Analysis and Applications 

About: Analysis and Applications is an academic journal published by World Scientific. The journal publishes majorly in the area(s): Nonlinear system & Computer science. It has an ISSN identifier of 0219-5305. Over the lifetime, 530 publications have been published receiving 8640 citations.

Global dissipative solutions of the camassa–holm equation

Abstract: This paper is devoted to the continuation of solutions to the Camassa–Holm equation after wave breaking. By introducing a new set of independent and dependent variables, the evolution problem is rewritten as a semilinear hyperbolic system in an L∞ space, containing a non-local source term which is discontinuous but has bounded directional variation. For a given initial condition, the Cauchy problem has a unique solution obtained as fixed point of a contractive integral transformation. Returning to the original variables, we obtain a semigroup of global dissipative solutions, defined for every initial data $\bar u\in H^1 ({\mathbb R})$, and continuously depending on the initial data. The new variables resolve all singularities due to possible wave breaking and ensure that energy loss occurs only through wave breaking.

Analytic regularity and polynomial approximation of parametric and stochastic elliptic pde's

Abstract: Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PDE's in a bounded domain D with coefficients depending on possibly countably many parameters. It shows that the dependence of the solution on the parameters in the diffusion coefficient is analytically smooth. This analyticity is then exploited to prove that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated by multivariate polynomials (in the parameters) with coefficients taking values in the Hilbert space of weak solutions of the elliptic problem with a controlled number of terms N. The convergence rate in terms of N does not depend on the number of parameters in V which may be countable, therefore breaking the curse of dimensionality. The discretization of the coefficients from a family of continuous, piecewise linear finite element functions in D is shown to yield finite dimensional approximations whose convergence rate in terms of the overall number Ndof of degrees of freedom is the minimum of the convergence rates afforded by the best N-term sequence approximations in the parameter space and the rate of finite element approximations in D for a single instance of the parametric problem.

Deep vs. shallow networks: An approximation theory perspective

Abstract: The paper briefly reviews several recent results on hierarchical architectures for learning from examples, that may formally explain the conditions under which Deep Convolutional Neural Networks perform much better in function approximation problems than shallow, one-hidden layer architectures. The paper announces new results for a non-smooth activation function — the ReLU function — used in present-day neural networks, as well as for the Gaussian networks. We propose a new definition of relative dimension to encapsulate different notions of sparsity of a function class that can possibly be exploited by deep networks but not by shallow ones to drastically reduce the complexity required for approximation and learning.

Estimating the approximation error in learning theory

Abstract: Let B be a Banach space and (ℋ,‖·‖ℋ) be a dense, imbedded subspace. For a ∈ B, its distance to the ball of ℋ with radius R (denoted as I(a, R)) tends to zero when R tends to infinity. We are interested in the rate of this convergence. This approximation problem arose from the study of learning theory, where B is the L2 space and ℋ is a reproducing kernel Hilbert space. The class of elements having I(a, R) = O(R-r) with r > 0 is an interpolation space of the couple (B, ℋ). The rate of convergence can often be realized by linear operators. In particular, this is the case when ℋ is the range of a compact, symmetric, and strictly positive definite linear operator on a separable Hilbert space B. For the kernel approximation studied in Learning Theory, the rate depends on the regularity of the kernel function. This yields error estimates for the approximation by reproducing kernel Hilbert spaces. When the kernel is smooth, the convergence is slow and a logarithmic convergence rate is presented for analytic kern...

Vector valued reproducing kernel hilbert spaces of integrable functions and mercer theorem

Abstract: We characterize the reproducing kernel Hilbert spaces whose elements are p-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for p = 2, we show that the spectral decomposition of this integral operator gives a complete description of the reproducing kernel, extending the Mercer theorem.