Ritz method

A method for solving problems in variational calculus and, in general, finite-dimensional extremal problems, based on optimization of a functional on finite-dimensional subspaces or manifolds.

Let the problem of finding a minimum point of a functional on a separable Banach space be posed, where is bounded from below. Let some system of elements , complete in (cf. Complete system), be given (a so-called coordinate system). In the Ritz method, the minimizing element in the -th approximation is sought in the linear hull of the first coordinate elements , i.e. the coefficients of the approximation

are defined by the condition that be minimal among the specified elements. Instead of a coordinate system one can specify a sequence of subspaces , not necessarily nested.

Let be a Hilbert space with scalar product , let be a self-adjoint positive-definite (i.e. : for all ), possibly unbounded, operator in , and let be the Hilbert space obtained by completing the domain of definition of with respect to the norm generated by the scalar product , . Let it be required to solve the problem

(1)

This is equivalent to the problem of finding a minimum point of the quadratic functional

which can be written in the form

where is a solution of equation (1). Let , be closed (usually, finite-dimensional) subspaces such that as for every , where is the orthogonal projection in projecting onto . By minimizing in one obtains a Ritz approximation to the solution of equation (1); moreover, as . If and is a basis in , then the coefficients of the element

(2)

are determined from the linear system of equations

(3)

One can also arrive at a Ritz approximation without making use of the variational statement of the problem (1). Namely, by defining the approximation (2) from the condition

(the Galerkin method), one arrives at the same system of equations (3). That is why the Ritz method for equation (1) is sometimes called the Ritz–Galerkin method.

Ritz's method is widely applied when solving eigenvalue problems, boundary value problems and operator equations in general. Let and be self-adjoint operators in . Moreover, let be positive definite, be positive, , and let the operator be completely continuous in (cf. Completely-continuous operator). By virtue of the above requirements, is self-adjoint and positive in , and the spectrum of the problem

(4)

consists of positive eigenvalues:

Ritz's method is based on a variational determination of eigenvalues. For instance,

by carrying out minimization only over the subspace one obtains Ritz approximations of . If is, as above, a basis in , then the Ritz approximations of , , are determined from the equation

and the vector of coefficients of the approximation

to is determined as a non-trivial solution of the linear homogeneous system . The Ritz method provides an approximation from above of the eigenvalues, i.e. , . If the -th eigenvalue of problem (4) is simple , then the convergence rate of the Ritz method is characterized by the following relations:

where as . Similar relations can be carried over to the case of multiple , but then they need certain refinements (see [2]). W. Ritz [4] proposed his method in 1908, but even earlier Lord Rayleigh had applied this method to solve certain eigenvalue problems. In this connection the Ritz method is often called the Rayleigh–Ritz method, especially if one speaks about solving an eigenvalue problem.

References

[1]	M.M. Vainberg, "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley (1973) (Translated from Russian)
[2]	M.A. Krasnosel'skii, G.M. Vainikko, P.P. Zabreiko, et al., "Approximate solution of operator equations" , Wolters-Noordhoff (1972) (Translated from Russian)
[3]	S.G. [S.G. Mikhlin] Michlin, "Variationsmethoden der mathematischen Physik" , Akademie Verlag (1962) (Translated from Russian)
[4]	W. Ritz, "Ueber eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik" J. Reine Angew. Math. , 135 (1908) pp. 1–61

References

[a1]	G.H. Golub, C.F. van Loan, "Matrix computations" , Johns Hopkins Univ. Press (1989)
[a2]	G.J. Fix, "An analyse of the finite element method" , Prentice-Hall (1973)
[a3]	J. Stoer, R. Bulirsch, "Einführung in die numerische Mathematik" , II , Springer (1978)
[a4]	P.G. Ciarlet, "The finite element method for elliptic problems" , North-Holland (1975)