A new approach to the convolution operator on a finite interval

Recently, [Y.Q. Bai, M. El Ghami and C. Roos, SIAM J. Opt. 15 (2004) 101–128] investigated a new class of kernel functions which differs from the class of self-regular kernel functions. The class is defined by some simple conditions on the growth and the barrier behavior of the kernel function. In this paper we generalize the analysis presented in the above paper for P*(κ) Linear Complementarity Problems (LCPs). The analysis for LCPs deviates significantly from the analysis for linear optimization. Several new tools and techniques are derived in this paper.

A class of polynomial primal-dual interior-point algorithms for P * (κ ) linear complementarity problems (LCPs) are presented. We generalize Ghami et al.'s[A polynomial-time algorithm for linear optimization based on a new class of kernel functions(2008)] algorithm for linear optimization (LO) problem to P * (κ ) LCPs. Our analysis is based on a class of finite kernel functions which have the linear and quadratic growth terms. Since P *(κ ) LCP is a generalization of LO problem, we lose the orthogonality of the vectors dx and ds . So our analysis is different from the one in Ghami et al's algorithm. Despite this, the favorable complexity result is obtained, namely, $O((1 + 2\kappa)n^{\frac{1}{1+p}}\log n \log(n/\epsilon))$, which is better than the usual large-update primal-dual algorithm based on the classical logarithmic barrier function for P * (κ ) LCP.

Kernel functions serve the central goal of creating new search directions for the primal-dual interiorpoint algorithm to solve linear optimization problems. A significantly improved primal-dual interior-point algorithm for linear optimization is presented based on a novel kernel function. We show a primal-dual interior-point technique for linear optimization based on a class of kernel functions that are eligible. This research presents a new efficient kernel function-based primal-dual IPM algorithm for semidefinite programming problems based on the Nesterov-Todd (NT) direction. With a new and simple technique, we propose a new kernel function to obtain an optimal solution of the perturbed problem (SDP)µ. We obtain the best-known complexity results,for smalland large-update namely O(pp+1/2p √nlog tr(X0S0)/ε) and O((pn)p+1/2p log trX0S0/ε) large update To prove the effectiveness of our proposed kernel function, we compare our numerical results with some alternatives presented by Touil et al. (2017).

The problem of face identification is considered as that of featureless similarity-based pattern recognition. In terms of the support object approach, we use a kernel function of two gray-level vectors for measuring the pair-wise proximity of face images. In addition to the usual kind of kernel functions called here rigid because of their being formed by way of immediate comparison of two gray-level distributions on the original image plane, we examine a new class of kernel functions which we call elastic ones. The principal idea of elastic kernel functions consists in an elastic transformation of the pixel grid of one image relative to that of the other, so that the positions of all identical elements in both faces would become the same.

Finite interval convolution operators with periodic kernel-functions are studied from the point of view of Fredholm properties and invertibility. These operators are associated with Wiener-Hopf operators with matrix-valued symbols defined on a space of functions whose domain is a contour consisting of two parallel straight-lines. For the Fredholm study a Wiener-Hopf operator is considered on a space of functions defined on a contour composed of two closed curves having a common multiple point. Invertibility of the finite interval operator is studied for a subclass of symbols related to the problem of wave diffraction by a strip grating.

Primal–dual interior-point algorithms for second-order cone optimization based on kernel functions

In this paper we construct sequences of estimators for a density function and its derivatives, which are not assumed to be uniformly bounded, using classes of kernel functions. Utilizing these estimators, a sequence of empirical Bayes estimators is proposed. It is found that this sequence is asymptotically optimal in the sense of Robbins (Ann. Math. Statist. 35 (1964) 1-20). The rates of convergence of the Bayes risks associated with the proposed empirical Bayes estimators are obtained. It is noted that the exact rate is $n^{-q}$ with $q \leqq \frac{1}{3}$. An example is given and an explicit kernel function is indicated.

