Rayleigh–Ritz method: Difference between revisions

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The '''Rayleigh–Ritz method''' is a direct numerical method of approximating [[eigenvalues and eigenvectors|eigenvalues]], originated in the context of solving physical [[Boundary value problem|boundary value problems]] and named after [[Lord Rayleigh]] and [[Walther Ritz]].

In this method, an infinite-dimensional [[linear operator]] is approximated by a finite-dimensional [[Dilation (operator theory)|compression]], on which we can use an [[eigenvalue algorithm]].

It is used in all applications that involve approximating [[eigenvalues and eigenvectors]], often under different names. In [[quantum mechanics]], where a system of particles is described using a [[Hamiltonian (quantum mechanics)|Hamiltonian]], the [[Ritz method]] uses [[ansatz|trial wave functions]] to approximate the ground state eigenfunction with the lowest energy. In the [[finite element method]] context, mathematically the same algorithm is commonly called the [[Ritz-Galerkin method]]. The Rayleigh–Ritz method or [[Ritz method]] terminology is typical in mechanical and structural engineering to approximate the [[Normal mode|eigenmodes]] and [[Resonance|resonant frequencies]] of a structure.

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The name of the method and its origin story have been debated by histroians.<ref name="Leissa">{{cite journal|last1=Leissa|first1=A.W.|title=The historical bases of the Rayleigh and Ritz methods|journal=Journal of Sound and Vibration|volume=287|issue=4–5|year=2005|pages=961–978| doi=10.1016/j.jsv.2004.12.021| bibcode=2005JSV...287..961L| url=https://www.sciencedirect.com/science/article/abs/pii/S0022460X05000362 |url-access=subscription}}</ref><ref name="Ilanko">{{cite journal|last1=Ilanko|first1=Sinniah|title=Comments on the historical bases of the Rayleigh and Ritz methods|journal=Journal of Sound and Vibration|volume=319|issue=1–2|year=2009|pages=731–733 | doi=10.1016/j.jsv.2008.06.001|bibcode=2009JSV...319..731I }}</ref> It has been called [[Ritz method]] after [[Walther Ritz]], since the numerical procedure has been published by [[Walther Ritz]] in 1908-1909. According to A. W. Leissa,<ref name="Leissa" /> [[Lord Rayleigh]] wrote a paper congratulating Ritz on his work in 1911, but stating that he himself had used Ritz's method in many places in his book and in another publication. This statement, although later disputed, and the fact that the method in the trivial case of a single vector results in the [[Rayleigh quotient]] make the case for the name ''Rayleigh–Ritz'' method. According to S. Ilanko,<ref name="Ilanko"/> citing [[Richard Courant]], both [[Lord Rayleigh]] and [[Walther Ritz]] independently conceived the idea of utilizing the equivalence between [[Boundary value problem|boundary value problems]] of [[partial differential equations]] on the one hand and problems of the [[calculus of variations]] on the other hand for numerical calculation of the solutions, by substituting for the variational problems simpler approximating extremum problems in which a finite number of parameters need to be determined; see the article [[Ritz method]] for details. Ironically for the debate, the modern justification of the algorithm drops the [[calculus of variations]] in favor of the simpler and more general approach of [[orthogonal projection]] as in [[Galerkin method]] named after [[Boris Galerkin]], thus leading also to the [[Ritz-Galerkin method]] naming.{{cn|reason=need historian reference here|date=June 2024}}

== Method ==

Let <math>T</math> be a [[linear operator]] on a [[Hilbert space]] <math>\mathcal{H}</math>, with [[inner product]] <math>(\cdot, \cdot)</math>. Now consider a finite set of functions <math>\mathcal{L} = \{\varphi_1, ...,\varphi_n\}</math>. Depending on the application these functions may be:

* A subset of the [[orthonormal basis]] of the original operator;<ref name=daviesplum>{{cite journal|last1=Davies|first1=E. B.|last2=Plum|first2=M.|title=Spectral Pollution|journal=IMA Journal of Numerical Analysis|url=https://arxiv.org/abs/math/0302145|author-link1=E. Brian Davies|year=2003}}</ref>

* A space of [[splines]] (as in the [[Galerkin method]]);<ref name=sulimayers>{{cite book|last1=Süli|first1=Endre|author-link1=Endre Süli|last2=Mayers|first2=David|title=An Introduction to Numerical Analysis|publisher=[[Cambridge University Press]]|isbn=0521007941|year=2003}}}}</ref>

* A set of functions which approximate the [[eigenfunctions]] of the operator.<ref name=levitinshargorodsky>{{cite journal|last1=Levitin|first1=Michael|last2=Shargorodsky|first2=Eugene|title=Spectral pollution and second order relative spectra for self-adjoint operators|journal=IMA Journal of Numerical Analysis|url=https://arxiv.org/abs/math/0212087|year=2004}}</ref>

One could use the orthonormal basis generated from the eigenfunctions of the operator, which will produce [[diagonal matrix|diagonal]] approximating matrices, but in this case we would have already had to calculate the spectrum.

