Finite-order Weights Research Articles

Tractability using periodized generalized Faure sequences

In Lemieux and Faure (2009), a construction closely related to generalized Faure sequences was introduced. Unlike generalized Faure sequences, this construction is extensible in the dimension and lends itself well to high-dimensional problems. In this paper, we consider a weighted space of functions for which this construction is particularly well suited and we use it to prove tractability in that space. This space of functions is a special case of one in which we have “finite-order weights”, which were studied recently by several authors. We compare our findings with these previously obtained results and provide numerical results to illustrate the practical potential of our approach.

Infinite-dimensional integration on weighted Hilbert spaces

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.

Liberating the dimension for function approximation

We study approximation of functions that may depend on infinitely many variables. We assume that such functions belong to a separable weighted Hilbert space that is the direct sum of tensor product Hilbert spaces of functions with finitely many variables. The weight sequence γ = { γ u } is used to moderate the influence of terms depending on finitely many variables from u . We consider algorithms that use finitely many arbitrary linear functionals. Each linear functional is an inner product whose cost depends on the number of active variables of the inner product generator. That is, if the generator has d active variables then the cost is $ ( d ) for a given non-decreasing function $. The error of an algorithm is defined in the worst case setting in a norm given by weighted L 2 norms for terms depending on finitely many variables. The ε -complexity is understood as the minimal cost among all algorithms with errors at most ε . We are especially interested in polynomial tractability, where the ε -complexity is bounded by a polynomial in ε − 1 , and weak tractability, where the ε -complexity is sub-exponential in ε − 1 . The results are as follows. • An algorithm whose cost is equal to the ε -complexity. It turns out the algorithm does not depend on the cost function $. • Necessary and sufficient conditions on polynomial tractability. It turns out that we may have polynomial tractability even when $ ( d ) is exponential in d . This holds since the minimal number of active variables that must be used to compute an ε -approximation may be surprisingly small, e.g., o ( ln ( ε − 1 ) ) or even smaller. • Necessary and sufficient conditions on weak tractability. It turns out that we have two quite different cases depending on whether the largest eigenvalue for the univariate case is simple or not. We may have weak tractability even when $ ( d ) is doubly exponential in d . • Specializing tractability conditions for product and finite-order weights.

Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

Abstract In this article we combine the favorable properties of efficient Gaussian type orbitals basis sets, which are applied with good success in conventional electronic structure methods, and tensor product multiscale bases, which provide guaranteed convergence rates and allow for adaptive resolution. To this end, we develop and study a new approach for the treatment of the electronic Schrödinger equation based on a modified adaptive sparse grid technique and a certain particle-wise decomposition with respect to one-particle functions obtained by a nonlinear rank-1 approximation. Here, we employ a multiscale Gaussian frame for the sparse grid spaces and we use Gaussian type orbitals to represent the rank-1 approximation. With this approach we are able to treat small atoms and molecules with up to six electrons.

Liberating the dimension

Many recent papers considered the problem of multivariate integration, and studied the tractability of the problem in the worst case setting as the dimensionality d increases. The typical question is: can we find an algorithm for which the error is bounded polynomially in d , or even independently of d ? And the general answer is: yes, if we have a suitably weighted function space. Since there are important problems with infinitely many variables, here we take one step further: we consider the integration problem with infinitely many variables–thus liberating the dimension–and we seek algorithms with small error and minimal cost. In particular, we assume that the cost for evaluating a function depends on the number of active variables. The choice of the cost function plays a crucial role in the infinite dimensional setting. We present a number of lower and upper estimates of the minimal cost for product and finite-order weights. In some cases, the bounds are sharp.

