Monotone Approximation Research Articles

Least squares monotonic unimodal approximations to successively updated data and an application to a Covid-19 outbreak

ABSTRACT Approximations to noisy data are constructed to obtain unimodality of successively updated data. Precisely, a method is developed that calculates least squares monotonic increasing - decreasing approximations to n data, for successive values of n. Having fixed n, the statement of the increasing-–decreasing constraints in terms of first differences of the data subject to one sign change gives rise to a discrete search calculation, because the position of the sign change is also an unknown of the optimization problem. Although this problem may have many local minima for each value of n, the method is made quite efficient by taking advantage of two properties of the optimization calculation. The first property, for fixed n, provides necessary and sufficient conditions that reduce the optimal selection of the position to solving two sequences of monotonic approximation problems to subranges of data in the least complexity O ( n ) . The sufficiency conditions is a new result. The second property shows that the optimal position increases as n increases. Hence, if this position of a fit to the first n−1 data is available, then in order to calculate the required position of a fit to the first n data there is no need to consider data that are before the current position. A Fortran program has been written and some timings of numerical results are given for up to n = 30,000 data and various levels of noise. They seem to be proportional to n. The method is applied to data of daily Covid-19 deaths of the United Kingdom for a period that exhibits a major outbreak, and obtains insights of the evolution of the process as new data enter the calculation. The results reveal certain features of the calculation that may be helpful to supporting policy making, when used in modelling epidemic processes. Further, a procedure is proposed where our method initializes the nonlinear least squares calculation of Richards epidemiological model, and the numerical results confirm some useful advantages over the plain use of this model. One benefit of our method for unimodal smoothing of successive data and estimation of the associated turning points is that it provides a property that appears in a wide range of processes from biology, economics, finance and social sciences, for instance.

Precise error bounds for numerical approximations of fractional HJB equations

Abstract We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $\sigma \in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $\mathcal{O}\big (h^{\frac{1}{2}}\big )$.

A monotone scheme for G-equations with application to the explicit convergence rate of robust central limit theorem

We propose a monotone approximation scheme for a class of fully nonlinear PDEs called G-equations. Such equations arise often in the characterization of G-distributed random variables in a sublinear expectation space. The proposed scheme is constructed recursively based on a piecewise constant approximation of the viscosity solution to the G-equation. We establish the convergence of the scheme and determine the convergence rate with an explicit error bound, using the comparison principles for both the scheme and the equation together with a mollification procedure. The first application is obtaining the convergence rate of Peng's robust central limit theorem with an explicit bound of Berry-Esseen type. The second application is an approximation scheme with its convergence rate for the Black-Scholes-Barenblatt equation.

An inexact proximal linearized DC algorithm with provably terminating inner loop

Standard approaches to difference-of-convex (DC) programs require exact solution to a convex subproblem at each iteration, which generally requires noiseless computation and infinite iterations of an inner iterative algorithm. To tackle these difficulties, inexact DC algorithms have been proposed, mostly by relaxing the convex subproblem to an approximate monotone inclusion problem. However, there is no guarantee that such relaxation can lead to a finitely terminating inner loop. In this paper, we point out the termination issue of existing inexact DC algorithms by presenting concrete counterexamples. Exploiting the notion of ϵ-subdifferential, we propose a novel inexact proximal linearized DC algorithm termed tPLDCA. Despite permission to a great extent of inexactness in computation, tPLDCA enjoys the same convergence guarantees as exact DC algorithms. Most noticeably, the inner loop of tPLDCA is guaranteed to terminate in finite iterations as long as the inner iterative algorithm converges to a solution of the proximal point subproblem, which makes an essential difference from prior arts. In addition, by assuming the first convex component of the DC function to be pointwise maximum of finitely many convex smooth functions, we propose a computational friendly surrogate of ϵ-subdifferential, whereby we develop a feasible implementation of tPLDCA. Numerical results demonstrate the effectiveness of the proposed implementation of tPLDCA.

