Sum Of Univariate Functions Research Articles

Sharp upper and lower estimates for the approximation of bivariate functions by sums of univariate functions

Sharp upper and lower estimates for the approximation of bivariate functions by sums of univariate functions

On piecewise linear approximations of bilinear terms: structural comparison of univariate and bivariate mixed-integer programming formulations

Bilinear terms naturally appear in many optimization problems. Their inherent non-convexity typically makes them challenging to solve. One approach to tackle this difficulty is to use bivariate piecewise linear approximations for each variable product, which can be represented via mixed-integer linear programming (MIP) formulations. Alternatively, one can reformulate the variable products as a sum of univariate functions. Each univariate function can again be approximated by a piecewise linear function and modelled via an MIP formulation. In the literature, heterogeneous results are reported concerning which approach works better in practice, but little theoretical analysis is provided. We fill this gap by structurally comparing bivariate and univariate approximations with respect to two criteria. First, we compare the number of simplices sufficient for an varepsilon -approximation. We derive upper bounds for univariate approximations and compare them to a lower bound for bivariate approximations. We prove that for a small prescribed approximation error varepsilon , univariate varepsilon -approximations require fewer simplices than bivariate varepsilon -approximations. The second criterion is the tightness of the continuous relaxations (CR) of corresponding sharp MIP formulations. Here, we prove that the CR of a bivariate MIP formulation describes the convex hull of a variable product, the so-called McCormick relaxation. In contrast, we show by a volume argument that the CRs corresponding to univariate approximations are strictly looser. This allows us to explain many of the computational effects observed in the literature and to give theoretical evidence on when to use which kind of approximation.

Reliability‐based design optimization of a car body using dimension‐ reduced Chebyshev polynomial

AbstractA reliability‐based design optimization (RBDO) method of a car body is presented on basis of dimension‐reduced Chebyshev polynomial method (DCM). To improve calculation efficiency and save computational time, complex models are often approximated by metamodels in reliability analysis. Traditional metamodels require a large number of sample points, which is time‐consuming. To improve the efficiency, DCM is proposed to approximate the performance function of the car body. First, the performance function is decomposed by the dimension‐reduction method into a sum of univariate functions, which are then fitted through Chebyshev polynomials. The reliability of the car body is predicted by the Taylor expansion method and the fourth‐moment method. Finally, the result of RBDO is obtained using an improved adaptive genetic algorithm. The proposed method saves on the calculation time with high precision. Besides, the improved adaptive genetic algorithm reduces the number of iterations in the car body optimization and improves the efficiency.

A dimension-reduction based Chebyshev polynomial method for uncertainty analysis in composite corrugated sandwich structures

A prediction method for the ultimate load of composite corrugated sandwich structures (CCSS) was established based on the dimension-reduction method and Chebyshev polynomial. First, an ultimate load response function of the CCSS was reduced in dimension on basis of the univariate method. The original response function was converted to the sum of univariate functions of all random variables. Chebyshev polynomial was used to fit all univariate functions, and the univariate Chebyshev approximate model (UCAM) of the response function was obtained. Second, the fitting accuracy of the UCAM was analytically compared with the Kriging model, radial basis function, and response surface methodology. Finally, the effect of the order on the fitting accuracy of UCAM was studied. The UCAM has higher fitting accuracy and calculation efficiency compared with the other three approximate models. When the order of UCAM is lower than ninth-order, the larger the order, the higher the fitting accuracy. And the order and the fitting accuracy are no longer positively correlated when the order exceeds ninth-order. The order is recommended to choose an odd-order when using UCAM to solve the uncertainty problem.

Reaction–diffusion equations on graphs: stationary states and Lyapunov functions

Reaction–diffusion equations on graphs (networks) serve as mathematical models of various phenomena in physics and biology. We study the existence of spatially heterogeneous stationary states, provided that the diffusion coefficients are sufficiently small. We provide an easily applicable criterion for determining which of them are nonnegative. Next, we consider the problem of constructing Lyapunov functions for reaction–diffusion equations on graphs, provided that a Lyapunov function for the corresponding non-diffusive system is known. We provide an easy-to-use result applicable in situations where the non-diffusive Lyapunov function is a sum of univariate functions with nondecreasing derivatives. The results are illustrated by means of several examples from mathematical biology.

Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

In 1951, Diliberto and Straus [5] proposed a levelling algorithm for the uniform approximation of a bivariate function, defined on a rectangle with sides parallel to the coordinate axes, by sums of univariate functions. In the current paper, we consider the problem of approximation of a continuous function defined on a compact Hausdorff space by a sum of two closed algebras containing constants. Under reasonable assumptions, we show the convergence of the Diliberto–Straus algorithm. For the approximation by sums of univariate functions, it follows that Diliberto–Straus’s original result holds for a large class of compact convex sets.

Network Reconstruction Using Nonparametric Additive ODE Models

Network representations of biological systems are widespread and reconstructing unknown networks from data is a focal problem for computational biologists. For example, the series of biochemical reactions in a metabolic pathway can be represented as a network, with nodes corresponding to metabolites and edges linking reactants to products. In a different context, regulatory relationships among genes are commonly represented as directed networks with edges pointing from influential genes to their targets. Reconstructing such networks from data is a challenging problem receiving much attention in the literature. There is a particular need for approaches tailored to time-series data and not reliant on direct intervention experiments, as the former are often more readily available. In this paper, we introduce an approach to reconstructing directed networks based on dynamic systems models. Our approach generalizes commonly used ODE models based on linear or nonlinear dynamics by extending the functional class for the functions involved from parametric to nonparametric models. Concomitantly we limit the complexity by imposing an additive structure on the estimated slope functions. Thus the submodel associated with each node is a sum of univariate functions. These univariate component functions form the basis for a novel coupling metric that we define in order to quantify the strength of proposed relationships and hence rank potential edges. We show the utility of the method by reconstructing networks using simulated data from computational models for the glycolytic pathway of Lactocaccus Lactis and a gene network regulating the pluripotency of mouse embryonic stem cells. For purposes of comparison, we also assess reconstruction performance using gene networks from the DREAM challenges. We compare our method to those that similarly rely on dynamic systems models and use the results to attempt to disentangle the distinct roles of linearity, sparsity, and derivative estimation.

Canonical Sets of BestL1-Approximation

In mathematics, the termapproximationusually means either interpolation on a point set or approximation with respect to a given distance. There is a concept, which joins the two approaches together, and this is the concept of characterization of the best approximants via interpolation. It turns out that for some large classes of functions the best approximants with respect to a certain distance can be constructed by interpolation on a point set that does not depend on the choice of the function to be approximated. Such point sets are calledcanonical sets of best approximation. The present paper summarizes results on canonical sets of bestL1-approximation with emphasis on multivariate interpolation and bestL1-approximation by blending functions. The bestL1-approximants are characterized as transfinite interpolants on canonical sets. The notion of a Haar-Chebyshev system in the multivariate case is discussed also. In this context, it is shown that some multivariate interpolation spaces share properties of univariate Haar-Chebyshev systems. We study also the problem of best one-sided multivariateL1-approximation by sums of univariate functions. Explicit constructions of best one-sidedL1-approximants give rise to well-known and new inequalities.

On error formulas for approximation by sums of univariate functions

The purpose of the paper is to develop a new method for obtaining explicit formulas for the error of approximation of bivariate functions by sums of univariate functions. It should be remarked that formulas of this type have been known only for functions defined on a rectangle with sides parallel to coordinate axes. Our method, based on a maximization process over closed bolts, allows the consideration of functions defined on a hexagon or octagon with sides parallel to coordinate axes.

Best one-sided L1-approximation of bivariate functions by sums of univariate ones

Let \(f \in C^{1,1} ([-1,1]^2), \partial ^2f/\partial x \partial y \geqq 0\). We characterize the unique best one-sided L 1-approximant h * to f from above (resp. h * from below) with respect to the subspace B 1,1 which consists of all bivariate functions which are sums of univariate functions. h * resp. h * are constructed by a Hermite type interpolation on the diagonal \(\{ (t,t) : t \in I \} \) resp. the anti-diagonal \(\{ (t,-t) : t \in I \} \), where \(I := [-1,1]\).

On the approximation of a bivariate function by the sum of univariate functions

On the approximation of a bivariate function by the sum of univariate functions