We now approximate <math>T</math> by <math>T_{\mathcal{L}}</math>, which is defined as the matrix with entries<ref name=daviesplum></ref>

<math display="block">(T_{\mathcal{L}})_{i,j} = (T \varphi_i, \varphi_j).</math>

and solve the eigenvalue problem <math>T_{\mathcal{L}}u = \lambda u</math>. It can be shown that the matrix <math>T_{\mathcal{L}}</math> is the [[Dilation (operator theory)|compression]] of <math>T</math> to <math>\mathcal{L}</math>.<ref name=daviesplum></ref>

For [[differential operators]] (such as [[Sturm-Liouville problem|Sturm-Liouville operators]]), the inner product <math>(\cdot, \cdot)</math> can be replaced by the [[weak formulation]] <math>\mathcal{A}(\cdot, \cdot)</math>.<ref name=sulimayers></ref><ref name=pryce>{{cite book|last1=Pryce|first1=John D.|title=Numerical Solution of Sturm-Liouville Problems|ISBN=0198534159|publisher=Oxford University Press|year=1994}}</ref>

If a subset of the orthonormal basis was used to find the matrix, the eigenvectors of <math>T_{\mathcal{L}}</math> will be [[linear combinations]] of orthonormal basis functions, and as a result they will be approximations of the eigenvectors of <math>T</math>.<ref name=arfkenweber>{{cite book|last1=Arfken|first1 = George B.|author-link1=George B. Arfken|last2 = Weber| first2 = Hans J.|year = 2005|title= Mathematical Methods For Physicists|url= https://books.google.com/books?id=tNtijk2iBSMC&pg=PA83|edition= 6th|publisher=Academic Press}}</ref>

== Properties ==

=== Spectral pollution ===

It is possible for the Rayleigh-Ritz method to produce values which do not converge to actual values in the spectrum of the operator as the truncation gets large. These values are known as spectral pollution.<ref name=daviesplum></ref><ref name=levitinshargorodsky></ref><ref>{{cite magazine|url=https://ima.org.uk/16912/unscrambling-the-infinite-can-we-compute-spectra/|last1=Colbrook|first1=Matthew|title=Unscrambling the Infinite: Can we Compute Spectra?|magazine=Mathematics Today|publisher=Institute of Mathematics and its Applications}}</ref> In some cases (such as for the [[Schrödinger equation]]), there is no approximation which both includes all eigenvalues of the equation, and contains no pollution.<ref>{{cite journal|last1=Colbrook|first1=Matthew|last2=Roman|first2=Bogdan|last3=Hansen|first3=Anders|title=How to Compute Spectra with Error Control|url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.250201|journal=Physical Review Letters|year=2019}}</ref>

The spectrum of the compression (and thus pollution) is bounded by the [[numerical range]] of the operator; in many cases it is bounded by a subset of the numerical range known as the [[essential numerical range]].<ref>{{cite journal|last1=Pokrzywa|first1=Andrzej|title=Method of orthogonal projections and approximation of the spectrum of a bounded operator|year=1979|journal=Studia Mathematica}}</ref><ref>{{cite journal|last1=Bögli|first1=Sabine|last2=Marletta|first2=Marco|last3=Tretter|first3=Christiane|title=The essential numerical range for unbounded linear operators|journal=Journal of Functional Analysis|year=2020|url=https://arxiv.org/abs/1907.09599}}</ref>

== For matrix eigenvalue problems ==

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</math>

Thus, for the given matrix <math>W</math> with its column-space that is spanned by two exact right singular vectors, we determine these right singular vectors, as well as the corresponding left singular vectors and the singular values, all exactly. For an arbitrary matrix <math>W</math>, we obtain approximate singular triplets which are optimal given <math>W</math> in the sense of optimality of the Rayleigh–Ritz method.

== Applications and examples ==

=== In quantum physics ===

In quantum physics, where the spectrum of the [[Hamiltonian (quantum mechanics)|Hamiltonian]] is the set of discrete energy levels allowed by a quantum mechanical system, the Rayleigh-Ritz method is used to approximate the energy states and wavefunctions of a complicated atomic or nuclear system.<ref name=arfkenweber></ref>

=== In dynamical systems ===

The [[Koopman operator]] allows a finite-dimensional [[nonlinear system]] to be encoded as an infinite-dimensional [[linear system]]. In general, both of these problems are difficult to solve, but for the latter we can use the Ritz-Galerkin method to approximate a solution.<ref>{{cite web|last1=Servadio|first1=Simone|last2=Arnas|first2=David|last3=Linares|first3=Richard|title=A Koopman Operator Tutorial with Orthogonal Polynomials|publisher=arXiv|url=https://arxiv.org/abs/2111.07485}}</ref>

== See also ==