We associate certain probability measures on R to geodesics in the space HL of positively curved metrics on a line bundle L, and to geodesics in the finite dimensional symmetric space of hermitian norms on H(X, kL). We prove that the measures associated to the finite dimensional spaces converge weakly to the measures related to geodesics in HL as k goes to infinity. The convergence of second or...

on the basis of a reproducing kernel space, an iterative algorithm for solving the inverse problem for heat equation with a nonlocal boundary condition is presented. the analytical solution in the reproducing kernel space is shown in a series form and the approximate solution vn is constructed by truncating the series to n terms. the convergence of vn to the analytical solution is also proved. ...

on the basis of a reproducing kernel space, an iterative algorithm for solving the one-dimensional linear and nonlinear schrödinger equations is presented. the analytical solution is shown in a series form in the reproducing kernel space and the approximate solution is constructed by truncating the series. the convergence of the approximate solution to the analytical solution is also proved. th...

We prove that the implicit time Euler scheme coupled with finite elements space discretization for 2D Navier–Stokes equations on torus subject to a random perturbation converges in $$L^2(\varOmega )$$ , and describe rate of convergence an $$H^1$$ -valued initial condition. This refines previous results which only established probability these numerical approximations. Using exponential moment e...

Let (S(t))t≥0 be a one-parameter family of positive integral operators on a locally compact space L. For a possibly non-uniform partition of [0, 1] define a finite measure on the path space CL[0, 1] by using a) S(∆t) for the transition between any two consecutive partition times of distance ∆t and b) a suitable continuous interpolation scheme (e.g. Brownian bridges or geodesics). If necessary n...

We consider the problem of estimating a regression function on the basis of empirical data. We use a Reproducing Kernel Hilbert Space (RKHS) as our hypothesis space, and we follow the methodology of Tikhonov regularization. We show that this leads to a learning scheme that is different from the one usually considered in Learning Theory. Subject to some regularity assumptions on the regression f...

Consider a nonlinear ill-posed operator equation F (u) = y where F is defined on a Banach space X. In this paper we analyze finite dimensional variational regularization, which takes into account operator approximations and noisy data. As shown in the literature, depending on the setting, convergence of the regularized solutions of the finite dimensional problems can be with respect to the stro...

A fast algorithm is developed to compute orthogonal polynomial expansions on sparse grids for a function of d variables in a weighted L space. The proposed algorithm combines the fast cosine transform, a fast transform from the Chebyshev orthogonal polynomial basis to the orthogonal polynomial basis for the weighted L space and a fast algorithm of computing hierarchically structured basis funct...

The qualitative natures of the transient behaviours of the waiting times in a GI/G/l system are studied first, when the waiting time of the initial customer is v. For instance, it is shown that the mean waiting time is increasing in n only when v < l for some l. Next this paper gives a way of calculating exact values of the mean waiting time of the n-th arrival in M/M/l and M/D/l and further pr...