Is Gauss quadrature optimal for analytic functions?

We consider the problem of optimal quadratures for integrands f: [ -1 , 1 ] → R which have an analytic extension f to an open disk D, of radius r about the origin such that |f| ≤ 1 on Dr. If r = 1, we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degree n rather than at optimal points, tends to infinity with n. In particular there is an "infinite" penalty for using Gauss quadrature. On the other hand, if r > l, Gauss quadrature is almost optimal. These results hold for both the worst-case and asymptotic settings

Quantum algorithm and circuit design solving the Poisson equation

The Poisson equation occurs in many areas of science and engineering. Here we focus on its numerical solution for an equation in d dimensions. In particular we present a quantum algorithm and a scalable quantum circuit design which approximates the solution of the Poisson equation on a grid with error \varepsilon. We assume we are given a supersposition of function evaluations of the right hand side of the Poisson equation. The algorithm produces a quantum state encoding the solution. The number of quantum operations and the number of qubits used by the circuit is almost linear in d and polylog in \varepsilon^{-1}. We present quantum circuit modules together with performance guarantees which can be also used for other problems.Comment: 30 pages, 9 figures. This is the revised version for publication in New Journal of Physic

What is the complexity of surface integration

We study the worst case complexity of computing ε-approximations of surface integrals. This problem has two sources of partial information: the integrand f and the function g defining the surface. The problem is nonlinear in its dependence on g. Here, f is an r times continuously differentiable scalar function of l variables, and g is an s times continuously differentiable injective function of d variables with l components. We must have d ≤ l and s ≥ 1 for surface integration to be well-defined. Surface integration is related to the classical integration problem for functions of d variables that are min{r, s − 1} times continuously differentiable. This might suggest that the complexity of surface integration should be �((1/ε) d / min{r,s−1}). Indeed, this holds when d < l and s = 1, in which case the surface integration problem has infinite complexity. However, if d ≤ l and s ≥ 2, we prove tha

What is the complexity of volume calculation

We study the worst case complexity of computing ε-approximations of volumes of d-dimensional regions g([0, 1] d), by sampling the function g. Here, g is an s times continuously differentiable injection from [0, 1] d to R d, where we assume that s ≥ 1. Since the problem can be solved exactly when d = 1, we concentrate our attention on the case d ≥ 2. This problem is a special case of the surface integration problem studied in [12]. Let c be the cost of one function evaluation. The results of [12] might suggest that the εcomplexity of volume calculation should be proportional to c(1/ε) d/s when s ≥ 2. However, using integration by parts to reduce the dimension, we show that if s ≥ 2, then the complexity is proportional to c(1/ε) (d−1)/s. Next, we consider the case s = 1, which is the minimal smoothness for which our volume problem is well-defined. We show that when s = 1, an ε-approximatio

Tractability of quasilinear problems I: general results

The tractability of multivariate problems has usually been studied only for the approximation of linear operators. In this paper we study the tractability of quasilinear multivariate problems. That is, we wish to approximate nonlinear operators Sd(·, ·) that depend linearly on the first argument and satisfy a Lipschitz condition with respect to both arguments. Here, both arguments are functions of d variables. Many computational problems of practical importance have this form. Examples inlude the solution of specific Dirichlet, Neumann, and Schrödinger problems. We show, under appropriate assumptions, that quasilinear problems, whose domain spaces are equipped with product or finite-order weights, are tractable or strongly tractable in the worst case setting. This paper is the first part in a series of papers. Here, we present tractability results for quasilinear problems under general assumptions on quasilinear operators and weights. In future papers, we shall verify these assumptions for quasilinear problems such as the solution of specific Dirichlet, Neumann, and Schrödinger problems

BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

Continuous complexity theory gets its name from the model of mathematical computation on which it is based. In the mathematics of the standard combinatorial computational complexity theory, modeling the inner workings of a computer takes center stage, and complexities of problems are measured in terms of bit operations using the Turing machine model, leading to hierarchies of complexity and to problems such as the famous P = NP conjecture. In the continuous complexity model, real number operations such as multiplication and function evaluation are primitives, and complexity analysis has a more analytic nature. The continuous (or real-number) model of computation is applicable for a number of reasons. One is the expectation that in current and future computations, fundamental analytic operations (such as addition or function evaluation) can be carried out in short enough times and within close enough time orders of each other that other higher order computational factors (of which there can be many) dominate complexity analyses. The real number model of computation is the basis of recent foundational work in computation theory [BSS], [BCSS], and of older work [T], [HS], [S