Some negative results for a monotonic approximation of a function that they have a fractional derivative of a complex order

Some negative results for a monotonic approximation of a function that they have a fractional derivative of a complex order

The Relation Between the Degree of Monotone Approximation and the Degree of Unconstrained Approximation

Many researchers related to the degree of unconstrained approximation to constrained approximation, and proved the inequality: For a continuous function on a closed interval we have where is a positive constant. The converse of the relation is not achieved, so we will obtain the converse of the relation under some conditions on which belonging to quasi normed spaces.

On monotone approximation of piecewise continuous monotone functions with the help of translations and dilations of the Laplace integral

For piecewise continuous monotone functions defined on a bounded interval $[-b;b]$, a monotone smooth approximation $Q(x)$ of any prescribed accuracy in the metric of the space $\mathbf{C}(\Pi)$ with as small as desired measure of the difference $[-b;b]\setminus\Pi$, $\Pi\subset[-b;b]$, is constructed using translations and dilations of the Laplace function (integral). In fact, this extends to the case of piecewise continuous monotone functions the result (obtained by the author formerly) on arbitrarily exact in the metric of the space $\mathbf{C}[-b;b]$ monotone approximation of continuous monotone functions with the help of translations and dilations of the Laplace integral. Besides, we suggest a new way of approximation in the form of linear combination of translations and dilations of the Laplace integral. Finally, we give and discuss concrete numerical examples of using approximation ways under study for a piecewise constant (stepwise) monotone function and for a piecewise continuous monotone function. Here, we also compare the results obtained for two discussed ways of approximation.

Узагальнення негативних результатів для інтерполяційного опуклого наближення функцій, що мають дробову похідну в просторі Соболєва

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function of Sobolev`s space by polynomials. The prob-lem of positive approximation is to estimate the pointwise degree of approxi-mation of a function of r times continuously differentiable and positive func-tions on [0, 1]. Estimates of the form (1) for positive approximation are known ([1, 2]). The problem of monotone approximation is that of estimating the de-gree of approximation of a monotone nondecreasing function bymonotone nondecreasing polynomials. Estimates of the form (1) for monotone approxi-mation were proved in [3, 4, 7]. In [3, 4] is consider r is natural and r not equal one. In [7] is consider r is real and rmore two. It was proved that for monotone approximation estimates of the form (1) are fails for r is real and r more two. The problem of convex approximation is that of estimating the degree of ap-proximation of a convex function by convex polynomials. The problem of con-vex approximation is consider in [8-10]. In [8] is consider r is natural and r not equal one. In [9] is consider r is real and r more two. It was proved that for con-vex approximation estimates of the form (1) are fails for r is real and r more two. In [10] the question of approximation of function of Sobolev`s space and convex by algebraic convex polynomial is consider, if the index of the Sobolev space is in the interval from three to four. It is proved that the estimate that gen-eralizes (1) is false This paper investigates the issue of approximation of con-vex functions from the Sobolev space by convex algebraic polynomials for a real index of the Sobolev space from the interval from two to three.Similarly to the paper [10], a counterexample is built, which shows that the estimate that generalizes the estimate (1) is false.This paper isthe generalization of results papers [9] and [11]. The main result is the analog of the theorem 2.3 in [11].

Методи розв’язування початкової задачі для нелінійних інтегро-диференціальних рівнянь з оцінкою локальної похибки

