This chapter introduces different notions of tractability and intractability. Most notably, the curse of dimensionality occurs when the solution of a multivariate problem would necessitate a number of observations growing exponentially with the number of variables involved. As illustrations, relying on reproducing-kernel techniques, it is shown on the one hand that the integration of trigonometric polynomials is an intractable problem, and on the other hand that integration in weighted Sobolev spaces is a tractable problem provided the weights decay fast enough.
This chapter introduces different notions of tractability and intractability. Most notably, the curse of dimensionality occurs when the solution of a multivariate problem would necessitate a number of observations growing exponentially with the number of variables involved. As illustrations, relying on reproducing-kernel techniques, it is shown on the one hand that the integration of trigonometric polynomials is an intractable problem, and on the other hand that integration in weighted Sobolev spaces is a tractable problem provided the weights decay fast enough.
We study d-variate approximation for a weighted unanchored Sobolev space having smoothness m ≥ 1. Folk wisdom would lead us to believe that this problem should become easier as its smoothness increases. This is true if we are only concerned with asymptotic analysis: the nth minimal error is of order n^-(m-δ) for any δ > 0. However, it is unclear how long we need to wait before this asymptotic behavior kicks in. How does this waiting period depend on d and m? We prove that no matter how the weights are chosen, the waiting period is at least m^d, even if the error demand ε is arbitrarily close to 1. Hence, for m ≥ 2, this waiting period is exponential in d, so that the problem suffers from the curse of dimensionality and is intractable. In other words, the fact that the asymptotic behavior improves with m is irrelevant when d is large. So, we will be unable to vanquish the curse of dimensionality unless m = 1 , i.e., unless the smoothness is minimal. We then show that our problem can be tractable if m = 1. That is, we can find an ε-approximation using polynomially-many (in d and ε^-1) information operations, even if only function values are permitted. When m = 1, it is even possible for the problem to be strongly tractable, i.e., we can find an ε-approximation using polynomially-many (in ε^-1) information operations, independent of d. These positive results hold when the weights of the Sobolev space decay sufficiently quickly or are bounded finite-order weights, i.e., the d-variate functions we wish to approximate can be decomposed as sums of functions depending on at most ω variables, where ω is independent of d.
In this dissertation we study the tractability of the information-based complexity n(ε,d) for d-variate function approximation problems. In the deterministic setting for many unweighted problems the curse of dimensionality holds, that means, for some fixed error tolerance ε>0 the complexity n(ε,d) grows exponentially in d. For integration problems one can usually break the curse with the standard Monte Carlo method. For function approximation problems, however, similar effects of randomization have been unknown so far. The thesis contains results on three more or less stand-alone topics. For an extended five page abstract, see the section "Introduction and Results". Chapter 2 is concerned with lower bounds for the Monte Carlo error for general linear problems via Bernstein numbers. This technique is applied to the L∞-approximation of certain classes of C∞-functions, where it turns out that randomization does not affect the tractability classification of the problem. Chapter 3 studies the L∞-approximation of functions from Hilbert spaces with methods that may use arbitrary linear functionals as information. For certain classes of periodic functions from unweighted periodic tensor product spaces, in particular Korobov spaces, we observe the curse of dimensionality in the deterministic setting, while with randomized methods we achieve polynomial tractability. Chapter 4 deals with the L1-approximation of monotone functions via function values. It is known that this problem suffers from the curse in the deterministic setting. An improved lower bound shows that the problem is still intractable in the randomized setting. However, Monte Carlo breaks the curse, in detail, for any fixed error tolerance ε>0 the complexity n(ε,d) grows exponentially in d only.
In this paper we study multivariate integration for a weighted Korobov space for which the Fourier coefficients of the functions decay exponentially fast. This implies that the functions of this space are infinitely times differentiable. Weights of the Korobov space monitor the influence of each variable and each group of variables. We show that there are numerical integration rules which achieve an exponential convergence of the worst-case integration error. We also investigate the dependence of the worst-case error on the number of variables s s , and show various tractability results under certain conditions on weights of the Korobov space. Tractability means that the dependence on s s is never exponential, and sometimes the dependence on s s is polynomial or there is no dependence on s s at all.
In a previous paper, we developed a general framework for establishing tractability and strong tractability for quasilinear multivariate problems in the worst case setting. One important example of such a problem is the solution of the heat equation ut = ∆u − qu in Id × (0,T), where I is the unit inter- val and T is a maximum time value. This problem is to be solved subject to homogeneous Dirichlet boundary conditions, along with the initial conditions u(·,0) = f over Id. The solution u depends linearly on f, but nonlinearly on q. Here, both f and q are d-variate functions from a reproducing kernel Hilbert space with finite-order weights of order ω. This means that, although d can be arbitrary large, f and q can be decomposed as sums of functions of at most ω variables, with ω independent of d. In this paper, we apply our previous general results to the heat equation. We study both the absolute and normalized error criteria. For either error criterion, we show that the problem is . That is, the number of evaluations of f and q needed to obtain an ε-approximation is polynomial in ε and d, with the degree of the polynomial depending linearly on ω. In addition, we want to know when the problem is strongly tractable, meaning that the dependence is polynomial only in ε, independently of d. We show that if the sum of the weights defining the weighted reproducing kernel Hilbert space is uniformly bounded in d and the integral of the univariate kernel is positive, then the heat equation is strongly tractable
In the theory of tractability of multivariate problems one usually studies problems with finite smoothness. Then we want to know which s-variate problems can be approximated to within ε by using, say, polynomially many in s and ε−1 function values or arbitrary linear functionals. There is a recent stream of work for multivariate analytic problems for which we want to answer the usual tractability questions with ε−1 replaced by 1+logε−1. In this vein of research, multivariate integration and approximation have been studied over Korobov spaces with exponentially fast decaying Fourier coefficients. This is work of J. Dick, G. Larcher, and the authors. There is a natural need to analyze more general analytic problems defined over more general spaces and obtain tractability results in terms of s and 1+logε−1. The goal of this paper is to survey the existing results, present some new results, and propose further questions for the study of tractability of multivariate analytic questions.
We study the numerical integration problem for functions with infinitely many variables which stem from a weighted reproducing kernel Hilbert space. We study the worst case ε \varepsilon -complexity which is defined as the minimal cost among all algorithms whose worst case error over the Hilbert space unit ball is at most ε \varepsilon . Here we assume that the cost of evaluating a function depends polynomially on the number of active variables. The infinite-dimensional integration problem is tractable if the ε \varepsilon -complexity is bounded by a constant times a power of 1 / ε 1/\varepsilon . The smallest such power is called the exponent of tractability. We provide improved lower bounds for the exponent of tractability for general finite-order weights and, with the help of multilevel algorithms, improved upper bounds for three newly defined classes of finite-order weights. The newly defined finite-intersection weights model the situation where each group of variables interacts with at most ρ \rho other groups of variables, ρ \rho some fixed number. For these weights we obtain sharp upper bounds for any decay of the weights and any polynomial degree of the cost function. For the other two classes of finite-order weights our upper bounds are sharp if, e.g., the decay of the weights is fast or slow enough. Furthermore, we deduce a lower bound for the exponent of tractability for arbitrary weights and a constructive upper bound for product weights.