Weighted Hilbert Space Research Articles

Correction to "Quasi-Monte Carlo methods for high-dimensional integration: The standard (weighted Hilbert space) setting and beyond"

Correction to "Quasi-Monte Carlo methods for high-dimensional integration: The standard (weighted Hilbert space) setting and beyond"

Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

This paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube \([0,1]^s\). It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost \(1/N\), where \(N\) is the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets. doi:10.1017/S1446181112000077

Compact composition operators on weighted Hilbert spaces of analytic functions

We characterize the compactness of composition operators acting on a large family of Hilbert spaces of analytic functions which lie between Bergman and Dirichlet spaces. Our characterization is given in terms of generalized Nevanlinna counting functions.

Separation of the Grushin differential operator in weighted Hilbert spaces

In this paper we investigate the separation property of the Grushin differential operator of the form $$Gu = - \frac{1} {2}\left( {\frac{{\partial ^2 u}} {{\partial x^2 }} + \frac{{x^4 }} {4}\frac{{\partial ^2 u}} {{\partial y^2 }}} \right) + Q(x,y)u(x,y), \forall (x,y) \in \Omega \subset R^2 $$ in the Hilbert space L2(Ω, H), as well as in the weighted Hilbert space L2,k(Ω, H), with the operator potential Q(x, y) ∈ C1(Ω, L(H)), where L(H) is the space of all bounded linear operators on the arbitrary Hilbert space H and Ω be an open subset of R2.

QUASI-MONTE CARLO METHODS FOR HIGH-DIMENSIONAL INTEGRATION: THE STANDARD (WEIGHTED HILBERT SPACE) SETTING AND BEYOND

AbstractThis paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N, whereNis the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.

Density Approximations for Multivariate Affine Jump-Diffusion Processes

We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert spaces for random variables which possess all polynomial moments. We establish parametric conditions which guarantee existence and differentiability of transition densities of affine models and show how they naturally fit into the approximation framework. Empirical applications in option pricing, credit risk, and likelihood inference highlight the usefulness of our expansions. The approximations are extremely fast to evaluate, and they perform very accurately and numerically stable.

The Fejer-Riesz type result for some weighted Hilbert spaces of analytic functions in the unit disc

In this note we prove Fejer-Riesz inequality type results for some weighted Hilbert spaces of analytic functions in the unit disc. We describe also a class of such spaces for which Fejer-Riesz inequality type results do not hold.

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.

Systems with Symmetry Breaking and Restoration

Statistical systems, in which spontaneous symmetry breaking can be accompanied by spontaneous local symmetry restoration, are considered. A general approach to describing such systems is formulated, based on the notion of weighted Hilbert spaces and configuration averaging. The approach is illustrated by the example of a ferroelectric with mesoscopic fluctuations of paraelectric phase. The influence of the local symmetry restoration on the system characteristics, such as sound velocity and Debye-Waller factor, is discussed.

Decreasing and fast solutions for a second-order difference equation related to Fisher–Kolmogorov's equation

We prove the existence of so-called fast solutions of a second-order difference equation related to traveling wave solutions of Fisher–Kolmogorov's equation. Variational approach is used and the fast solutions are obtained as minimizers of an energy functional on a weighted Hilbert space. Numerical experiments are presented.

Constructions of general polynomial lattices for multivariate integration

We study a construction algorithm for certain polynomial lattice rules modulo arbitrary polynomials. The underlying polynomial lattices are special types of digital nets as introduced by Niederreiter. Dick, Kuo, Pillichshammer and Sloan recently introduced construction algorithms for polynomial lattice rules modulo irreducible polynomials which yield a small worst-case error for integration of functions in certain weighted Hilbert spaces. Here, we generalize these results to the case where the polynomial lattice rules are constructed moduloarbitrarypolynomials.

Asymptotic behavior of stochastic discrete complex Ginzburg–Landau equations

The complex Ginzburg–Landau equation is one of the most-studied equations in applied mathematics. We consider the discretization of complex Ginzburg–Landau equations on one dimensional lattice driven by a general Gaussian random field including the translation invariant one. The long time behavior of the sample paths and the distributions of solutions are studied respectively. Under the gauge nonlinear interaction, the dynamical behavior for the sample paths of the system is described by a global random attractor which is a random compact invariant set in a weighted Hilbert space. Furthermore the distributions of the system exponentially converge to the unique invariant measure of the system, that is the system is ergodic. The asymptotic compactness and dissipative method are important in our approach.

