Transcript Lecture 1

Lecture 1
Linear Variational
Problems (Part I)
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 Newton’s method provides a systematic way to
reduce the solution of some nonlinear problems to the
solution of a sequence of linear problems.
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 Newton’s 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 Newton’s 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.
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 Newton’s 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 Newton’s 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 Newton’s method.
2. The Newton’s method in Hilbert spaces
Let V be a Hilbert space (real for simplicity); we denote by
V’ the dual space of V and by <. , .> 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
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 above relation suggest the following algorithm, classically
known as the Newton-Raphson method:
(1)
u0 is given in V ;
then, for n ≥ 0, un  un+1 by
(2)
un+1 = un – A’(un)–1A (un).
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+∞ 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)(un+1– un) = – A(un),
or even better (and equivalently) as:
un+1– un  V,
(VF)
<A’(un)(un+1– un),v> = – <A(un),v>,  v  V.
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}  < A’(un)v,w> : V×VR
and the linear functional
v < A(un),v> : 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
considering a triple {V, a, L} verifying the following
conditions:
(i) V is a real Hilbert space for the scalar (inner) product
(.,.) and the associated norm || . ||.
(ii) a: V×V  R is bilinear, continuous and V-elliptic (this
last property meaning that there exists  > 0 such that
a(v,v) ≥  ||v||2,  v V ).
(iii) L: V  R is linear and continuous.
Remark 1: We do not assume that the bilinear functional
a is symmetric.
Remark 2: If V is a finite dimensional space, the Vellipticity 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.
Why do we consider such a family of linear problems ?
Among the many reasons, let us mention the main ones:
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).
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 Newton’s
method and suppose that A = J’ where J is a strongly
convex twice differentiable functional (i.e.,   > 0,
such that <J’(w) – J’(v), w – v> ≥  ||w – v||2,  v, w),
then the linear problem that we have to solve at each
iteration to obtain un +1 belongs to the (LVP) family*.
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 1960’s) many authors
use the terminology variational equations (and of course
inequalities) even when a(.,.) is not symmetric.
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) ≥  ||v||2,  v  V.
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:
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.
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 < ρ < 2 ||A||– 2 the mapping v  v – ρ(Av – l) is a
uniformly strict contraction of V.
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 < ρ < 2 ||A||– 2 the mapping v  v – ρ(Av – l) is a
uniformly strict contraction of V.
(3) (LVP) and Au = l are equivalent.
It follows from observations (1)-(3) that if ρ verifies
0 < ρ < 2 ||A||– 2
we have geometric convergence of the following
algorithm,  u0  V :
(1) u0 is given in V,
n ≥ 0, un  un+1 by
(2) un+1 = un – ρ(Aun – l).
The practical interest of (1)-(2) as written above is limited
by the fact that in general A and l are unknown.
A more practical form of (1)-(2) is obtained by replacing
relation (2) by the following equivalent one:
un+1  V,
(2)’
(un+1,v) = (un,v) – ρ[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 ρ
is not known in general can be overcome by using various
techniques using a sequence { ρn }n instead of a fixed ρ.
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.