Continuous Piecewise Linear Functions Research Articles

Exact analytical formulae for linearly distributed vortex and source sheets in uence computation in 2D vortex methods

We consider the methodology of numerical schemes development for two-dimensional vortex method. We describe two different approaches to deriving integral equation for unknown vortex sheet intensity. We simulate the velocity of the surface line of an airfoil as the influence of attached vortex and source sheets. We consider a polygonal approximation of the airfoil and assume intensity distributions of free and attached vortex sheets and attached source sheet to be approximated with piecewise constant or piecewise linear (continuous or discontinuous) functions. We describe several specific numerical schemes that provide different accuracy and have a different computational cost. The study shows that a Galerkin-type approach to solving boundary integral equation requires computing several integrals and double integrals over the panels. We obtain exact analytical formulae for all the necessary integrals, which makes it possible to raise significantly the accuracy of vortex sheet intensity computation and improve the quality of velocity and vorticity field representation, especially in proximity to the surface line of the airfoil. All the formulae are written down in the invariant form and depend only on the geometric relationship between the positions of the beginnings and ends of the panels.

Full groups of Cuntz–Krieger algebras and Higman–Thompson groups

In this paper, we will study representations of the continuous full group \Gamma_A of a one-sided topological Markov shift (X_A,\sigma_A) for an irreducible matrix A with entries in \{0,1\} as a generalization of Higman–Thompson groups V_N, 1 < N \in {\mathbb{N}} . We will show that the group \Gamma_A can be represented as a group \Gamma_A^{\operatorname{tab}} of matrices, called A -adic tables, with entries in admissible words of the shift space X_A , and a group \Gamma_A^{\operatorname{PL}} of right continuous piecewise linear functions, called A -adic PL functions, on [0,1] with finite singularities.

Optimal retrofit of heat exchanger networks: A stepwise approach

This paper introduces a new stepwise approach for the optimal retrofit of heat exchanger networks. At each iteration, the proposed method involves superstructures which embed retrofit alternatives as well as numerical optimization to minimize the project’s total annualized cost. In order to reduce calculation times, new algorithms are proposed to thin out superstructures. These are based on thermodynamic criteria, namely on the concept of heat path, and on new graph theoretical results. To carry out the numerical optimization tasks, novel mixed integer nonlinear models are developed. These models support any combination of constant, polynomial and (continuous) piecewise linear enthalpy functions of temperature. Numerical examples are presented for the retrofit of the preheat train of an atmospheric crude unit and for the retrofit of a fluid catalytic cracking plant. The new method is shown to reduce calculation times and total annualized costs reported in the literature.

Versatile and Accurate Schemes of Discretization in the Scattering Analysis of 2-D Composite Objects With Penetrable or Perfectly Conducting Regions

The method-of-moment discretization of boundary integral equations in the scattering analysis of closed infinitely long (2-D) objects, perfectly conducting (PEC) or penetrable, is traditionally carried out with continuous piecewise linear basis functions, which embrace pairs of adjacent segments. This is numerically advantageous because the discretization of the transversal component of the scattered fields, electric (TE) or magnetic (TM), becomes free from hypersingular Kernel contributions. In the analysis of composite objects, though, the imposition of the continuity requirement around junction nodes, where the boundaries of several regions intersect, becomes especially awkward. In this paper, we present, for the scattering analysis of composite objects, a new combined discretization of the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) integral equation, for homogeneous dielectric regions, and the electric-field integral equation, for PEC regions, such that the basis functions are defined strictly on each segment, with no continuity constraint between adjacent segments. We show the improved observed accuracy with the proposed TE-PMCHWT implementation on several dielectric objects with sharp edges and corners and moderate or high contrasts. Furthermore, we illustrate the versatility of these schemes in the analysis of 2-D composite piecewise homogeneous objects without sacrificing accuracy with respect to the conventional implementations.

Semiparametric methods for estimation of a nonlinear exposure-outcome relationship using instrumental variables with application to Mendelian randomization.

ABSTRACTMendelian randomization, the use of genetic variants as instrumental variables (IV), can test for and estimate the causal effect of an exposure on an outcome. Most IV methods assume that the function relating the exposure to the expected value of the outcome (the exposure‐outcome relationship) is linear. However, in practice, this assumption may not hold. Indeed, often the primary question of interest is to assess the shape of this relationship. We present two novel IV methods for investigating the shape of the exposure‐outcome relationship: a fractional polynomial method and a piecewise linear method. We divide the population into strata using the exposure distribution, and estimate a causal effect, referred to as a localized average causal effect (LACE), in each stratum of population. The fractional polynomial method performs metaregression on these LACE estimates. The piecewise linear method estimates a continuous piecewise linear function, the gradient of which is the LACE estimate in each stratum. Both methods were demonstrated in a simulation study to estimate the true exposure‐outcome relationship well, particularly when the relationship was a fractional polynomial (for the fractional polynomial method) or was piecewise linear (for the piecewise linear method). The methods were used to investigate the shape of relationship of body mass index with systolic blood pressure and diastolic blood pressure.

