Elsevier

Applied Mathematics and Computation

Volume 245, 15 October 2014, Pages 357-371
Applied Mathematics and Computation

Randomized and quantum complexity of nonlinear two-point BVPs

https://doi.org/10.1016/j.amc.2014.07.106Get rights and content

Abstract

We deal with the complexity of nonlinear BVPs with nonlinear two-point boundary conditions. We consider the randomized and quantum models of computation. We assume that the right-hand side function is r times differentiable with all derivatives bounded by a constant. We show that the ε-complexity is roughly of order ε-1/(r+1/2) in the randomized setting, and ε-1/(r+1) in the quantum setting. We compare our results with known results in the deterministic setting. The speed-up of the randomized computations with respect to the deterministic computations is by 1/(r(2r+1)) in the exponent of 1/ε, and the speed-up of the quantum computations by 1/(r(r+1)) in the exponent.

Introduction

In this paper we deal with the complexity of nonlinear two-point boundary-value problems. We consider two settings: the randomized setting and the quantum setting. We derive upper and lower bounds on the complexity of this problem.

Differential equations arises in many fields and are important for numerical analysis. These problems are nonlinear and there are only few general results on the complexity of such problems. Initial-value problems are well studied in literature in different models of computation. Boundary-value problems are harder since the complexity analysis requires besides initial-value problems, nonlinear equations.

For many computational problems the randomized and quantum computations yield a speed-up over the deterministic ones. Many continuous problems has already been studied in these settings. Examples include the integration [12], approximation [5], [6], initial-value problems [1], [10], [7] and linear two-point boundary-value problems [2].

In [2] scalar linear two-point boundary-value problems of order k are considered. Three models of computation are investigated: the deterministic with standard and linear information, the randomized and the quantum. There are determined the upper and the lower complexity bounds in the class of right-hand side functions with bounded derivatives up to order r. The ε-complexity bounds are of order (1/ε)1/r in the deterministic setting with standard information. When linear information is allowed, the upper complexity bound is of order (1/ε)1/(r+k-q), where q is the highest order of the derivative of the solution appearing in the right-hand side. The lower complexity bound is of order (1/ε)1/(r+k). In the randomized setting the complexity bounds are roughly of order (1/ε)1/(r+1/2), and in the quantum setting of order (1/ε)1/(r+1).

Nonlinear boundary-value problems in the deterministic setting were investigated by Kacewicz in [8]. Kacewicz considered two type of information: standard information consisting of evaluations of the right-hand side function and its derivatives, and general linear information consisting of arbitrary linear functionals of the right-hand side function. Two problems are analyzed. The first one is a general nonlinear two-point boundary-value problem with nonlinear boundary conditions and the second one is a scalar second-order problem with separated boundary conditions. There are proposed algorithms solving these problems and determined the upper and the lower ε-complexity bounds. Kacewicz considered the class of r-times differentiable right-hand side functions with bounded derivatives. For the general nonlinear BVP the complexity bounds are roughly of order (1/ε)1/r when only standard information is allowed, and (1/ε)1/(r+1) for linear information, where the upper and the lower complexity bounds differ by a factor of loglog1/ε. The complexity bounds for the second order BVP differ from the bounds for the general nonlinear BVP only for linear information and are roughly of order (1/ε)1/(r+2).

We study the same problems as in [8]. However, in this paper we consider the randomized and quantum settings. We construct two algorithms for solving these problems: a randomized and a quantum. We derive upper bounds for the error and the cost for these algorithms. We also derive the lower complexity bounds for these problems. We show that the constructed algorithms are almost optimal and the ε-complexity for both, the general and the second order BVPs, is roughly of order (1/ε)1/(r+1/2) in the randomized setting, and (1/ε)1/(r+1) in the quantum setting. These bounds along with the bounds of Kacewicz in the deterministic setting are presented in Table 1. The upper bounds in the randomized and quantum settings differ from the lower bounds by an arbitrarily small parameter γ>0 in the exponent. These results yield that randomized computations yield a speed-up of 1/(r(2r+1)) in the exponent of ε-1 over the deterministic setting with standard information, and the quantum computers a speed-up of 1/(r(r+1)) in the exponent. The complexity bounds in the randomized and quantum settings are of the same order as for the linear boundary-value problems presented in [2].

This paper is organized as follows: in Section 2 the problem is formulated and basic definitions are presented. In Section 3, known results treating this problem are presented together with some results needed in the analysis of the algorithms and in the proof of the lower bounds. Section 4 contains the definitions of the algorithms solving the nonlinear two-point boundary-value problems, and also the analysis of these algorithms and the upper complexity bounds. In Section 5 the lower complexity bounds are derived. In Appendix A the proofs of some auxiliary lemmas and theorems are presented.

Section snippets

Problem formulation and basic definitions

We consider the following nonlinear two-point boundary-value problem with nonlinear boundary conditions:z(x)=f(x,z(x)),x[a,b],p(z(a),z(b))=0,where a<b,f:[a,b]×RdRd,z:[a,b]Rd, and p:Rd×RdRd. We assume that problem (1) has a unique solution. We assume that function f belongs to the class Fdr of r-times differentiable functions with all partial derivatives bounded by a constant, i.e.Fdr={f:[a,b]×RdRd|fCr([a,b]×Rd),D(i)fD,i=0,1,r},where r1,D is a positive constant, D(i) run through all

Auxiliary results

There are known complexity bounds for problem (1) in the deterministic setting with standard and linear information. Kacewicz in [8] proved the following theorem dealing with the upper and the lower complexity bounds for this problem.

Theorem 1

[8]

For problem (1) we have:

  • for all p(s,w) satisfying Assumption 1 and r2 there exists a positive constant K1 such that for arbitrary sufficiently small ε>0 we havecompεdet-st(Fdr)K11ε1/rloglog1ε,compεdet-lin(Fdr)K11ε1/(r+1)loglog1ε

  • for p(s,w)=s-η and r1 there

Upper bounds for problem (1)

The upper complexity bounds for problem (1) in the randomized and quantum settings are given in the following theorem.

Theorem 7

For any γ(0,1) and for all p satisfying Assumption 1 there exist the numbers C1(γ),C2(γ) and ε0(γ) (depending on γ,a,b, the class parameters and the function p), such that for any ε(0,ε0(γ)),δ(0,1/2) the ε-complexity of problem (1) satisfiescompεrand(Fdr)C1(γ)1ε1/(r+1/2-γ),compεquant(Fdr,δ)C2(γ)1ε1/(r+1-γ)log1δ.

We will prove this theorem by constructing two algorithms, a

Lower bounds

We state the lower complexity bounds for the general boundary-value problems (1). Consider the function p(s,w)=s-α. For such defined function p, problem (1) becomes the initial-value problem with initial condition z(a)=α. Such p satisfies Assumption 1 and F=s-α. Function F satisfies Assumption 2, so for such p, class Fdr is equivalent the class Fdr. We use the lower bounds on the complexity of initial-value problems from [10]. From Theorem 5 we get the following lower bounds on the complexity

Summary of the paper

We have studied the complexity of the nonlinear boundary-value problem of the form (1), and the scalar second order problem of the form (4). We have considered the randomized and quantum settings. For each setting we have constructed the algorithm solving problem (1) (which also solves (4)). We have determined the upper and the lower complexity bounds for these problems. These bounds along with known bounds in the deterministic setting are presented in Table 1 in Introduction (for clarity we

Acknowledgments

This research was partly supported by the Polish NCN Grant – decision No. DEC-2013/09/B/ST1/04275 and by Polish Ministry of Science and Higher Education.

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