One of the modern scientific methods of researching phenomena and pro-cesses is mathematical modeling, which in many cases allows replacing the real process and makes it possible to obtain both a qualitative and a quantitative pic-ture of the process. Since the exact solutions of such models can be found in very individual cases, it is necessary to use approximate methods. In applied mathematics, fractional-rational approximations, which under appropriate con-ditions give a high rate of convergence of algorithms, bilateral and monotonic approximations have become widely used.In this work, using the technique of constructing one-step methods for solv-ing the initial problem for ordinary differential equations and developing the sought solution into a finite continued fraction, a numerical method for solving the Cauchy problem for nonlinear integro-differential equations of the Volterra type is proposed. The values of the parameters at which the nonlinear method of the first and second order of accuracy is obtained are found.Computational formulas are proposed, which at each integration step allow obtaining an upper and lower approximation to the exact solution without additional references to the right-hand side of the integro-differential equation. Calculation formulas, in which the main terms of the local error differ only in sign, form a two-sided method. We take the half-sum of bilateral approximations to the exact solution as the approximate solution at the given integration point, and the absolute value of the half-difference determines the error of the obtained result.The modular nature of the proposed algorithms makes it possible to ob-tain several approximations to the exact solution of the initial problem for the nonlinear integro-differential equation at each point of integration. The comparison of these approximations gives useful information in the matter of choosing the integration step or in assessing the accuracy of the result

The $L^{\infty }$-positivity Preserving Property and Stochastic Completeness

We say that a Riemannian manifold satisfies the $L^p$-positivity preserving property if $(-\Delta + 1)u\ge 0$ in a distributional sense implies $u \ge 0$ for all $ u \in L^p$.While geodesic completeness of the manifold at hand ensures the $L^p$-positivity preserving property for all $p \in (1, +\infty)$, when $p = + \infty$ some assumptions are needed. In this paper we show that the $L^\infty$-positivity preserving property is in fact equivalent to stochastic completeness, i.e., the fact that the minimal heat kernel of the manifold preserves probability. The result is achieved via some monotone approximation results for distributional solutions of $-\Delta + 1 \ge 0$, which are of independent interest.

N- Monotone Approximation in Weighted Space

The purpose of this paper is to discuss the degree of best monotone multi- approximation of unbounded functions weighted space in terms the modules of smoothness by using some algebraic linear operators. In addition, introduced ,proofs some properties of the modulus of smoothness.

Piecewise Monotone Approximation of Unbounded Functions In Weighted Space LP,w([-,

In this paper, investigate the approximation of unbounded functions in weighted space, by using trigonometric polynomials considered. We introduced type of polynomials piecewise monotone having same local monotonicity as unbounded functions without affecting the order of huge error have a finite number of max. and min. unbounded functions that amount. In addition, we established not included any of extreme points of this functions, of and closed subset γ on closed intervals then there exist class of polynomials such that the best of approximation has high or order of and such that for sufficiently great of the polynomials and functions have the same monotonicity at each of γ.

Transient Stability Analysis of Direct Drive Wind Turbine in DC-Link Voltage Control Timescale during Grid Fault

Transient stability during grid fault is experienced differently in modern power systems, especially in wind-turbine-dominated power systems. In this paper, transient behavior and stability issues of a direct drive wind turbine during fault recovery in DC-link voltage control timescale are studied. First, the motion equation model that depicts the phase and amplitude dynamics of internal voltage driven by unbalanced active and reactive power is developed to physically depict transient characteristics of the direct drive wind turbine itself. Considering transient switch control induced by active power climbing, the two-stage model is employed. Based on the motion equation model, transient behavior during fault recovery in a single machine infinite bus system is studied, and the analysis is also divided into two stages: during and after active power climbing. During active power climbing, a novel approximate analytical expression is proposed to clearly reveal the frequency dynamics of the direct drive wind turbine, which is identified as approximate monotonicity at excitation of active power climbing. After active power climbing, large-signal oscillation behavior is concerned. A novel analysis idea combining time-frequency analysis based on Hilbert transform and high order modes is employed to investigate and reveal the nonlinear oscillation, which is characterized by time-varying oscillation frequency and amplitude attenuation ratio. It is found that the nonlinear oscillation and even stability are related closely to the final point during active power climbing. With a large active power climbing rate, the nonlinear oscillation may lose stability. Simulated results based on MATLAB® are also presented to verify the theoretical analysis.

