Optimal solution of ordinary differential equations
Journal of Complexity
Quantum summation with an application to integration
Journal of Complexity
Randomized and quantum algorithms yield a speed-up for initial-value problems
Journal of Complexity
Improved bounds on the randomized and quantum complexity of initial-value problems
Journal of Complexity
Quantum lower bounds by entropy numbers
Journal of Complexity
The randomized complexity of initial value problems
Journal of Complexity
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 establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These algorithms yield new upper complexity bounds, which differ from known lower bounds by only an arbitrarily small positive parameter in the exponent, and a logarithmic factor. In both the randomized and quantum settings, initial-value problems turn out to be essentially as difficult as scalar integration.