Tractability of the Helmholtz equation with non-homogeneous Neumann boundary conditions: The relation to the [formula omitted]-approximation

We want to compute a worst case ε -approximation to the solution of the Helmholtz equation − Δ u + q u = f over the unit d -cube I d , subject to Neumann boundary conditions ∂ ν u = g on ∂ I d . We measure error in the H 1 ( I d ) -norm. Let card ( ε , d ) denote the minimal number of evaluations of f , g , and q needed to compute an absolute or normalized ε -approximation, assuming that f , g , and q vary over balls of weighted reproducing kernel Hilbert spaces. This problem is said to be weakly tractable if card ( ε , d ) grows subexponentially in ε − 1 and d . It is said to be polynomially tractable if card ( ε , d ) is polynomial in ε − 1 and d , and strongly polynomially tractable if this polynomial is independent of d . We have previously studied tractability for the homogeneous version g = 0 of this problem. In this paper, we investigate the tractability of the non-homogeneous problem, with general g . Using new perturbation estimates having explicit constants, we are able to relate the tractability of this problem to that of the L 2 ( I d ) -approximation problem. First, suppose that we use product weights, in which the role of any variable is moderated by its particular weight. We then find that if the sum of the weights is sublinearly bounded, then the problem is weakly tractable; moreover, this condition is more or less necessary. We then show that the problem is polynomially tractable if the sum of the weights is logarithmically or uniformly bounded, and we estimate the exponents of tractability for these two cases. Next, we turn to finite-order weights of fixed order ω , in which a d -variate function can be decomposed as a sum, each term depending on at most ω variables. We show that the problem is always polynomially tractable for finite-order weights, and we give estimates for the exponents of tractability. Since our results so far have established nothing stronger than polynomial tractability, we look more closely at whether strong polynomial tractability is possible. We show that our problem is never strongly polynomially tractable for the absolute error criterion. Moreover, we believe that the same is true for the normalized error criterion, but we have been able to prove this lack of strong tractability only when certain conditions hold on the weights. Finally, we use the Korobov and min kernels, along with product weights, to illustrate our results.

Tractability of Multivariate Approximation over a Weighted Unanchored Sobolev Space

We study d-variate L2-approximation for a weighted unanchored Sobolev space having smoothness m≥1. This space is equipped with an unusual norm which is, however, equivalent to the norm of the d-fold tensor product of the standard Sobolev space. One might hope that the problem should become easier as its smoothness increases. This is true for our problem 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? It is easy to prove that no matter how the weights are chosen, the waiting period is at least md, even if the error demand e 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. In this paper, we prove the more difficult fact that our problem can be tractable if m=1. That is, we can find an e-approximation using polynomially-many (in d and e−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 e-approximation using polynomially-many (in e−1) information operations, independently 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.

Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights: Average Case Setting

Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights: Average Case Setting

A note on the complexity and tractability of the heat equation

We wish to solve the heat equation u t = Δ u - qu in I d × ( 0 , T ) , where I is the unit interval and T is a maximum time value, subject to homogeneous Dirichlet boundary conditions and to initial conditions u ( · , 0 ) = f over I d . We show that this problem is intractable if f belongs to standard Sobolev spaces, even if we have complete information about q. However, if f and q belong to a reproducing kernel Hilbert space with finite-order weights, we can show that the problem is tractable, and can actually be strongly tractable.

Finite-order Integration Weights Can be Dangerous

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

Good lattice rules based on the general weighted star discrepancy

We study the problem of constructing rank-1 lattice rules which have good bounds on the star Here the non-negative weights are general weights rather than the product weights considered in most earlier works. In order to show the existence of such good lattice rules, we use an averaging argument, and a similar argument is used later to prove that these lattice rules may be obtained using a component-by-component (CBC) construction of the generating vector. Under appropriate conditions on the weights, these lattice rules satisfy strong tractability bounds on the weighted star discrepancy. Particular classes of weights known as order-dependent and finite-order weights are then considered and we show that the cost of the construction can be very much reduced for these two classes of weights.