On Uniform Monotone Approximation of Continuous Monotone Functions with the Help of Translations and Dilations of the Laplace Integral

On Uniform Monotone Approximation of Continuous Monotone Functions with the Help of Translations and Dilations of the Laplace Integral

Univariate simultaneous high order abstract fractional monotone approximation with applications

Univariate simultaneous high order abstract fractional monotone approximation with applications

Abstract Fractional Monotone Approximation with Applications

Here we extended our earlier fractional monotone approximation theory to abstract fractional monotone approximation, with applications to Prabhakar fractional calculus and non-singular kernel fractional calculi. We cover both the left and right sides of this constrained approximation. Let f∈Cp−1,1, p≥0 and let L be a linear abstract left or right fractional differential operator such that Lf≥0 over 0,1 or −1,0, respectively. We can find a sequence of polynomials Qn of degree ≤n such that LQn≥0 over 0,1 or −1,0, respectively. Additionally f is approximated quantitatively with rates uniformly by Qn with the use of first modulus of continuity of fp.

CHARACTERIZATION OF APPROXIMATE MONOTONE OPERATORS

Results concerning local boundedness of operators have a long history. In 1994, Vesel´y connected the concept of approximate monotonicity of an operator with local boundedness of that. It is our desire in this note to characterize an approximate monotone operator. Actually, we show that a well-known property of monotone operators, namely representing by convex functions, remains valid for the larger subclass of operators. In this general framework we establish the similar results by Fitzpatrick. Also, celebrated results of Mart´ inez-Legaz and Th´era inspired us to prove that the set of maximal e -monotone operators between a normed linear space X and its continuous dual X ∗ can be identified as some subset of convex functions on X × X ∗ .

On the nature of Hawking’s incompleteness for the Einstein-vacuum equations: The regime of moderately spatially anisotropic initial data

In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately monotonic curvature singularities can dynamically form along a spacelike hypersurface. In this article, we study the Cauchy problem and give a rigorous proof of this phenomenon in sufficiently high dimensions, thereby providing the first constructive proof of stable curvature-blowup (without symmetry assumptions) along a spacelike hypersurface as an effect of pure gravity. Our proof applies to an open subset of regular initial data satisfying the assumptions of Hawking’s celebrated “singularity” theorem, which shows that the solution is geodesically incomplete but does not reveal the nature of the incompleteness. Specifically, our main result is a proof of the dynamic stability of the Kasner curvature singularity for a subset of Kasner solutions whose metrics exhibit only moderately (as opposed to severely) spatially anisotropic behavior. Of independent interest is our method of proof, which is more robust than earlier approaches in that (i) it does not rely on approximate monotonicity identities and (ii) it accommodates the possibility that the solution develops very singular high-order spatial derivatives, whose blowup-rates are allowed to be, within the scope of our bootstrap argument, much worse than those of the base-level quantities driving the fundamental blowup. For these reasons, our approach could be used to obtain similar blowup-results for various Einstein-matter systems in any number of spatial dimensions for solutions corresponding to an open set of moderately spatially anisotropic initial data, thus going beyond the nearly spatially isotropic regime treated in earlier works.

Gâteaux differentiability and uniform monotone approximation of convex functions in Banach spaces

Gâteaux differentiability and uniform monotone approximation of convex functions in Banach spaces

Nearly Monotone Neural Approximation with Quadratic Activation Function

Quadratic functions give good rates of approximation when used as activation functions of feedforward neural networks. Also, monotonicity is important to describe the function behavior, so the behavior of its constrained approximation. Previously, the degree of approximation by feedforward neural networks with quadratic activation function is proved to be within no less than the second order modulus of smoothness. In this paper, we discuss whether the improvement of the above estimates for Lebesgue integrable functions is possible or not. By nearly monotone approximation, it is possible to talk about a higher order modulus of smoothness, while it is not for just monotone functions. We get a nearly monotone function approximation by splitting the interval [0,1] into a partition with infinitely small lengths and then excluding intervals near the endpoints of the partition’s subintervals. However, counter examples cut hope for any more improvement outside that restricted interval. All the results are proved in the Lp-space with p < 1.