Information-based complexity
The computational complexity of differential and integral equations: an information-based approach
The computational complexity of differential and integral equations: an information-based approach
The complexity of definite elliptic problems with noisy data
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
Quantum complexity of integration
Journal of Complexity
An Introduction to Quantum Computing Algorithms
An Introduction to Quantum Computing Algorithms
Quantum computation and quantum information
Quantum computation and quantum information
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
Quantum summation with an application to integration
Journal of Complexity
Quantum integration in Sobolev classes
Journal of Complexity
Path Integration on a Quantum Computer
Quantum Information Processing
From Monte Carlo to quantum computation
Mathematics and Computers in Simulation - Special issue: 3rd IMACS seminar on Monte Carlo methods - MCM 2001
Quantum approximation I. Embeddings of finite-dimensional Lp spaces
Journal of Complexity
Quantum approximation II. Sobolev embeddings
Journal of Complexity
Quantum complexity of parametric integration
Journal of Complexity
Randomized and quantum algorithms yield a speed-up for initial-value problems
Journal of Complexity
Monte Carlo approximation of weakly singular integral operators
Journal of Complexity
The randomized information complexity of elliptic PDE
Journal of Complexity
Optimal approximation of elliptic problems by linear and nonlinear mappings I
Journal of Complexity - Special issue: Algorithms and complexity for continuous problems Schloss Dagstuhl, Germany, September 2004
Optimal approximation of elliptic problems by linear and nonlinear mappings II
Journal of Complexity
Optimal integration error on anisotropic classes for restricted Monte Carlo and quantum algorithms
Journal of Approximation Theory
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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 ⊂ Rd with smooth coefficients and homogeneous boundary conditions. We seek to approximate the solution on a smooth submanifold M ⊆ Q of dimension 0 ≤ d1 ≤ d. With the right-hand side belonging to Cr (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)/d1,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 d1, while in the classical randomized setting it is (up to logarithmic factors) n-min((r+2m)/d1,r/d+1/2).