Tractability of quasilinear problems II: Second-order elliptic problems

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 Helmholtz equation -Au + qu = f in the d-dimensional unit cube, in which 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 w. This means that, although d can be arbitrarily 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 Helmholtz equation, subject to either Dirichlet or Neumann homogeneous boundary conditions. We study both the absolute and normalized error criteria. For all four possible combinations of boundary conditions and error criteria, we show that the problem is tractable. That is, the number of evaluations of f and q needed to obtain an e-approximation is polynomial in e -1 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 e -1 , 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 Helmholtz equation is strongly tractable for three of the four possible combinations of boundary conditions and error criteria, the only exception being the Dirichlet boundary condition under the normalized error criterion.

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.

Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights: Worst Case Setting

We consider approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces with finite-order weights. This means we consider functions of d variables that can be represented as sums of functions of at most q* variables. Here, q* is fixed (and presumably small) and d may be arbitrarily large. For the univariate problem, d = 1, we assume we know algorithms A1,ε that use O(ε−p) function or linear functional evaluations to achieve an error ε in the worst case setting. Based on these algorithms A1,ε, we provide a construction of polynomial-time algorithms Ad,ε for the general d-variate problem with the number of evaluations bounded roughly by ε−pdq* to achieve an error ε in the worst case setting.

Finite-order weights imply tractability of linear multivariate problems

We study the minimal number n ( ɛ , d ) of information evaluations needed to compute a worst case ɛ -approximation of a linear multivariate problem. This problem is defined over a weighted Hilbert space of functions f of d variables. One information evaluation of f is defined as the evaluation of a linear continuous functional or the value of f at a given point. Tractability means that n ( ɛ , d ) is bounded by a polynomial in both ɛ - 1 and d. Strong tractability means that n ( ɛ , d ) is bounded by a polynomial only in ɛ - 1 . We consider weighted reproducing kernel Hilbert spaces with finite-order weights. This means that each function of d variables is a sum of functions depending only on q * variables, where q * is independent of d. We prove that finite-order weights imply strong tractability or tractability of linear multivariate problems, depending on a certain condition on the reproducing kernel of the space. The proof is not constructive if one uses values of f.

Finite-order weights imply tractability of multivariate integration

Multivariate integration of high dimension s occurs in many applications. In many such applications, for example in finance, integrands can be well approximated by sums of functions of just a few variables. In this situation the superposition (or effective) dimension is small, and we can model the problem with finite-order weights, where the weights describe the relative importance of each distinct group of variables up to a given order (where the order is the number of variables in a group), and ignore all groups of variables of higher order. In this paper we consider multivariate integration for the anchored and unanchored (non-periodic) Sobolev spaces equipped with finite-order weights. Our main interest is tractability and strong tractability of QMC algorithms in the worst-case setting. That is, we want to find how the minimal number of function values needed to reduce the initial error by a factor ε depends on s and ε −1. If there is no dependence on s, and only polynomial dependence on ε −1, we have strong tractability, whereas with polynomial dependence on both s and ε −1 we have tractability. We show that for the anchored Sobolev space we have strong tractability for arbitrary finite-order weights, whereas for the unanchored Sobolev space we have tractability for all bounded finite-order weights. In both cases, the dependence on ε −1 is quadratic. We can improve the dependence on ε −1 at the expense of polynomial dependence on s. For finite-order weights, we may achieve almost linear dependence on ε −1 with a polynomial dependence on s whose degree is proportional to the order of the weights. We show that these tractability bounds can be achieved by shifted lattice rules with generators computed by the component-by-component (CBC) algorithm. The computed lattice rules depend on the weights. Similar bounds can also be achieved by well-known low discrepancy sequences such as Halton, Sobol and Niederreiter sequences which do not depend on the weights. We prove that these classical low discrepancy sequences lead to error bounds with almost linear dependence on n −1 and polynomial dependence on d. We present explicit worst-case error bounds for shifted lattice rules and for the Niederreiter sequence. Better tractability and error bounds are possible for finite-order weights, and even for general weights if they satisfy certain conditions. We present conditions on general weights that guarantee tractability and strong tractability of multivariate integration.