GKN theory for linear Hamiltonian systems

In this paper, we give the definition of maximal and minimal operators for linear Hamiltonian systems and investigate the relationship between the conjugate scalar product in a weighted Hilbert space and the skew-symmetric boundary form of the associated singular Hamiltonian operator, namely, the one-to-one correspondence between the set of self-adjoint extensions of the minimal operator and the set of Lagrangian symplectic subspaces. These results extend and improve the classical Glazman–Krein–Naimark (GKN) theory for quasi-differential operators.

The weighted b-adic diaphony

The author introduces a new numerical measure for uniform distribution of sequences in [ 0 , 1 ) s , called weighted b -adic diaphony. It is proved that the computing complexity of the weighted b -adic diaphony of an arbitrary net, composed of N points in [ 0 , 1 ) s , is O ( sN 2 ) . As special cases of the weighted b -adic diaphony we obtain some well-known kinds of the diaphony. An analogy of the inequality of Erdös–Turan–Koksma is given. We introduce the notion of limiting weighted b -adic diaphony, based on the Walsh functional system over finite groups as a characteristic of the behaviour of point nets in [ 0 , 1 ) ∞ . A general lower bound of the limiting weighted b -adic diaphony of an arbitrary net of N points in [ 0 , 1 ) ∞ is proved. We introduce a class of weighted Hilbert space and prove a connection between the worst-case error of the integration of this space and the weighted b -adic diaphony.

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights. We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules. We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

Strong tractability of integration using scrambled Niederreiter points

We study the randomized worst-case error and the randomized error of scrambled quasi-Monte Carlo (QMC) quadrature as proposed by Owen. The function spaces considered in this article are the weighted Hilbert spaces generated by Haar-like wavelets and the weighted Sobolev-Hilbert spaces. Conditions are found under which multivariate integration is strongly tractable in the randomized worst-case setting and the randomized setting, respectively. The e-exponents of strong tractability are found for the scrambled Niederreiter nets and sequences. The sufficient conditions for strong tractability for Sobolev spaces are more lenient for scrambled QMC quadratures than those for deterministic QMC net quadratures.

Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces

We introduce a weighted reproducing kernel Hilbert space which is based on Walsh functions. The worst-case error for integration in this space is studied, especially with regard to ( t , m , s ) -nets. It is found that there exists a digital ( t , m , s ) -net, which achieves a strong tractability worst-case error bound under certain condition on the weights. We also investigate the worst-case error of integration in weighted Sobolev spaces. As the main tool we define a digital shift invariant kernel associated to the kernel of the weighted Sobolev space. This allows us to study the mean square worst-case error of randomly digitally shifted digital ( t , m , s ) -nets. As this digital shift invariant kernel is almost the same as the kernel for the Hilbert space based on Walsh functions, we can derive results for the weighted Sobolev space based on the analysis of the Walsh function space. We show that there exists a ( t , m , s ) -net which achieves the best possible convergence order for integration in weighted Sobolev spaces and are strongly tractable under the same condition on the weights as for lattice rules.

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.

Separation of the general second order elliptic differential operator with an operator potential in the weighted Hilbert spaces

The purpose of this paper is to study the separation for the general second order elliptic differential operator G=G 0+V(x),x∈R n, in the weighted Hilbert space H= L 2, k ( R n , H 1), where G 0=−∑ i, j=1 n a ij ( x) D i j is the differential operator with the real positive coefficients a ij ( x)∈ C 2( R n ) and D i j= ∂ 2 ∂x i ∂x j ,i,j=1,…,n . The operator potential V( x)∈ C 1( R n , L( H 1)), where L( H 1) is the space of all bounded linear operators on the arbitrary Hilbert space H 1. Moreover, we study the existence and uniqueness of the solution of the second order differential equation −∑ i, j=1 n a ij ( x) D i j u( x)+ V( x) u( x)= f( x), where f( x)∈ H, in the weighted Hilbert space H= L 2, k ( R n , H 1).

Large Bergman spaces: invertibility, cyclicity, and subspaces of arbitrary index

In a wide class of weighted Bergman spaces, we construct invertible non-cyclic elements. These are then used to produce z-invariant subspaces of index higher than one. In addition, these elements generate non-trivial bilaterally invariant subspaces in anti-symmetrically weighted Hilbert spaces of sequences.