The quantum query complexity of elliptic PDE

Heinrich, Stefan

doi:10.1016/j.jco.2006.04.005

Cited by 18 publications

(10 citation statements)

References 25 publications

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“…The complexity of solving elliptic PDE in the quantum model of computation will be the topic of a subsequent paper [11].…”

Section: Introductionmentioning

confidence: 99%

The randomized information complexity of elliptic PDE

Heinrich

2006

Journal of Complexity

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We study the information complexity in the randomized setting of solving a general elliptic PDE of order 2m in a smooth, bounded domain Q ⊂ R d with smooth coefficients and homogeneous boundary conditions. The solution is sought on a smooth submanifold M ⊆ Q of dimension 0 d 1 d, the right-hand side is supposed to be in C r (Q), the error is measured in the L ∞ (M) norm. We show that the nth minimal error is (up to logarithmic factors) of order n − min((r+2m)/d 1 , r/d+1/2) .For comparison, in the deterministic setting the nth minimal error is of order n −r/d , for all d 1 .

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“…The complexity of solving elliptic PDE in the quantum model of computation will be the topic of a subsequent paper [11].…”

Section: Introductionmentioning

confidence: 99%

The randomized information complexity of elliptic PDE

Heinrich

2006

Journal of Complexity

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“…A referee pointed out that with the help of Proposition 5 in [14] the proof of Theorem 2 can be simplified by using a more elegant reduction technique. Now we present the key parts of this technique.…”

Section: The Proof Of Resultsmentioning

confidence: 99%

“…The following Proposition that is used for our reductions is a direct consequence of Proposition 5 in [14].…”

Section: S(f ) =S • R(f ) (42)mentioning

confidence: 99%

Optimal integration error on anisotropic classes for restricted Monte Carlo and quantum algorithms

2008

Journal of Approximation Theory

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We study restricted Monte Carlo integration for anisotropic Hölder-Nikolskii classes. The results show that with clog 2 n random bits we have the same optimal order for the nth minimal Monte Carlo integration error as with arbitrary random numbers. We also study the computation of integration on anisotropic Sobolev classes in the quantum setting and present the optimal bound of nth minimal query error. The results show that the error bound of quantum algorithms is much smaller than that of deterministic and randomized algorithms.

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“…The randomized setting, where also the use of random numbers is allowed, is studied by Heinrich [51]. The complexity of solving elliptic PDE in the quantum model of computation (where one can use a certain nonclassical randomness) is studied in [52].…”

Section: Remark 16mentioning

confidence: 99%

Optimal approximation of elliptic problems by linear and nonlinear mappings II

Dahlke

Novak

Sickel

2006

Journal of Complexity

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We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: (a) linear mappings of rank n; (b) n-term approximation with respect to a Riesz basis; (c) approximation based on linear information about the right-hand side f; (d) continuous mappings. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space B r q (L p ( )) and ⊂ R d is a bounded Lipschitz domain; the error is always measured in the H s -norm. The respective widths are the linear widths (or approximation numbers), the nonlinear widths, the Gelfand widths, and the manifold widths. As a technical tool, we also study the Bernstein numbers. Our main results are the following. If p 2 then the order of convergence is the same for all four classes of approximations. In particular, the best linear approximations are of the same order as the best nonlinear ones. The best linear approximation can be quite difficult to realize as a numerical algorithm since the optimal Galerkin space usually depends on the operator and on the shape of the domain . For p < 2 there is a difference, nonlinear approximations are better than linear ones. However, in this case, it turns out that linear information about the right-hand side f is again optimal. Our main theoretical tool is the best n-term approximation with respect to an optimal Riesz basis and related nonlinear widths. These general results are used to study the Poisson equation in a polygonal domain. It turns out that best n-term wavelet approximation is (almost) optimal. The main results of this paper are about approximation, not about computation. However, we also discuss consequences of the results for the numerical complexity of operator equations.

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The quantum query complexity of elliptic PDE

Cited by 18 publications

References 25 publications

The randomized information complexity of elliptic PDE

The randomized information complexity of elliptic PDE

Optimal integration error on anisotropic classes for restricted Monte Carlo and quantum algorithms

Optimal approximation of elliptic problems by linear and nonlinear mappings II

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