On the existence of positive solutions and their continuous dependence on functional parameters for some class of elliptic problems

We discuss the existence and the dependence on functional parameters of solutions of the Dirichlet problem for a kind of the generalization of the balance of a membrane equation. Since we shall propose an approach based on variational methods, we treat our equation as the Euler-Lagrange equation for a certain integral functional J. We will not impose either convexity or coercivity of the functional. We develop a duality theory which relates the infimum on a special set X of the energy functional associated with the problem, to the infimum of the dual functional on a corresponding set X^d. The links between minimizers of both functionals give a variational principle and, in consequence, their relation to our boundary value problem. We also present the numerical version of the variational principle. It enables the numerical characterization of approximate solutions and gives a measure of a duality gap between primal and dual functional for approximate solutions of our problem.     

Functional regression modeling via regularized Gaussian basis expansions

We consider the problem of constructing functional regression models for scalar responses and functional predictors, using Gaussian basis functions along with the technique of regularization. An advantage of our regularized Gaussian basis expansions to functional data analysis is that it creates a much more flexible instrument for transforming each individual’s observations into functional form. In constructing functional regression models there remains the problem of how to determine the number of basis functions and an appropriate value of a regularization parameter. We present model selection criteria for evaluating models estimated by the method of regularization in the context of functional regression models. The proposed functional regression models are applied to Canadian temperature data. Monte Carlo simulations are conducted to examine the efficiency of our modeling strategies. The simulation results show that the proposed procedure performs well especially in terms of flexibility and stable estimates.     

Coupled intervals in the discrete calculus of variations: necessity and sufficiency

In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: The nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [R. Hilscher, V. Zeidan, Comput. Math. Appl., to appear], where the right endpoint is fixed.     

InfoVis Is So Much More: A Comment on Gelman and Unwin and an Invitation to Consider the Opportunities

We propose new tools for visualizing large amounts of functional data in the form of smooth curves. The proposed tools include functional versions of the bagplot and boxplot, which make use of the first two robust principal component scores, Tukey’s data depth and highest density regions.

By-products of our graphical displays are outlier detection methods for functional data. We compare these new outlier detection methods with existing methods for detecting outliers in functional data, and show that our methods are better able to identify outliers.

An R-package containing computer code and datasets is available in the online supplements.     

An adaptive least‐squares mixed finite element method for the Signorini problem

We present and analyze a least squares formulation for contact problems in linear elasticity which employs both, displacements and stresses, as independent variables. As a consequence, we obtain stability and high accuracy of our discretization also in the incompressible limit. Moreover, our formulation gives rise to a reliable and efficient a posteriori error estimator. To incorporate the contact constraints, the first‐order system least squares functional is augmented by a contact boundary functional which implements the associated complementarity condition. The bilinear form related to the augmented functional is shown to be coercive and therefore constitutes an upper bound, up to a constant, for the error in displacements and stresses in . This implies the reliability of the functional to be used as an a posteriori error estimator in an adaptive framework. The efficiency of the use of the functional as an a posteriori error estimator is monitored by the local proportion of the boundary functional term with respect to the overall functional. Computational results using standard conforming linear finite elements for the displacement approximation combined with lowest‐order Raviart‐Thomas elements for the stress tensor show the effectiveness of our approach in an adaptive framework for two‐dimensional and three‐dimensional Hertzian contact problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 276–289, 2017     

Path integrals as analysis on path space by time slicing approximation

This is a survey of our papers [3, 4]. We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, translation, real linear transformation and functional differentiation. The invariance under translation and orthogonal transformation, the interchange of the order with Riemann-Stieltjes integrals and some limits, the semiclassical approximation, the integration by parts and the Taylor expansion formula with respect to functional differentiation, and the fundamental theorem of calculus hold in Feynman path integral. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Estimate of solutions for differential and difference functional equations with applications to difference methods

The theorems on the estimate of solutions for nonlinear second-order partial differential functional equations mainly of parabolic type with Dirichlet’s condition and for the suitable explicit finite difference functional schemes are proved. The proofs are based on the comparison technique. The convergent difference method given is considered without an assumption of the global generalized Perron condition on the functional variable but with local one in some sense only. It is a consequence of our estimate theorems. The functional dependence is of the Volterra type.     

Dissipativity of delay functional differential equations with bounded lag

Delay functional differential equations are essentially different from ordinary differential equations because their phase space is infinite dimensional. We first establish a sufficient condition for delay functional differential equations with bounded lag to be dissipative. Then we construct a one-leg θ-method to solve such dissipative equations and prove that it is dissipative if θ=1. One numerical example is given to confirm our theoretical result.     

Massively Parallel Nonparametric Regression,With an Application to Developmental Brain Mapping

A penalized approach is proposed for performing large numbers of parallel nonparametric analyses of either of two types: restricted likelihood ratio tests of a parametric regression model versus a general smooth alternative, and nonparametric regression. Compared with naïvely performing each analysis in turn, our techniques reduce computation time dramatically. Viewing the large collection of scatterplot smooths produced by our methods as functional data, we develop a clustering approach to summarize and visualize these results. Our approach is applicable to ultra-high-dimensional data, particularly data acquired by neuroimaging; we illustrate it with an analysis of developmental trajectories of functional connectivity at each of approximately 70,000 brain locations. Supplementary materials, including an appendix and an R package, are available online.