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 $ J : U \rightarrow \mathbf R $ on a separable Banach space $ U $ be posed, where $ J $ is bounded from below. Let some system of elements $ \{ \phi _ {n} \} _ {1} ^ \infty \subset U $, complete in $ U $ (cf. Complete system), be given (a so-called coordinate system). In the Ritz method, the minimizing element in the $ n $-th approximation is sought in the linear hull of the first $ n $ coordinate elements $ \phi _ {1} \dots \phi _ {n} $, i.e. the coefficients $ c _ {1} ^ {( n)} \dots c _ {n} ^ {( n)} $ of the approximation

$$ u _ {n} = \sum _ { j= 1} ^ { n } c _ {j} ^ {( n)} \phi _ {j} $$

are defined by the condition that $ J ( u _ {n} ) $ be minimal among the specified elements. Instead of a coordinate system one can specify a sequence of subspaces $ U _ {n} \subset U $, not necessarily nested.

Let $ H $ be a Hilbert space with scalar product $ ( u , v ) $, let $ A $ be a self-adjoint positive-definite (i.e. $ \exists \gamma > 0 $: $ ( Au, u) \geq \gamma \| u \| ^ {2} $ for all $ u \in D( A) $), possibly unbounded, operator in $ H $, and let $ H _ {A} $ be the Hilbert space obtained by completing the domain of definition $ D ( A) \subseteq H $ of $ A $ with respect to the norm $ \| u \| _ {A} $ generated by the scalar product $ ( u , v ) _ {A} = ( Au , v ) $, $ u , v \in D ( A) $. Let it be required to solve the problem

$$ \tag{1 } Au = f . $$

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

$$ \Phi ( u) = ( Au , u ) - ( u , f ) - ( f , u ) , $$

which can be written in the form

$$ \Phi ( u) = \| u - u _ {0} \| _ {A} ^ {2} - \| u _ {0} \| _ {A} ^ {2} ,\ \ u \in H _ {A} , $$

where $ u _ {0} = A ^ {- 1} f $ is a solution of equation (1). Let $ H _ {n} \subset H _ {A} $, $ n = 1 , 2 \dots $ be closed (usually, finite-dimensional) subspaces such that $ \| u - P _ {n} u \| _ {A} \rightarrow 0 $ as $ n \rightarrow \infty $ for every $ u \in H _ {A} $, where $ P _ {n} $ is the orthogonal projection in $ H _ {A} $ projecting onto $ H _ {n} $. By minimizing $ \Phi $ in $ H _ {n} $ one obtains a Ritz approximation $ u _ {n} = P _ {n} u _ {0} $ to the solution of equation (1); moreover, $ \| u _ {n} - u _ {0} \| _ {A} = \| u _ {0} - P _ {n} u _ {0} \| _ {A} \rightarrow 0 $ as $ n \rightarrow \infty $. If $ \mathop{\rm dim} H _ {n} = n $ and $ \phi _ {1} ^ {( n)} \dots \phi _ {n} ^ {( n)} $ is a basis in $ H _ {n} $, then the coefficients of the element

$$ \tag{2 } u _ {n} = \ \sum _ { j= 1} ^ { n } c _ {j} ^ {( n)} \phi _ {j} ^ {( n)} $$

are determined from the linear system of equations

$$ \tag{3 } \sum _ { j= 1} ^ { n } ( \phi _ {j} ^ {( n)} , \phi _ {i} ^ {( n)} ) _ {A} c _ {j} ^ {( n)} = \ ( f , \phi _ {i} ^ {( n) }) ,\ \ i = 1 \dots n . $$

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

$$ ( A u _ {n} - f , \phi _ {i} ^ {( n)} ) = 0 ,\ \ i = 1, \dots, n $$

(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 $ A $ and $ B $ be self-adjoint operators in $ H $. Moreover, let $ A $ be positive definite, $ B $ be positive, $ D ( A) \subseteq D ( B) $, and let the operator $ A ^ {- 1} B $ be completely continuous in $ H _ {A} $ (cf. Completely-continuous operator). By virtue of the above requirements, $ A ^ {- 1} B $ is self-adjoint and positive in $ H _ {A} $, and the spectrum of the problem

$$ \tag{4 } Au = \lambda Bu $$

consists of positive eigenvalues:

$$ A u _ {k} = \lambda _ {k} B u _ {k} ,\ \ 0 < \lambda _ {1} \leq \lambda _ {2} \leq \dots ; \ \ \lambda _ {k} \rightarrow \infty \textrm{ as } k \rightarrow \infty . $$

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

$$ \lambda _ {1} = \ \inf _ {u \in H _ {A} } \ \frac{( Au , u ) }{( Bu , u ) } ; $$

by carrying out minimization only over the subspace $ H _ {n} \subset H _ {A} $ one obtains Ritz approximations $ \lambda _ {1n} , u _ {1n} $ of $ \lambda _ {1} , u _ {1} $. If $ \phi _ {1} ^ {( n)} \dots \phi _ {n} ^ {( n)} $ is, as above, a basis in $ H _ {n} $, then the Ritz approximations $ \lambda _ {kn} $ of $ \lambda _ {k} $, $ k = 1 \dots n $, are determined from the equation

$$ \mathop{\rm det} ( A _ {n} - \lambda B _ {n} ) = 0 , $$

$$ A _ {n} = \{ ( A \phi _ {j} ^ {( n)} , \phi _ {i} ^ {( n)} ) \} _ {i , j = 1 } ^ {n} ,\ B _ {n} = \{ ( B \phi _ {j} ^ {( n)} , \phi _ {i} ^ {( n)} ) \} _ {i , j = 1 } ^ {n} , $$

and the vector of coefficients $ \mathbf c _ {k , n } = ( c _ {1k} ^ {( n)} \dots c _ {nk} ^ {( n)} ) $ of the approximation

$$ u _ {kn} = \ \sum _ { j= 1} ^ { n } c _ {jk} ^ {( n)} \phi _ {j} ^ {( n)} $$

to $ u _ {k} $ is determined as a non-trivial solution of the linear homogeneous system $ ( A _ {n} - \lambda _ {kn} B _ {n} ) \mathbf c _ {kn} = 0 $. The Ritz method provides an approximation from above of the eigenvalues, i.e. $ \lambda _ {kn} \geq \lambda _ {k} $, $ k = 1, \dots, n $. If the $ k $-th eigenvalue of problem (4) is simple $ ( \lambda _ {k- 1} < \lambda _ {k} < \lambda _ {k+ 1} ) $, then the convergence rate of the Ritz method is characterized by the following relations:

$$ \lambda _ {kn} - \lambda _ {k} = \ \lambda _ {k} ( 1 + \epsilon _ {kn} ) \ \| u _ {k} - P _ {n} u _ {k} \| _ {A} ^ {2} , $$

$$ \| u _ {k} \| _ {A} = 1 ,\ \| u _ {kn} - u _ {k} \| _ {A} = ( 1 + \epsilon _ {kn} ^ \prime ) \ \| u _ {k} - P _ {n} u _ {k} \| _ {A} , $$

$$ \| u _ {kn} \| _ {A} = \| u \| _ {A} = 1 , $$

where $ \epsilon _ {kn} , \epsilon _ {kn} ^ \prime \rightarrow 0 $ as $ n \rightarrow \infty $. Similar relations can be carried over to the case of multiple $ \lambda _ {k} $, 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)