un1,v) (un,v) ?a(un,v) L(v), ? v
  ? V.
• Problem (2) is also from the (LVP) family, with
  the role
• of a(.,.) played by the V-scalar product (.,.).
  The fact that ?
• is not known in general can be overcome by using
  various
• techniques using a sequence ?n n instead of a
  fixed ?.
• If a(.,.) is symmetric, conjugate gradient
  provides a
• more efficient alternative to algorithm (1)-(2),
  at little extra
• cost. Conjugate gradient will be discussed next.