Introduction to Mathematical Physics/Some mathematical problems and their solution/Boundary problems, variational methods

Variational formulationEdit

secvafor

Let us consider a boundary problem:

Problem:

Find   such that:

eqLufva

 

 

Let us suppose that is a unique solution   of this problem. For a functional space   sufficiently large the previous problem, may be equivalent to the following:

Problem:

Let us coonsider aboundey problem\

Find   such that:

eqvari

 

This is the variational form of the boundary problem. To obtain the equality eqvari, we have just multiplied scalarly by a "test function"   equation eqLufva. A "weak form" of this problem can be found using Green formula type formulas: the solution space is taken larger and as a counterpart, the test functions space is taken smaller. let us illustrate those ideas on a simple example:

Example:

Find   in   such that:

 

 

The variational formulation is: Find   in   such that:

 

Using the Green equality (see appendix secappendgreeneq)\index{Green formula} and the boundary conditions we can reformulate the problem in: find  , such that:

 

One can show that this problem as a unique solution in   where

 

is the Sobolev space of order 1 on   and   is the adherence of   in the space   (space of infinitely differentiable functions on  , with compact support in  ).

It may be that, as for the previous example, the solution function space and the test function space are the same. This is the case for most of linear operators   encountered in physics. In this case, one can associate to   a bilinear form  . The (weak) variational problem is thus:

provari2

Problem:

Find   such that

 

Example:

In this previous example, the bilinear form is:

 


There exist theorems (Lax-Milgram theorem for instance) that proove the existence and unicity of the solution of the previous problem provari2 , under certain conditions on the bilinear form  .

Finite elements approximationEdit

secvarinum


Let us consider the problem provari2 :

Problem:

Find   such that:

 

The method of approximation consists in choosing a finite dimension subspace   of   and the problem to solve becomes:

provari3

Problem: Find   such that

 


A base of   of   is chosen to satisfy boundary conditions. The problem is reduced to finding the components of  :

 

If   is a bilinear form

 

and to find the  's is equivalent to solve a linear system (often close to a diagonal system)that can be solved by classical algorithms ([ma:equad:Ciarlet88],[ma:compu:Press92]) which can be direct methods (Gauss,Cholesky) or iterative (Jacobi, Gauss-Seidel, relaxation). Note that if the vectors of the basis of   are eigenvectors of  , then the solving of the system is immediate (diagonal system). This is the basis of the spectral methods for solving evolution problems. If   is not linear, we have to solve a nonlinear system. Let us give an example of a basis  .

Example:

When  , an example of   that can be chosen ([ma:equad:Ciarlet88]) is the set of piecewise-linear continuous functions that are zero in   and   (for Dirichlet boundary conditions). More precisely,   can be approximated by the space of piecewise linear continuous functions on intervals  ,   going from zero to  , that are zero in   and  . The basis of such a space is made by functions   defined by:

 

 

and zero anywhere else (see figure figapproxesp).

figapproxesp

 
Space   can be approximated by pice wise linear continuous functions.

Finite difference approximationEdit

Finite difference method is one of the most basic method to tackle PDE problems. It is not strictly speaking a variational approximation. It is rather a sort of variational method where weight functions   are Dirac functions  . Indeed, when considering the boundary problem,

eqfini

 

instead of looking for an approximate solution   which can be decomposed on a basis of weight functions  :

 

the action of   on   is directly expressed in terms of Dirac functions, as well as the right hand term of equation eqfini:

eqfini2

 

\begin{rem} If   contains derivatives, the following formulas are used : right formula, order 1:

 

 

right formula order 2:

 

 

Left formulas can be written in a similar way. Centred formulas, second order are:

 

 

Centred formulas, fourth order are:

 

 

\end{rem} One can show that the equation eqfini2 is equivalent to the system of equations:

eqfini3

 

One can see immediately that equation eqfini2 implies equation eqfini3 in choosing "test" functions   of support   and such that  .

