Almost optimal solution of initial-value problems by randomized and quantum algorithms

Kacewicz, Bolesław

doi:10.1016/j.jco.2006.03.001

Cited by 62 publications

(58 citation statements)

References 6 publications

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“…For instance, we mention the results in [7,19,20,22,30,31] and the references therein. These randomized methods are usually found to be superior over corresponding deterministic methods in the sense that the resulting discretization error decays already with order O(N −γ− 1 2 ) under the same smoothness assumptions as sketched above.…”

Section: Introductionmentioning

confidence: 99%

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Kruse

2017

Computational Methods in Applied Mathematics

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Abstract. This paper contains an error analysis of two randomized explicit Runge-Kutta schemes for ordinary differential equations (ODEs) with timeirregular coefficient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coefficient functions are only integrable with respect to the time variable but are not assumed to be continuous. A further field of application are ODEs with coefficient functions that contain weak singularities with respect to the time variable.The main result consists of precise bounds for the discretization error with respect to the L p (Ω; R d )-norm. In addition, convergence rates are also derived in the almost sure sense. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. The theoretical results are illustrated through a few numerical experiments.

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Section: Introductionmentioning

confidence: 99%

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Kruse

2017

Computational Methods in Applied Mathematics

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“…We can consider N to be an arbitrary positive integer and take above l = N 1/d . We have shown the following Theorem 1 Let S be a linear continuous functional on X r,ρ , whose extention toX r,ρ satisfies (6). There exists a constant C such that for all N ≥ 1 there is a quantum (randomized) algorithm (defined by (17) with A( f ) computed in the respective setting as described above), with the cost not greater than C N , and the error bounded by…”

Section: Quantum and Randomized Algorithms And Their Performancementioning

confidence: 99%

“…LetŜ be a linear continuous functional on C [0, 1] d whose extension to the space of piecewise continuous functions satisfies (6). That is, let B be an interval of the form (3), and u be given by…”

Section: A Linear Continuous Functional S On F Can Be Defined By Anotmentioning

confidence: 99%

“…In the last decade a number of continuous problems have been studied in the quantum setting, see e.g. [4] for the approximation problem, [13] for quantum computing of path integrals or [6] for quantum solving of ODEs. For many continuous problems the speed-up of quantum algorithms over classical randomized ones is possible due to a proper application of a fast quantum summation algorithm by Brassard et al [1].…”

Section: Introductionmentioning

confidence: 99%

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On the quantum and randomized approximation of linear functionals on function spaces

Kacewicz

2010

Quantum Inf Process

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We deal with quantum and randomized algorithms for approximating a class of linear continuous functionals. The functionals are defined on a Hölder space of functions f of d variables with r continuous partial derivatives, the r th derivative being a Hölder function with exponent ρ. For a certain class of such linear problems (which includes the integration problem), we define algorithms based on partitioning the domain of f into a large number of small subdomains, and making use of the well-known quantum or randomized algorithms for summation of real numbers. For N information evaluations (quantum queries in the quantum setting), we show upper bounds on the error of order N −(γ +1) in the quantum setting, and N −(γ +1/2) in the randomized setting, where γ = (r + ρ)/d is the regularity parameter. Hence, we obtain for a wider class of linear problems the same upper bounds as those known for the integration problem. We give examples of functionals satisfying the assumptions, among which we discuss functionals defined on the solution of Fredholm integral equations of the second kind, with complete information about the kernel. We also provide lower bounds, showing in some cases sharpness of the obtained results, and compare the power of quantum, randomized and deterministic algorithms for the exemplary problems.

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“…Our analysis of the query complexity is carried out in the quantum setting of information-based complexity theory, as developed in [11]. For a study of other basic numerical problems in this framework we refer to [24,12,14,15,19,21,27,32], see also the surveys [13,16]. For general background on quantum computation we refer to the surveys [2,8,26], and the monographs [25,9,22].…”

Section: Introductionmentioning

confidence: 99%

The quantum query complexity of elliptic PDE

Heinrich

2006

Journal of Complexity

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The query complexity of the following numerical problem is studied in the quantum model of computation: consider a general elliptic partial differential equation of order 2m in a smooth, bounded domain Q ⊂ R d with smooth coefficients and homogeneous boundary conditions. We seek to approximate the solution on a smooth submanifold M ⊆ Q of dimension 0 d 1 d. With the right-hand side belonging to C r (Q), and the error being measured in the L ∞ (M) norm, we prove that the nth minimal quantum error is (up to logarithmic factors) of order n − min((r+2m)/d 1 ,r/d+1) .For comparison, in the classical deterministic setting the nth minimal error is known to be of order n −r/d , for all d 1 , while in the classical randomized setting it is (up to logarithmic factors) n − min((r+2m)/d 1 ,r/d+1/2) .

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Almost optimal solution of initial-value problems by randomized and quantum algorithms

Cited by 62 publications

References 6 publications

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

Error Analysis of Randomized Runge–Kutta Methods for Differential Equations with Time-Irregular Coefficients

On the quantum and randomized approximation of linear functionals on function spaces

The quantum query complexity of elliptic PDE

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