Forecasting the real prices of crude oil: A robust weighted least squares approach

Introduction Well tests are performed to acquire qualitative and quantitative knowledge of the well and the reservoir being, tested. Typically, a well test involves modification of the rate or the pressure at one or more wells in the reservoir and observation of the resultant reservoir response (a change in pressure or rate, respectively) at the perturbed well and/or adjacent wells. The reservoir response is then used to construct a well test interpretation model from which well and reservoir parameters, such as permeability and skin, can be calculated. The well test interpretation model describes the pressure or rate behavior of the actual well/reservoir system during the test and must be identified from the shape of the reservoir response. This information is different from, but complementary to, the information provided by other interpretation models that can be derived, for example, from log measurements or geologic observations. A well test interpretation model indicates primarily how many media with significantly different permeabilities and porosities are involved in the flow process and how these media interact. One possible well test interpretation model is the double-porosity model, which describes double-porosity behavior. Double-porosity behavior is obtained when two different media are involved in the flow process: a higher-permeability medium that produces fluid into the well and a lower-permeability medium that recharges the higher-permeability medium. Double-porosity behavior is typical of fissured reservoirs and multilayered reservoirs with high permeability contrast between layers. Different double-porosity behaviors are possible, depending on the degree of interaction, or interporosity flow, between the two constitutive media. The two extremes are (1) restricted. or "pseudosteady-state," interporosity flow, obtained when there is a significant impediment to flow, or interporosity skin, between the most-permeable and the least-permeable media, and (2) unrestricted, or "transient," interporosity flow, obtained when there is no interporosity skin. The following describes the various techniques available for identifying double-porosity behavior from well test pressure data. Conventional Analysis Conventional analysis involves plotting test pressure data vs. some function of time on a semilog plot (Figs. 1a and 1b). In theory, double-porosity behavior yields two parallel straight lines on a semilog plot, provided there are no near-wellbore or outer-boundary effects. Such a semilog plot is schematically represented in Fig. la. The first semilog straight line represents the homogeneous behavior of the most-permeable medium before the least-permeable medium starts recharging. As Fig. 1a indicates, this first straight line lasts longer for restricted interporosity flow than for unrestricted interporosity flow. The second semilog straight line represents the homogeneous behavior of both media when recharge from the least-permeable medium is fully established. The two parallel straight lines are separated by a transition zone that corresponds to the onset of interporosity flow. The transition can be a straight line in the case of unrestricted interporosity flow. The slope of such a transition straight line is equal to half that of the two parallel straight lines. In practice, however, the two parallel straight lines may or may not be present. This depends on the condition of the well, the composition of the reservoir fluid, and the duration of the test. As a result, the same well may yield different responses in different tests. Fig. 1b illustrates a case of double-porosity behavior where only the last semilog straight line exists. This straight line represents the homogeneous behavior of the total system and is not characteristic of double-porosity behavior. Thus a semilog plot is not an efficient tool for identifying double-porosity behavior. More generally, straight-line analysis techniques are not valid as diagnostic tools, because an apparent straight line through a range of data does not necessarily prove the existence of a specific flow regime. Log-Log Analysis Log-log analysis involves a log-log plot of pressure change vs. elapsed time (Fig. 2). Double-porosity behavior yields an S-shaped log-log pressure curve on a log-log plot, as illustrated in Fig. 2. The initial portion of the curve represents a homogeneous behavior resulting from depletion in only the most-permeable medium. This corresponds to the region labeled "homogeneous behavior (most-permeable system)" in Figs. la and lb. A transition follows, corresponding to interporosity flow, during which pressure in the two media tends to equilibrate. Finally, homogeneous behavior resumes again, as a result of depletion in both constitutive media at the same time. JPT P. 631^

