462 research outputs found
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 of
SDEs and functionals 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 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 with values in a separable Hilbert space and Lipschitz
continuous functionals . 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 -th
minimal error in terms of the decay of the eigenvalues of the covariance
operator of . 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 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 samples, and the functional of an infinite number of variables is approximated by a function of only variables. A multilevel algorithm employing a sum of sample averages, each with different truncated dimensions, , and different sample sizes, , yields faster convergence than a single level algorithm. This talk presents results in the worst-case error setting
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