How to increase the order to get minimal-error algorithms for systems of ODE

Kacewicz, Bolesław

doi:10.1007/bf01379663

Cited by 34 publications

(34 citation statements)

References 2 publications

(6 reference statements)

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“…Theorem A (Kacewicz [3]). Let yðx; sÞ be the solution of (12) This result allows us to establish a bound on the global error from the knowledge about the behavior of the local errors.…”

Section: Results For Initial-value Problemsmentioning

confidence: 97%

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Complexity of Nonlinear Two-Point Boundary-Value Problems

Kacewicz

2002

Journal of Complexity

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We study upper and lower bounds on the worst-case e-complexity of nonlinear two-point boundary-value problems. We deal with general systems of equations with general nonlinear boundary conditions, as well as with second-order scalar problems. Two types of information are considered: standard information defined by the values or partial derivatives of the right-hand-side function, and linear information defined by arbitrary linear functionals. The complexity depends significantly on the problem being solved and on the type of information allowed. We define algorithms based on standard or linear information, using perturbed Newton's iteration, which provide upper bounds on the e-complexity. The upper and lower bounds obtained differ by a factor of log log 1=e: Neglecting this factor, for general problems the e-complexity for the right-hand-side functions having r ðr52Þ continuous bounded partial derivatives turns out to be of order ð1=eÞ 1=r for standard information, and ð1=eÞ 1=ðrþ1Þ for linear information. For second-order scalar problems, linear information is even more powerful. The e-complexity in this case is shown to be of order ð1=eÞ 1=ðrþ2Þ ; while for standard information it remains at the same level as in the general case. # 2002 Elsevier Science (USA)

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“…Theorem A (Kacewicz [3]). Let yðx; sÞ be the solution of (12) This result allows us to establish a bound on the global error from the knowledge about the behavior of the local errors.…”

Section: Results For Initial-value Problemsmentioning

confidence: 97%

“…Assumptions about f and p that we adopt must reflect the relation between problems (1) and (11). It is not surprising that these assumptions are more demanding than the conditions imposed on f for initial-value problems [3].…”

Section: Problem Formulation and Definitionsmentioning

confidence: 99%

Complexity of Nonlinear Two-Point Boundary-Value Problems

Kacewicz

2002

Journal of Complexity

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“…This problem has been considered in a number of papers, see, e.g., [7] or [8]. The following result is a straightforward modification of Corollary 4.1 from [7].…”

Section: Article In Pressmentioning

confidence: 89%

“…For dX2 (and also in the case of nonautonomous equations with d ¼ 1), finding lower bounds on the complexity of (1) can be reduced to the similar problem for integration. For autonomous problems with d ¼ 1 a straightforward reduction to the integration problem is not possible, and the question about lower bounds is then still open (a similar difficulty is successfully faced in the worst-case setting in [7]). …”

Section: Article In Pressmentioning

confidence: 96%

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Randomized and quantum algorithms yield a speed-up for initial-value problems

Kacewicz

2004

Journal of Complexity

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Quantum algorithms and complexity have recently been studied not only for discrete, but also for some numerical problems. Most attention has been paid so far to the integration and approximation problems, for which a speed-up is shown in many important cases by quantum computers with respect to deterministic and randomized algorithms on a classical computer. In this paper, we deal with the randomized and quantum complexity of initial-value problems. For this nonlinear problem, we show that both randomized and quantum algorithms yield a speed-up over deterministic algorithms. Upper bounds on the complexity in the randomized and quantum settings are shown by constructing algorithms with a suitable cost, where the construction is based on integral information. Lower bounds result from the respective bounds for the integration problem. r 2004 Elsevier Inc. All rights reserved.

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Quantum Algorithms for Continuous Problems and Their Applications

Papageorgiou

Traub

2014

Advances in Chemical Physics

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Many problems in science and engineering are formulated using continuous mathematical models. Usually they can only be solved numerically and therefore approximately. Since they are often difficult to solve on a classical computer it's interesting to investigate whether they can be solved faster on a quantum computer.After a brief introduction to quantum algorithms we report on a wide range of applications including high dimensional integration, path integration, Hamiltonian simulation, and ground state energy estimation. We provide a rather extensive bibliography.

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How to increase the order to get minimal-error algorithms for systems of ODE

Cited by 34 publications

References 2 publications

Complexity of Nonlinear Two-Point Boundary-Value Problems

Complexity of Nonlinear Two-Point Boundary-Value Problems

Randomized and quantum algorithms yield a speed-up for initial-value problems

Quantum Algorithms for Continuous Problems and Their Applications

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