Objective variation simplex algorithm for continuous piecewise linear programming

This paper works on a modified simplex algorithm for the local optimization of Continuous PieceWise Linear (CPWL) programming with generalization of hinging hyperplane objective and linear constraints. CPWL programming is popular since it can be equivalently transformed into difference of convex functions programming or concave optimization. Inspired by the concavity of the concave CPWL functions, we propose an Objective Variation Simplex Algorithm (OVSA), which is able to find a local optimum in a reasonable time. Computational results are presented for further insights into the performance of the OVSA compared with two other algorithms on random test problems.

On convergent finite difference schemes for variational–PDE-based image processing

We study an adaptive anisotropic Huber functional based image restoration scheme. By using a combination of L2-L1 regularization functions, an adaptive Huber functional based energy minimization model provides denoising with edge preservation in noisy digital images. We study a convergent finite difference scheme based on continuous piecewise linear functions and use a variable splitting scheme, namely the Split Bregman, to obtain the discrete minimizer. Experimental results are given in image denoising and comparison with additive operator splitting, dual fixed point, and projected gradient schemes illustrate that the best convergence rates are obtained for our algorithm.

Boundary non-crossing probabilities for Slepian process

In this contribution we derive an explicit formula for the boundary non-crossing probability of Slepian processes associated with a continuous piecewise linear boundary function. Furthermore, we use this formula to develop an approximation result to the boundary non-crossing probability for general continuous boundaries.

Exploring the use of two-dimensional piecewise-linear functions as an alternative model for representing and processing grayscale-images

Traditionally, a grayscale image is represented as a rectangular array whose internal values describe a discrete level of intensity or luminancedenoted as pixel. Due to its structure and complete compatibility with matrix operators, this representation is the most widely used in imageprocessing. Although the strong robustness of this standard is not in question, it is always enriching to have an alternative description format inorder to provide not only a different image representation scheme but also an additional approach to image processing. Motivated by this fact, in thispaper the viability of using continuous piecewise linear functions of two spatial variables as an alternative model description of grayscale imagesis explored. Moreover, the possibility of applying this type of representation in image processing is also examined by using mapping variabletransformations, here denominated as functional filters. Furthermore, it is also shown that such alternative image model can also be used in morecomplex tasks like tridimensional volume estimation. To verify this proposal, illustrative examples are reported, showing not only an inherentimprovement in the visual perception of image, but also a practical image processing capability.

Second main theorem in the tropical projective space

Tropical Nevanlinna theory, introduced by Halburd and Southall as a tool to analyze integrability of ultra-discrete equations, studies the growth and complexity of continuous piecewise linear real functions. The purpose of this paper is to extend tropical Nevanlinna theory to n-dimensional tropical projective spaces by introducing a natural characteristic function for tropical holomorphic curves, and by proving a tropical analogue of Cartan's second main theorem. It is also shown that in the 1-dimensional case this result implies a known tropical second main theorem due to Laine and Tohge.

Towards the best constant in front of the Ditzian-Totik modulus of smoothness

We give accurate estimates for the constants $$ K\bigl(\mathcal{A}(I), n, x\bigr)=\sup_{f\in\mathcal{A}(I)}\frac{|L_{n} f(x)-f(x)|}{\omega_{\sigma}^{2} ( f; 1/\sqrt{n} )}, \quad x\in I, n=1,2,\ldots, $$ where $I=\mathbb {R}$ or $I=[0,\infty)$ , $L_{n}$ is a positive linear operator acting on real functions f defined on the interval I, $\mathcal{A}(I)$ is a certain subset of such function, and $\omega _{\sigma}^{2} (f;\cdot)$ is the Ditzian-Totik modulus of smoothness of f with weight function σ. This is done under the assumption that σ is concave and satisfies some simple boundary conditions at the endpoint of I, if any. Two illustrative examples closely connected are discussed, namely, Weierstrass and Szasz-Mirakyan operators. In the first case, which involves the usual second modulus, we obtain the exact constants when $\mathcal{A}(\mathbb {R})$ is the set of convex functions or a suitable set of continuous piecewise linear functions.

Approximation properties of Fourier sums for 2pi-periodic piecewise linear continuous functions

Approximation properties of Fourier sums for 2pi-periodic piecewise linear continuous functions

Extremal problems for hypotheses testing with set-valued decisions

We consider a class of extremal problems for multiple hypothesis testing with set-valued decisions and given total variation distances between hypotheses. The quality of a test is measured by an arbitrary piecewise linear continuous function of the error probabilities. We show that the extremal value of the test quality may be found as a solution of some linear programming problem, so the original infinite-dimensional problem is reduced to a certain finite-dimensional one.

Solving the value-at-risk minimisation model with linear programming techniques

This study considers portfolio selection problems in which risk is measured using value-at-risk (VaR). Because VaR is generally a non-convex and non-smooth function, conventional optimisation methods fail to solve portfolio selection models based on VaR. This has been considered an open problem since the 1990s. The study first proves that VaR is a continuous piecewise linear function of position when it is estimated with the scenario simulation method, and then, proposes a method for solving VaR minimisation models. The proposed method can yield a good local minimum or even a global optimum, although it has no theoretical guarantee. Because the proposed method uses only linear programming techniques, common linear programming solution software can be employed to solve VaR minimisation models of practical sizes in a short period of time. To illustrate the performance of the proposed method, we use it to solve a VaR minimisation model with 30 equities and 1,000 scenarios. We also compare the precision and computing time with another method. The results show that both methods can yield the optimal solution, but the proposed method is much faster.