Minimization problemsEdit

A minimization problem can be written as follows:

promini

Problem: Let   a functional space, a   a functional. Find  , such that:

 

The solving of minimization problems depends first on the nature of the functional   and on the space  . As usually, the functional   is often approximated by a function of several variables   where the  's are the coordinate of   in some base   that approximates  . The methods to solve minimization problems can be classified into two categories: One can distinguish problems without constraints (see Fig. figcontraintesans) and problems with constraints (see Fig. figcontrainteavec). Minimization problems without constraints can be tackled theoretically by the study of the zeros of the differential function   if exists. Numerically it can be less expensive to use dedicated methods. There are methods that don't use derivatives of   (downhill simplex method, direction-set method) and methods that use derivative of   (conjugate gradient method, quasi-Newton methods). Details are given in ([ma:compu:Press92]). Problems with constraints reduce the functional space   to a set   of functions that satisfy some additional conditions. Note that those sets   are not vectorial spaces: a linear combination of vectors of   are not always in  . Let us give some example of constraints: Let   a functional space. Consider the space

 

where   are   functionals. This is a first example of constraints. It can be solved theoretically by using Lagrange multipliers ([ma:equad:Ciarlet88]), \index{constraint}. A second example of constraints is given by

 

where   are   functionals. The linear programming (see example exmplinepro) problem is an example of minimization problem with such constraints (in fact a mixing of equalities and inequalities constraints).

figcontraintesans

 
Minimization of a function of two variables.

figcontrainteavec]]

 
Minimization of a function of two variables with constraints. Here, space is reduced to a disk in the plane  .

figcontrainteaveclag

 
Illustration of the Lagrange multiplier. At point   tangent vector on the surface and on the constraint curve are not paralele:   does not correspond to an extremum. At point   both tangent vector are colinear: we have an extremum.

Example:

Let us consider of a first class of functional   that are important for PDE problems. Consider again the bilinear form introduced at section secvafor. If this bilinear form   is symmetrical, {\it i.e.}

 

the problem can be written as a minimization problem by introducing the functional:

 

One can show that (under certain conditions) solving the variational problem:\\ Find   such that:

 

is equivalent to solve the minimization problem:\\ Find  , such that:

 

Physical principles have sometimes a natural variational formulation (as natural as a PDE formulation). We will come back to the variational formulations at the section on least action principle (see section secprinmoindreact ) and at the section secpuisvirtu on the principle of virtual powers.

exmplinepro

Example: Another example of functional is given by the linear programming problem. Let us consider a function   that can be located by its coordinates   in a basis  ,  :

 

In linear programming, the functional to minimize can be written

 

and is subject to   primary constraints:

 

and   additional constraints:

 

 

 

The numerical algorithm to solve this type of problem is presented in ([ma:compu:Press92]). It is called the simplex method.

exmpsimul

Example: When the variables   can take only discrete values, one speaks about discrete or combinatorial optimization. The function to minimize can be for instance (travelling salesman problem):

 

where   are the coordinates of a city number  . The coordinates of the cities are numbers fixed in advance but the order in which the   cities are visited ({\it i.e} the permutation   of  ) is to be find to minimize  . Simulated annealing is a method to solve this problem and is presented in ([ma:compu:Press92]).

Lagrange multipliersEdit

Lagrange multipliers method is an interesting approach to solve the minimization problem of a function of   variables with constraints, {\it i. e. } to solve the following problem: \index{contrainte}\index{Lagrange multiplier}

Problem:

Find   in a space   of dimension   such that

 

with   constraints,  

Lagrange multipliers method is used in statistical physics (see sectionchapphysstat). In a problem without any constraints, solution   verifies:

 

In the case with constraints, the   coordinates   of   are not independent. Indeed, they should verify the relations:

 

Lagrange multipliers method consists in looking for   numbers   called Lagrange multipliers such that:

 

One obtains the following equation system: