Finite-dimensional Counterpart Research Articles

Randomized complexity of parametric integration and the role of adaption II. Sobolev spaces

We study the complexity of randomized computation of integrals depending on a parameter, with integrands from Sobolev spaces. That is, for r,d1,d2∈N, 1≤p,q≤∞, D1=[0,1]d1, and D2=[0,1]d2 we are given f∈Wpr(D1×D2) and we seek to approximate(Sf)(s)=∫D2f(s,t)dt(s∈D1), with error measured in the Lq(D1)-norm. Information is standard, that is, function values of f. Our results extend previous work of Heinrich and Sindambiwe (1999) [10] for p=q=∞ and Wiegand (2006) [15] for 1≤p=q<∞. Wiegand's analysis was carried out under the assumption that Wpr(D1×D2) is continuously embedded in C(D1×D2) (embedding condition). We also study the case that the embedding condition does not hold. For this purpose a new ingredient is developed – a stochastic discretization technique. In Part I a basic problem of Information-Based Complexity on the power of adaption for linear problems in the randomized setting was solved. Here a further aspect of this problem is settled.

Counterexamples in isometric theory of symmetric and greedy bases

We continue the study initiated in Albiac and Wojtaszczyk (2006) of properties related to greedy bases in the case when the constants involved are sharp, i.e., in the case when they are equal to 1. Our main goal here is to provide an example of a Banach space with a basis that satisfies Property (A) but fails to be 1-suppression unconditional, thus settling Problem 4.4 from Albiac and Ansorena (2017). In particular, our construction demonstrates that bases with Property (A) need not be 1-greedy even with the additional assumption that they are unconditional and symmetric. We also exhibit a finite-dimensional counterpart of this example, and show that, at least in the finite-dimensional setting, Property (A) does not pass to the dual. As a by-product of our arguments, we prove that a symmetric basis is unconditional if and only if it is total, thus generalizing the well-known result that symmetric Schauder bases are unconditional.

Moderate deviations for systems of slow–fast stochastic reaction–diffusion equations

The goal of this paper is to study the moderate deviation principle for a system of stochastic reaction–diffusion equations with a time-scale separation in slow and fast components and small noise in the slow component. Based on weak convergence methods in infinite dimensions and related stochastic control arguments, we obtain an exact form for the moderate deviations rate function in different regimes as the small noise and time-scale separation parameters vanish. Many issues that appear due to the infinite dimensionality of the problem are completely absent in their finite-dimensional counterpart. In comparison to corresponding large deviation principles, the moderate deviation scaling necessitates a more delicate approach to establishing tightness and properly identifying the limiting behavior of the underlying controlled problem. The latter involves regularity properties of a solution of an associated elliptic Kolmogorov equation on Hilbert space along with a finite-dimensional approximation argument.

Statistical inference on the Hilbert sphere with application to random densities

The infinite-dimensional Hilbert sphere S∞ has been widely employed to model density functions and shapes, extending the finite-dimensional counterpart. We consider the Fréchet mean as an intrinsic summary of the central tendency of data lying on S∞. For sound statistical inference, we derive properties of the Fréchet mean on S∞ by establishing its existence and uniqueness as well as a root-n central limit theorem (CLT) for the sample version, overcoming obstructions from infinite-dimensionality and lack of compactness on S∞. Intrinsic CLTs for the estimated tangent vectors and covariance operator are also obtained. Asymptotic and bootstrap hypothesis tests for the Fréchet mean based on projection and norm are then proposed and are shown to be consistent. The proposed two-sample tests are applied to make inference for daily taxi demand patterns over Manhattan, modeled as densities, of which the square root densities are analyzed on the Hilbert sphere. Numerical properties of the proposed hypothesis tests which utilize the spherical geometry are studied in the real data application and simulations, where we demonstrate that the tests based on the intrinsic geometry compare favorably to those based on an extrinsic or flat geometry.

Continuum limit and preconditioned Langevin sampling of the path integral molecular dynamics

We investigate the continuum limit that the number of beads goes to infinity in the ring polymer representation of thermal averages. Studying the continuum limit of the trajectory sampling equation sheds light on possible preconditioning techniques for sampling ring polymer configurations with large number of beads. We propose two preconditioned Langevin sampling dynamics, which are shown to have improved stability and sampling accuracy. We present a careful mode analysis of the preconditioned dynamics and show their connections to the normal mode, the staging coordinate and the Matsubara mode representation for ring polymers. In the case where the potential is quadratic, we show that the continuum limit of the preconditioned mass modified Langevin dynamics converges to its equilibrium exponentially fast, which suggests that the finite dimensional counterpart has a dimension-independent convergence rate. In addition, the preconditioning techniques can be naturally applied to the multi-level quantum systems in the nonadiabatic regime, which are compatible with various numerical approaches.

Identification of critical node clusters for consensus in large-scale spatial networks

In this work, we address the problem of identifying a set of nodes that are critical to the rate of convergence of consensus dynamics in large-scale spatial networks. By assuming that nodes are uniformly distributed over a spatial region and that these can communicate with others in an infinitesimally small neighborhood, we start by formulating the consensus problem by means of a partial differential equation involving the Laplace operator, subject to Neumann boundary condition. As with its finite-dimensional counterpart, we observe how the performance of these dynamics is directly related to the second smallest eigenvalue of the Laplace operator over the domain of interest. We then reduce the critical node set identification problem to that of finding a ball of fixed radius, whose removal minimizes the rate of convergence over the residual domain. This leads us to consider two functional optimization problems. First, we treat the problem of determining the second smallest eigenvalue for a fixed domain by minimizing an energy functional. We characterize the critical points of the energy functional, and then construct the gradient dynamics that converge to the set of critical points. We then prove that the only locally asymptotically stable critical point is the second eigenfunction of the Laplace operator. Building on these results, we consider the critical ball identification problem, provide a characterization of the critical points, and define gradient dynamics to converge asymptotically to these points.

Hamel’s Formalism for Infinite-Dimensional Mechanical Systems

In this paper, we introduce Hamel’s formalism for infinite-dimensional mechanical systems and in particular consider its applications to the dynamics of nonholonomically constrained systems. This development is a nontrivial extension of its finite-dimensional counterpart. The analysis is applied to several continuum mechanical systems of interest, including coupled systems and systems with infinitely many constraints.

Mass concentration in a nonlocal model of clonal selection.

Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To better understand this impact, we propose a mathematical model describing the dynamics of a continuum of cell clones structured by the self-renewal potential. The model is an extension of the finite multi-compartment models of interactions between normal and cancer cells in acute leukemias. It takes a form of a system of integro-differential equations with a nonlinear and nonlocal coupling which describes regulatory feedback loops of cell proliferation and differentiation. We show that this coupling leads to mass concentration in points corresponding to the maxima of the self-renewal potential and the solutions of the model tend asymptotically to Dirac measures multiplied by positive constants. Furthermore, using a Lyapunov function constructed for the finite dimensional counterpart of the model, we prove that the total mass of the solution converges to a globally stable equilibrium. Additionally, we show stability of the model in the space of positive Radon measures equipped with the flat metric (bounded Lipschitz distance). Analytical results are illustrated by numerical simulations.

LOCALIZATION AND PROJECTIONS ON BI-PARAMETER BMO

We prove that for any operator $T$ on bi--parameter BMO the identity factors through $T$ or $I - T$. Bourgain's localization method provides the conceptual framework of our proof. It consists in replacing the factorization problem on the non--separable bi--parameter BMO by its localized, finite dimensional counterpart. We solve the resulting finite dimensional factorization problems by exploiting the geometry and combinatorics of colored dyadic rectangles.

Dissipativity of Pseudorational Behaviors

This paper studies dissipativity for a class of infinite-dimensional systems, called pseudorational, in the behavioral context. Extending the finite-dimensional counterpart, we show that a pseudorational behavior is dissipative if and only if it admits a storage function or a dissipation function. For its proof, we derive a new necessary and sufficient condition for entire functions of exponential type in the so called Paley-Wiener class to allow a symmetric factorization. Characterization of dissipative behaviors and linear quadratic optimal behaviors are also given.

Electromagnetic integral equation approach based on contraction operator and solution optimization in Krylov subspace

SUMMARY The paper presents a new code for modelling electromagnetic fields in complicated 3-D environments and provides examples of the code application. The code is based on an integral equation (IE) for the scattered electromagnetic field, presented in the form used by the Modified Iterative Dissipative Method (MIDM). This IE possesses contraction properties that allow it to be solved iteratively. As a result, for an arbitrary earth model and any source of the electromagnetic field, the sequence of approximations converges to the solution at any frequency. The system of linear equations that represents a finite-dimensional counterpart of the continuous IE is derived using a projection definition of the system matrix. According to this definition, the matrix is calculated by integrating the Green's function over the ‘source’ and ‘receiver’ cells of the numerical grid. Such a system preserves contraction properties of the continuous equation and can be solved using the same iterative technique. The condition number of the system matrix and, therefore, the convergence rate depends only on the physical properties of the model under consideration. In particular, these parameters remain independent of the numerical grid used for numerical simulation. Applied to the system of linear equations, the iterative perturbation approach generates a sequence of approximations, converging to the solution. The number of iterations is significantly reduced by finding the best possible approximant inside the Krylov subspace, which spans either all accumulated iterates or, if it is necessary to save the memory, only a limited number of the latest iterates. Optimization significantly reduces the number of iterates and weakens its dependence on the lateral contrast of the model. Unlike more traditional conjugate gradient approaches, the iterations are terminated when the approximate solution reaches the requested relative accuracy. The number of the required iterates, which for simple iterations, is approximately proportional to the square root of the lateral contrast of conductivity, in case of optimization, becomes approximately proportional to the logarithm of the lateral contrast. The effect of optimization increases with the complexity of the model under consideration. The new code is applied to a number of problems of subsurface, borehole and marine geophysics. Some of the results are compared with independent computations using different approaches.

Proximal Methods with Penalization Effects in Banach Spaces

In this paper we introduce a boundary coercive condition on the regularizing function of the proximal point method in Banach spaces, which has a penalization effect and is useful for solving certain variational inequality problems. As its finite dimensional counterpart (proximal point method with Bregman distances) it avoids explicit consideration of the constrains, because the feasibility is taken care of by the regularizing function, whose derivative diverges on the boundary of the feasible set. The algorithm is therefore an interior point one. We show that this effect is guaranteed even for an inexact version of the method, namely the hybrid extragradient proximal point method proposed by Solodov and Svaiter for finite dimensional spaces. We give a full convergence analysis, and examples of regularizing functions satisfying the required conditions, for the cases of the feasible set being a closed ball or a polyhedron.

Kergin interpolation in Banach spaces

We show that Kergin interpolation, a generalized Lagrange–Hermite polynomial interpolation, may be defined on mappings between general Banach spaces. Like its finite-dimensional counterpart, Kergin interpolation in this setting is an affine-invariant projector. We obtain an error formula which we use to approximate holomorphic mappings. As an application we give a convergence theorem applicable to, for instance, operators on Banach algebras, such as the algebra of square matrices with complex coefficients.

Characterization of spaces admitting a homotopy dense embedding into a Hilbert manifold

We characterize spaces admitting a homotopy dense embedding (= embedding with locally homotopy negligible complement) into a Hilbert manifold. A finite-dimensional counterpart of this characterization is also considered.

Phase operators via group contraction

The problem of quantization of the classical phase of a harmonic oscillator (HO) is solved here in two steps. First, polar decomposition of the step operators of the u(2) algebra is performed. Second, the method of group contraction is used through which, in the limit j→∞, â,â° is passed to for the quantized HO and its Hermitian phase operators. Also, phase states, i.e., states with sharply defined phase, are constructed and the dynamical aspects of the contraction limit between the Jaynes–Cummings model (JCM) and a finite-dimensional counterpart with increasing j parameter are studied. Finally, the old problem of the phase operators is discussed in the wider frame of rank-1 algebras and classify the previous works in this frame.