Tractability Of Multivariate Problems Research Articles

On the Power of Standard Information for Tractability for L<sub>2</sub>- Approximation in the Randomized Setting

We study the approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted L2 norm. We consider algorithms that use standard information Λstd consisting of function values or general linear information Λall consisting of arbitrary linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability in the randomized setting for Λstd and Λall for the normalized or absolute error criterion. For the normalized error criterion, we show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and some notions of algebraic tractability without any condition. For the absolute error criterion, we show that the power of Λstd is the same as that of Λall for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 98, 101, 102 and almost solve Open Problem 100 as posed by E.Novak and H.Wo´zniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Zürich, 2012.

On the power of standard information for [formula omitted]-approximation in the average case setting

This paper studies the equivalence of tractability for classes of Λstd of function evaluations and Λall of all continuous linear functionals for L2-approximation in the average-case setting. For the normalized error criterion, we show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and algebraic tractability, in the sense that there exist algorithms using function values at some deterministic points which enjoy the same tractability properties as those using optimally chosen linear functionals. Moreover, for the absolute error criterion, we also show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and algebraic tractability under certain assumptions on the trace of the covariance operators. In particular, we give a partial answer to three problems raised by E. Novak and H. Woźniakowski in the book “Tractability of multivariate problems” (Open problems 116-118, EMS Tracts in Mathematics, vol.18, Zürich, 2012), proving that the same results hold under weaker conditions. Finally, we apply our results to the tensor product L2-approximation.

ABC on IBC

This is a tutorial introducing basic concepts of Information-Based Complexity (IBC) with the emphasis on tractability of multivariate problems. IBC deals with the complexity of computing approximate solutions of continuous problems. No prior knowledge of IBC is required from the reader, however, some knowledge of basic concepts of functional analysis and measure theory will be very helpful.This text is based on tutorials delivered by the author during the workshop “IBC-Paris 2017: Information-Based Complexity, High-Dimensional Problems” in Paris, March 2017, and the mini-semester at the Erwin Schrödinger Institute “Information-Based Complexity and Tractability of Multivariate Problems” in Vienna, September–October 2017.

Quasi-polynomial tractability

Tractability of multivariate problems has become a popular research subject. Polynomial tractability means that the solution of a d -variate problem can be solved to within ε with polynomial cost in ε − 1 and d . Unfortunately, many multivariate problems are not polynomially tractable. This holds for all non-trivial unweighted linear tensor product problems. By an unweighted problem we mean the case when all variables and groups of variables play the same role. It seems natural to ask what is the “smallest” non-exponential function T : [ 1 , ∞ ) × [ 1 , ∞ ) → [ 1 , ∞ ) for which we have T -tractability of unweighted linear tensor product problems; that is, when the cost of a multivariate problem can be bounded by a multiple of a power of T ( ε − 1 , d ) . Under natural assumptions, it turns out that this function is T qpol ( x , y ) = exp ( ( 1 + ln x ) ( 1 + ln y ) ) for all x , y ∈ [ 1 , ∞ ) . The function T qpol goes to infinity faster than any polynomial although not “much” faster, and that is why we refer to T qpol -tractability as quasi-polynomial tractability. The main purpose of this paper is to promote quasi-polynomial tractability, especially for the study of unweighted multivariate problems. We do this for the worst case and randomized settings and for algorithms using arbitrary linear functionals or only function values. We prove relations between quasi-polynomial tractability in these two settings and for the two classes of algorithms.

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 S d ( · , · ) 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 include 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.