Lower Bounds and Non-Uniform Time Discretization for Approximation of Stochastic Heat Equations

We study algorithms for approximation of the mild solution of stochastic heat equations on the spatial domain ]0,1[^d. The error of an algorithm is defined in L_2-sense. We derive lower bounds for the error of every algorithm that uses a total of N evaluations of one-dimensional components of the driving Wiener process W. For equations with additive noise we derive matching upper bounds and we construct asymptotically optimal algorithms. The error bounds depend on N and d, and on the decay of eigenvalues of the covariance of W in the case of nuclear noise. In the latter case the use of non-uniform time discretizations is crucial

Random Bit Multilevel Algorithms for Stochastic Differential Equations

We study the approximation of expectations \E(f(X)) for solutions $X$ of SDEs and functionals $f \colon C([0,1],\R^r) \to \R$ by means of restricted Monte Carlo algorithms that may only use random bits instead of random numbers. We consider the worst case setting for functionals $f$ from the Lipschitz class w.r.t.\ the supremum norm. We construct a random bit multilevel Euler algorithm and establish upper bounds for its error and cost. Furthermore, we derive matching lower bounds, up to a logarithmic factor, that are valid for all random bit Monte Carlo algorithms, and we show that, for the given quadrature problem, random bit Monte Carlo algorithms are at least almost as powerful as general randomized algorithms

Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces

We study the approximation of expectations \E(f(X)) for Gaussian random elements $X$ with values in a separable Hilbert space $H$ and Lipschitz continuous functionals $f \colon H \to \R$. We consider restricted Monte Carlo algorithms, which may only use random bits instead of random numbers. We determine the asymptotics (in some cases sharp up to multiplicative constants, in the other cases sharp up to logarithmic factors) of the corresponding $n$-th minimal error in terms of the decay of the eigenvalues of the covariance operator of $X$. It turns out that, within the margins from above, restricted Monte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms, and suitable random bit multilevel algorithms are optimal. The analysis of this problem leads to a variant of the quantization problem, namely, the optimal approximation of probability measures on $H$ by uniform distributions supported by a given, finite number of points. We determine the asymptotics (up to multiplicative constants) of the error of the best approximation for the one-dimensional standard normal distribution, for Gaussian measures as above, and for scalar autonomous SDEs

On the Complexity of Parabolic Initial Value Problems with Variable Drift

We consider linear parabolic initial value problems of second order in several dimensions. The initial condition is supposed to be fixed and we investigate the comutational complexity if the coefficients of the parabolic equations may vary in certain function spaces. Using the parametrix method (or Neumann series), we prove that lower bounds for the error of numerical methods are related to lower bounds for integration problems. On the other hand, approximating the Neumann series with Smolyak\u27s method, we show that the problem is not much harder than a certain approximation problem. For Hölder classes on compact sets, e.g., lower and upper bounds are close together, such that we have an almost optimal method

Properties of discontinuous and nova-amplified mass transfer in CVs

We investigate the effects of discontinuous mass loss in recurrent outburst events on the long-term evolution of cataclysmic variables (CVs). Similarly we consider the effects of frictional angular momentum loss (FAML), i.e. interaction of the expanding nova envelope with the secondary. Numerical calculations of CV evolution over a wide range of parameters demon- strate the equivalence of a discontinuous sequence of nova cycles and the corresponding mean evolution (replacing envelope ejection by a continuous wind), even close to mass transfer instability. A formal stability analysis of discontinuous mass transfer confirms this, independent of details of the FAML model. FAML is a consequential angular momentum loss which amplifies the mass transfer rate driven by systemic angular momentum losses such as magnetic braking. We show that for a given v_exp and white dwarf mass the amplification increases with secondary mass and is significant only close to the largest secondary mass consistent with mass transfer stability. The amplification factor is independent of the envelope mass ejected during the outburst, whereas the mass transfer amplitude induced by individual nova outbursts is proportional to it.Comment: 16 pages, 19 figures; to appear in MNRA

Evaluating Expectations of Functionals of Brownian Motions: a Multilevel Idea

Prices of path dependent options may be modeled as expectations of functions of an infinite sequence of real variables. This talk presents recent work on bounding the error of such expectations using quasi-Monte Carlo algorithms. The expectation is approximated by an average of $n$ samples, and the functional of an infinite number of variables is approximated by a function of only $d$ variables. A multilevel algorithm employing a sum of sample averages, each with different truncated dimensions, $d_l$, and different sample sizes, $n_l$, yields faster convergence than a single level algorithm. This talk presents results in the worst-case error setting