Information-based complexity
Error analysis of a randomized numerical method
Numerische Mathematik
A quasi-randomized Runge-Kutta method
Mathematics of Computation
Quasi-randomized numerical methods for systems with coefficients of bounded variation
Mathematics and Computers in Simulation - IMACS sponsored Special issue on the second IMACS seminar on Monte Carlo methods
Randomized and quantum algorithms yield a speed-up for initial-value problems
Journal of Complexity
Monte Carlo approximation of weakly singular integral operators
Journal of Complexity
The randomized information complexity of elliptic PDE
Journal of Complexity
Improved bounds on the randomized and quantum complexity of initial-value problems
Journal of Complexity
Almost optimal solution of initial-value problems by randomized and quantum algorithms
Journal of Complexity - Special issue: Information-based complexity workshops FoCM conference Santander, Spain, July 2005
Optimal adaptive solution of initial-value problems with unknown singularities
Journal of Complexity
On the randomized solution of initial value problems
Journal of Complexity
Optimal solution of a class of non-autonomous initial-value problems with unknown singularities
Journal of Computational and Applied Mathematics
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We study the complexity of randomized solution of initial value problems for systems of ordinary differential equations (ODE). The input data are assumed to be @c-smooth (@c=r+@r: the rth derivatives satisfy a @r-Holder condition). Recently, the following almost sharp estimate of the order of the nth minimal error was given by Kacewicz [Almost optimal solution of initial-value problems by randomized and quantum algorithms, J. Complexity 22 (2006) 676-690, see also ]:c"1n^-^@c^-^1^/^2=0. We present a Taylor Monte Carlo method and show that it has error rate n^-^@c^-^1^/^2, this way establishing the exact order of the randomized nth minimal error.