Free-Knot Spline Approximation of Stochastic Processes

We study optimal approximation of stochastic processes by polynomial splines with free knots. The number of free knots is either a priori fixed or may depend on the particular trajectory. For the $s$-fold integrated Wiener process as well as for scalar diffusion processes we determine the asymptotic behavior of the average $L_p$-distance to the splines spaces, as the (expected) number $k$ of free knots tends to infinity.Comment: 23 page

On the probabilistic continuous complexity conjecture

In this paper we prove the probabilistic continuous complexity conjecture. In continuous complexity theory, this states that the complexity of solving a continuous problem with probability approaching 1 converges (in this limit) to the complexity of solving the same problem in its worst case. We prove the conjecture holds if and only if space of problem elements is uniformly convex. The non-uniformly convex case has a striking counterexample in the problem of identifying a Brownian path in Wiener space, where it is shown that probabilistic complexity converges to only half of the worst case complexity in this limit

A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations

In this paper the numerical solution of non-autonomous semilinear stochastic evolution equations driven by an additive Wiener noise is investigated. We introduce a novel fully discrete numerical approximation that combines a standard Galerkin finite element method with a randomized Runge-Kutta scheme. Convergence of the method to the mild solution is proven with respect to the $L^p$-norm, $p \in [2,\infty)$. We obtain the same temporal order of convergence as for Milstein-Galerkin finite element methods but without imposing any differentiability condition on the nonlinearity. The results are extended to also incorporate a spectral approximation of the driving Wiener process. An application to a stochastic partial differential equation is discussed and illustrated through a numerical experiment.Comment: 31 pages, 1 figur

Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations

The simulation of the expectation of a stochastic quantity E[Y] by Monte Carlo methods is known to be computationally expensive especially if the stochastic quantity or its approximation Y_n is expensive to simulate, e.g., the solution of a stochastic partial differential equation. If the convergence of Y_n to Y in terms of the error |E[Y - Y_n]| is to be simulated, this will typically be done by a Monte Carlo method, i.e., |E[Y] - E_N[Y_n]| is computed. In this article upper and lower bounds for the additional error caused by this are determined and compared to those of |E_N[Y - Y_n]|, which are found to be smaller. Furthermore, the corresponding results for multilevel Monte Carlo estimators, for which the additional sampling error converges with the same rate as |E[Y - Y_n]|, are presented. Simulations of a stochastic heat equation driven by multiplicative Wiener noise and a geometric Brownian motion are performed which confirm the theoretical results and show the consequences of the presented theory for weak error simulations.Comment: 16 pages, 5 figures; formulated Section 2 independently of SPDEs, shortened Section 3, added example of geometric Brownian motion in Section