Parametric differentiation and estimation of eigenvalues by the Monte Carlo method (Q1288119)

The following integral equation of the second kind in \(L_\infty({\mathbb R}^n)\) with parameter \(\lambda\) is considered: \[ \varphi(x,\lambda)=\int_{{\mathbb R}^n}k(x,y,\lambda) \varphi(y,\lambda)dy+h(x,\lambda). \] A Monte Carlo estimator is constructed for the derivatives of the solution to this equation with respect to \(\lambda\). Since calculation of the parametric derivatives for the problem \[ \Delta u-cu=0,\quad \alpha u+\beta\frac{\partial u}{\partial\ell}\Big| _\Gamma =1 \] realizes iterations of the resolvent operator \([\Delta-c]^{-1}\) under the homogeneous boundary condition, the authors construct the Monte Carlo estimator for the eigenvalues of the Laplace operator.