Information-based complexity
The complexity of two-point boundary-value problems with piecewise analytic data
Journal of Complexity
Solving Ordinary Differential Equations with Discontinuities
ACM Transactions on Mathematical Software (TOMS)
Iterated Defect Correction for the Solution of Singular Initial Value Problems
SIAM Journal on Numerical Analysis
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
The randomized complexity of initial value problems
Journal of Complexity
Adaptive Itô-Taylor algorithm can optimally approximate the Itô integrals of singular functions
Journal of Computational and Applied Mathematics
Optimal solution of a class of non-autonomous initial-value problems with unknown singularities
Journal of Computational and Applied Mathematics
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The optimal solution of initial-value problems in ODEs is well studied for smooth right-hand side functions. Much less is known about the optimality of algorithms for singular problems. In this paper, we study the (worst case) solution of scalar problems with a right-hand side function having r continuous bounded derivatives in R, except for an unknown singular point. We establish the minimal worst case error for such problems (which depends on r similarly as in the smooth case), and define optimal adaptive algorithms. The crucial point is locating an unknown singularity of the solution by properly adapting the grid. We also study lower bounds on the error of an algorithm for classes of singular problems. In the case of a single singularity with nonadaptive information, or in the case of two or more singularities, the error of any algorithm is shown to be independent of r.