Tropical variants of some complex analysis results

Tropical Nevanlinna theory studies value distribution of continuous piecewise linear functions of a real variable. In this paper, we use the reasoning from tropical Nevanlinna theory to present tropical counterparts of some classical complex results related to Fermat type equations, Hayman conjecture and Bruck conjecture.

A stabilized finite element method for the convection dominated diffusion optimal control problem

In this paper, a stabilized finite element method for optimal control problems governed by a convection dominated diffusion equation is investigated. The state and the adjoint variables are approximated by piecewise linear continuous functions with bubble functions. The control variable either is approximated by piecewise linear functions (called the standard method) or is not discretized directly (called the variational discretization method). The stabilization term only depends on bubble functions, and the projection operator can be replaced by the difference of two local Gauss integrations. A priori error estimates for both methods are given and numerical examples are presented to illustrate the theoretical results.

Application of a Compensation Inspection Method in Geodynamic Monitoring

The paper proves the application of a compensation testing method for geodynamic monitoring when using multi-pole electrical systems. The transfer functions of a geoelectric section are presented as a system of equations, whose coefficients are determined at the initial setup of the measuring system. The block diagram of the compensation method application for geodynamic monitoring based on a multi-pole electrical system is given. Approximation in terms of continuous piecewise-linear functions will be used to distinguish the geodynamic offset vector of the geoelectric section. A system of equations for defining the geodynamic offset vector through the approximation vector by continuous piecewise-linear functions on a recorded geoelectric signal error is considered.

Sharp eigenvalue enclosures for the perturbed angular Kerr–Newman Dirac operator

We examine a certified strategy for determining sharp intervals of enclosure for the eigenvalues of matrix differential operators with singular coefficients. The strategy relies on computing the second-order spectrum relative to subspaces of continuous piecewise linear functions. For smooth perturbations of the angular Kerr–Newman Dirac operator, explicit rates of convergence linked to regularity of the eigenfunctions are established. Numerical tests which validate and sharpen by several orders of magnitude the existing benchmarks are also included.

Łukasiewicz Games

Boolean games provide a simple, compact, and theoretically attractive abstract model for studying multiagent interactions in settings where players will act strategically in an attempt to achieve individual goals. A standard critique of Boolean games, however, is that the strictly dichotomous nature of the preference relations induced by Boolean goals inevitably trivialises the nature of such strategic interactions: a player is assumed to be indifferent between all outcomes that satisfy her goal, and indifferent between all outcomes that do not satisfy her goal. While various proposals have been made to overcome this limitation, many of these proposals require the inclusion of nonlogical structures into games to capture nondichotomous preferences. In this article, we introduce Łukasiewicz games, which overcome this limitation by allowing goals to be specified using Łukasiewicz logics . By expressing goals as formulae of Łukasiewicz logics, we can express a much richer class of utility functions for players than is possible using classical Boolean logic: we can express every continuous piecewise linear polynomial function with rational coefficients over [0, 1] n as well as their finite-valued restrictions over {0, 1/ k , …, ( k − 1)/ k , 1} n . We thus obtain a representation of nondichotomous preference structures within a purely logical framework. After introducing the formal framework of Łukasiewicz games, we present a number of detailed worked examples to illustrate the framework, and then investigate some of their theoretical properties. In particular, we present a logical characterisation of the existence of Nash equilibria in finite and infinite Łukasiewicz games. We conclude by briefly discussing issues of computational complexity.

Identification of Major Risk Sources for Surface Water Pollution by Risk Indexes (RI) in the Multi-Provincial Boundary Region of the Taihu Basin, China.

Environmental safety in multi-district boundary regions has been one of the focuses in China and is mentioned many times in the Environmental Protection Act of 2014. Five types were categorized concerning the risk sources for surface water pollution in the multi-provincial boundary region of the Taihu basin: production enterprises, waste disposal sites, chemical storage sites, agricultural non-point sources and waterway transportations. Considering the hazard of risk sources, the purification property of environmental medium and the vulnerability of risk receptors, 52 specific attributes on the risk levels of each type of risk source were screened out. Continuous piecewise linear function model, expert consultation method and fuzzy integral model were used to calculate the integrated risk indexes (RI) to characterize the risk levels of pollution sources. In the studied area, 2716 pollution sources were characterized by RI values. There were 56 high-risk sources screened out as major risk sources, accounting for about 2% of the total. The numbers of sources with high-moderate, moderate, moderate-low and low pollution risk were 376, 1059, 101 and 1124, respectively, accounting for 14%, 38%, 5% and 41% of the total. The procedure proposed could be included in the integrated risk management systems of the multi-district boundary region of the Taihu basin. It could help decision makers to identify major risk sources in the risk prevention and reduction of surface water pollution.