## Sunday, 28 July 2013

### An alternative variational principle

In the previous post I talked about one of the less quoted results in the variational theory of Sturm-Liouville eigenvalue problems which I owe to the book on differential equations by Erich Kamke and which I reworked for the purposes of my dissertation.

The other result that I also found in Kamke's book states an alternative variational principle which can be used to obtain estimates of the eigenvalues. This method gives a weaker upper bound than the Rayleigh quotient, however it requires smaller computational effort. In return, one can widen the class of trial functions without severely complicating the calculations.

$$J\left\{ \phi\right\}=\frac{\left\langle \phi,L\left[\phi\right]\right\rangle }{\left\langle \phi,\phi\right\rangle }$$
Then using Cauchy-Schwarz inequality we obtain.
$$\left(\left\langle \phi,L\left[\phi\right]\right\rangle \right)^{2}\leq\left\langle \phi,\phi\right\rangle \left\langle L\left[\phi\right],L\left[\phi\right]\right\rangle \frac{\left\langle \phi,L\left[\phi\right]\right\rangle }{\left\langle \phi,\phi\right\rangle }\leq\frac{\left\langle L\left[\phi\right],L\left[\phi\right]\right\rangle }{\left\langle \phi,L\left[\phi\right]\right\rangle }$$
Written explicitly
$$\frac{\int_{\Omega}\rho\phi L\left[\phi\right]d\boldsymbol{x}}{\int_{\Omega}\rho\phi^{2}d\boldsymbol{x}}\le\frac{\int_{\Omega}\rho\left(L\left[\phi\right]\right)^{2}d\boldsymbol{x}}{\int_{\Omega}\rho\phi L\left[\phi\right]d\boldsymbol{x}}$$
Hence we can replace the original problem with the following one
$$K\left\{ \phi\right\} =\frac{\left\langle L\left[\phi\right],L\left[\phi\right]\right\rangle }{\left\langle \phi,L\left[\phi\right]\right\rangle }\to\min,\qquad\phi\in\mathfrak{A}$$
Example.
Consider the eigenvalue problem
$$-y''=\lambda y,\quad y\left(0\right)=y\left(1\right)=0$$
Take the trial function
$$y_{1}=\frac{x}{12}-\frac{x^{3}}{6}+\frac{x^{4}}{12}$$
First we perform the calculation using the Rayleigh quotient
$$J\left\{ y_{1}\right\} =\frac{\left\langle y_{1},L\left[y_{1}\right]\right\rangle }{\left\langle y_{1},y_{1}\right\rangle }=-\frac{\int_{0}^{1}y_{1}y_{1}''dx}{\int_{0}^{1}y_{1}^{2}dx}$$
\begin{aligned}\int_{0}^{1}y_{1}y_{1}''dx & =\int_{0}^{1}\left[\left(\frac{x}{12}-\frac{x^{3}}{6}+\frac{x^{4}}{12}\right)\left(-x+x^{2}\right)\right]dx\\ & =\int_{0}^{1}\left(\frac{x^{6}}{12}-\frac{x^{5}}{4}+\frac{x^{4}}{6}+\frac{x^{3}}{12}-\frac{x^{2}}{12}\right)dx\\ & =-\frac{17}{5040} \end{aligned}
The factor evaluated above will be common to both variational quotients
\begin{aligned}\int_{0}^{1}y_{1}^{2}dx & =\int_{0}^{1}\left(\frac{x^{8}}{144}-\frac{x^{7}}{36}+\frac{x^{6}}{36}+\frac{x^{5}}{72}+\frac{x^{2}}{144}\right)dx\\ & =\frac{31}{90720} \end{aligned}
$$\lambda_{1}\le J\left\{ y_{1}\right\} =\frac{17\cdot90720}{5040\cdot31}\approx9.871$$
Now we use the alternative variational principle
$$K\left\{ y_{1}\right\} =\frac{\left\langle L\left[y_{1}\right],L\left[y_{1}\right]\right\rangle }{\left\langle y_{1},L\left[y_{1}\right]\right\rangle }=-\frac{\int_{0}^{1}y_{1}''^{2}dx}{\int_{0}^{1}y_{1}y_{1}''dx}$$
Obviously, the announced gain in computational efficiency comes from replacing $$y^{2}$$ with $$\left(L\left[y\right]\right)^{2}$$ which will be a polynomial of the order less by 4, than $$y^{2}$$.
$$\int_{0}^{1}y_{1}''^{2}dx=\int_{0}^{1}\left(x^{4}-2x^{3}+x^{2}\right)dx=\frac{1}{30}$$
$$\lambda_{1}\le K\left\{ y_{1}\right\} =\frac{5040}{17\cdot30}\approx9.882$$
Hence
$$K\left\{ y_{1}\right\} >J\left\{ y_{1}\right\} >\lambda_{1}=\pi^{2}$$
as expected, however $$K\left\{ y_{1}\right\}$$ takes less operations to evaluate.