View source for Ritz method

A method for solving problems in [[Variational calculus|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r0825001.png" /> on a separable [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r0825002.png" /> be posed, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r0825003.png" /> is bounded from below. Let some system of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r0825004.png" />, complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r0825005.png" /> (cf. [[Complete system|Complete system]]), be given (a so-called coordinate system). In the Ritz method, the minimizing element in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r0825006.png" />-th approximation is sought in the linear hull of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r0825007.png" /> coordinate elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r0825008.png" />, i.e. the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r0825009.png" /> of the approximation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250010.png" /></td> </tr></table>

are defined by the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250011.png" /> be minimal among the specified elements. Instead of a coordinate system one can specify a sequence of subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250012.png" />, not necessarily nested.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250013.png" /> be a Hilbert space with scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250014.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250015.png" /> be a self-adjoint positive-definite (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250016.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250017.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250018.png" />), possibly unbounded, operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250019.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250020.png" /> be the Hilbert space obtained by completing the domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250022.png" /> with respect to the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250023.png" /> generated by the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250025.png" />. Let it be required to solve the problem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250027.png" /></td> </tr></table>

which can be written in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250028.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250029.png" /> is a solution of equation (1). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250031.png" /> be closed (usually, finite-dimensional) subspaces such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250032.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250033.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250035.png" /> is the orthogonal projection in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250036.png" /> projecting onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250037.png" />. By minimizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250038.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250039.png" /> one obtains a Ritz approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250040.png" /> to the solution of equation (1); moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250041.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250042.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250044.png" /> is a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250045.png" />, then the coefficients of the element

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

are determined from the linear system of equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250047.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250048.png" /></td> </tr></table>

(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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250050.png" /> be self-adjoint operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250051.png" />. Moreover, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250052.png" /> be positive definite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250053.png" /> be positive, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250054.png" />, and let the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250055.png" /> be completely continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250056.png" /> (cf. [[Completely-continuous operator|Completely-continuous operator]]). By virtue of the above requirements, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250057.png" /> is self-adjoint and positive in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250058.png" />, and the spectrum of the problem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250059.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

consists of positive eigenvalues:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250060.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250061.png" /></td> </tr></table>

by carrying out minimization only over the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250062.png" /> one obtains Ritz approximations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250063.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250064.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250065.png" /> is, as above, a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250066.png" />, then the Ritz approximations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250067.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250069.png" />, are determined from the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250070.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250071.png" /></td> </tr></table>

and the vector of coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250072.png" /> of the approximation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250073.png" /></td> </tr></table>

to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250074.png" /> is determined as a non-trivial solution of the linear homogeneous system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250075.png" />. The Ritz method provides an approximation from above of the eigenvalues, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250077.png" />. If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250078.png" />-th eigenvalue of problem (4) is simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250079.png" />, then the convergence rate of the Ritz method is characterized by the following relations:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250080.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250081.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250082.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250083.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250084.png" />. Similar relations can be carried over to the case of multiple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082500/r08250085.png" />, but then they need certain refinements (see [[#References|[2]]]). W. Ritz [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.M. Vainberg,   "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Krasnosel'skii,   G.M. Vainikko,   P.P. Zabreiko,   et al.,   "Approximate solution of operator equations" , Wolters-Noordhoff  (1972)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.G. [S.G. Mikhlin] Michlin,   "Variationsmethoden der mathematischen Physik" , Akademie Verlag  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Ritz,   "Ueber eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik"  ''J. Reine Angew. Math.'' , '''135'''  (1908)  pp. 1–61</TD></TR></table>



====Comments====


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