We provide the maximum number of limit cycles for continuous and discontinuous planar piecewise differential systems formed by linear Hamiltonian saddles and separated either by one or two parallel straight lines. We show that when these piecewise differential systems are either continuous or discontinuous and are separated by one straight line, or are continuous and are separated by two parallel straight lines, they do not have limit cycles. On the other hand, when these systems are discontinuous and separated by two parallel straight lines, we prove that the maximum number of limit cycles that they can have is one and that this maximum is reached by providing an example of such a system with one limit cycle. When the line of discontinuity of the piecewise differential system is formed by one straight line, the symmetry of the problem allows to take this straight line without loss of generality as the line x=0. Similarly, when the line of discontinuity of the piecewise differential system is formed by two parallel straight lines due to the symmetry of the problem, we can assume without loss of generality that these two straight lines are x=±1.

This paper presents the modelling of naturally fractured reservoirs using a multi-porosity multi-permeability thermal simulator. An iterative solution technique that allows connection between any pair of blocks is used.Dual-porosity and triple-porosity systems are simulated. Results follow those of the analytical methods. Simulation of a thermal recovery scheme using a horizontal steam injector and a horizontal producer in a naturally fractured heavy oil reservoir is presented as an example.

A sufficient condition for the convergence of the mean shift algorithm with Gaussian kernel

In this paper we propose new primal-dual interior point methods (IPMs) for <TEX>$P_*(\kappa)$</TEX> linear complementarity problems (LCPs) and analyze the iteration complexity of the algorithm. New search directions and proximity measures are defined based on a class of kernel functions, <TEX>$\psi(t)=\frac{t^2-1}{2}-{\int}^t_1e{^{q(\frac{1}{\xi}-1)}d{\xi}$</TEX>, <TEX>$q\;{\geq}\;1$</TEX>. If a strictly feasible starting point is available and the parameter <TEX>$q\;=\;\log\;\(1+a{\sqrt{\frac{2{\tau}+2{\sqrt{2n{\tau}}+{\theta}n}}{1-{\theta}}\)$</TEX>, where <TEX>$a\;=\;1\;+\;\frac{1}{\sqrt{1+2{\kappa}}}$</TEX>, then new large-update primal-dual interior point algorithms have <TEX>$O((1\;+\;2{\kappa})\sqrt{n}log\;n\;log\;{\frac{n}{\varepsilon}})$</TEX> iteration complexity which is the best known result for this method. For small-update methods, we have <TEX>$O((1\;+\;2{\kappa})q{\sqrt{qn}}log\;{\frac{n}{\varepsilon}})$</TEX> iteration complexity.

The first systematic account of the theory of parallels to come down to us is contained in Euclid’s Elements [173] dating from about 300 D.C. Euclid worked in Alexandria under Ptolemy I and was the head of the Museion, the most eminent scientific center of antiquity, which was founded at that time. Euclid’s Elements is a revised version of a number of Greek works from the fifth and fourth centuries D.C., namely the Elements attributed to Hippocrates of Chios (books I-IV and XI), the arithmetic works of the Pythagoreans (books VII-IX), Eudoxus ‘theory of similarity and ratios’ (books V-VI) and his method of exhaustion (book XII), and Theaetetus’ works on quadratic irrationalities (book X) and on regular polyhedra (book XIII). The Elements opens with a list of 23 definitions, many of which bear traces of ancient traditions. The last of these definitions deals with parallel lines: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction [173, vol. 1, p. 154].

Euclid’s definition of two parallel straight lines seems to be the correct one for the absolute ideal of a plane and of a straight line. The conception of a line at infinity, it would seem, should be stated if used in some such words as these. Points may be supposed to exist at such a distance from the points generally considered in a plane that their distances from all such points in the plane may be regarded as equal, i.e . the differences of these distances may in general be regarded as negligible. These points, as a first approximation, may, under certain circumstances, be regarded, when considered in relationship to points at a finite distance, as lying on a straight line. Parallel straight lines may be regarded—for most practical purposes—as far as points at a finite distance are concerned, as intersecting at one of these points.