Erich Novak

High dimensional polynomial interpolation on sparse grids

We study polynomial interpolation on a d-dimensional cube, where d is large. We suggest to use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The polynomial exactness of this method is almost optimal. Our error bounds show that the method is universal, i.e., almost optimal for many different function spaces. We report on numerical experiments for d = 10 using up to 652 065 interpolation points.

Tractability of Multivariate Problems

Multivariate problems occur in many applications. These problems are defined on spaces of d-variate functions and d can be huge – in the hundreds or even in the thousands. Some high-dimensional problems can be solved efficiently to within ε, i.e., the cost increases polynomially in ε−1 and d. However, there are many multivariate problems for which even the minimal cost increases exponentially in d. This exponential dependence on d is called intractability or the curse of dimensionality. This is the first of a three-volume set comprising a comprehensive study of the tractability of multivariate problems. It is devoted to algorithms using linear information consisting of arbitrary linear functionals. The theory for multivariate problems is developed in various settings: worst case, average case, randomized and probabilistic. A problem is tractable if its minimal cost is not exponential in ε−1 and d. There are various notions of tractability, depending on how we measure the lack of exponential dependence. For example, a problem is polynomially tractable if its minimal cost is polynomial in ε−1 and d. The study of tractability was initiated about 15 years ago. This is the first research monograph on this subject. Many multivariate problems suffer from the curse of dimensionality when they are defined over classical (unweighted) spaces. But many practically important problems are solved today for huge d in a reasonable time. One of the most intriguing challenges of theory is to understand why this is possible. Multivariate problems may become tractable if they are defined over weighted spaces with properly decaying weights. In this case, all variables and groups of variables are moderated by weights. The main purpose of this book is to study weighted spaces and to obtain conditions on the weights that are necessary and sufficient to achieve various notions of tractability. The book is of interest for researchers working in computational mathematics, especially in approximation of highdimensional problems. It may be also suitable for graduate courses and seminars. The text concludes with a list of thirty open problems that can be good candidates for future tractability research.

Optimal approximation of elliptic problems by linear and nonlinear mappings I

We study the optimal approximation of the solution of an operator equation Au=f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs on an arbitrary bounded Lipschitz domain. Here we prove that approximation by linear mappings is as good as the best n-term approximation with respect to an optimal Riesz basis. Our results are concerned with approximation, not with computation. Our goal is to understand better the possibilities of nonlinear approximation.

Quantum Complexity of Integration

It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Holder classes Fk, ?d on 0, 1]d and define ? by ?=(k+?)/d. The known optimal orders for the complexity of deterministic and (general) randomized methods are comp(Fk, ?d, ?)???1/? and comprandom(Fk, ?d, ?)???2/(1+2?). For a quantum computer we prove compquantquery(Fk, ?d, ?)???1/(1+?) and compquant(Fk, ?d, ?)?C??1/(1+?)(log??1)1/(1+?). For restricted Monte Carlo (only coin tossing instead of general random numbers) we prove compcoin(Fk, ?d, ?)?C??2/(1+2?)(log??1)1/(1+2?). To summarize the results one can say that ?there is an exponential speed-up of quantum algorithms over deterministic (classical) algorithms, if ? is small; ?there is a (roughly) quadratic speed-up of quantum algorithms over randomized classical methods, if ? is small.

Intractability Results for Integration and Discrepancy

We mainly study multivariate (uniform or Gaussian) integration defined for integrand spaces Fd such as weighted Sobolev spaces of functions of d variables with smooth mixed derivatives. The weight ?j moderates the behavior of functions with respect to the jth variable. For ?j?1, we obtain the classical Sobolev spaces whereas for decreasing ?js the weighted Sobolev spaces consist of functions with diminishing dependence on the jth variables. We study the minimal errors of quadratures that use n function values for the unit ball of the space Fd. It is known that if the smoothness parameter of the Sobolev space is one, then the minimal error is the same as the discrepancy. Let n(?, Fd) be the smallest n for which we reduce the initial error, i.e., the error with n=0, by a factor ?. The main problem studied in this paper is to determine whether the problem is intractable, i.e., whether n(?, Fd) grows faster than polynomially in ??1 or d. In particular, we prove intractability of integration (and discrepancy) if limsupd?dj=1?j/lnd=∞. Previously, such results were known only for restricted classes of quadratures. For ?j?1, the, following bounds hold for discrepancy ?with boundary conditions1.0628d(1+o(1))?n(?, Fd)?1.5d??2,asd?∞,?without boundary conditions1.0463d(1+o(1))?n(?, Fd)?1.125d??2,asd?∞. These results are obtained by analyzing arbitrary linear tensor product functionals Id defined over weighted tensor product reproducing kernel Hilbert spaces Fd of functions of d variables. We introduce the notion of a decomposable kernel. For reproducing kernels that have a decomposable part we prove intractability of all Id with non-zero subproblems with respect to the decomposable part of the kernel, as long as the weights satisfy the condition mentioned above.

The Curse of Dimension and a Universal Method For Numerical Integration

Many high dimensional problems are difficult to solve for any numerical method. This curse of dimension means that the computational cost must increase exponentially with the dimension of the problem. A high dimension, however, can be compensated by a high degree of smoothness. We study numerical integration and prove that such a compensation is possible by a recently invented method. The method is shown to be universal, i.e., simultaneously optimal up to logarithmic factors, on two different smoothness scales. The first scale is defined by isotropic smoothness conditions, while the second scale involves anisotropic smoothness and is related to partially separable functions.

Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers

Abstract We study the approximation problem (or problem of optimal recovery in the

On the power of adaption

L_2

Simple Monte Carlo and the Metropolis algorithm

-norm) for weighted Korobov spaces with smoothness parameter

Global Optimization Using Hyperbolic Cross Points

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