Piecewise Constant Functions Research Articles

Two-grid methods of finite element approximation for parabolic integro-differential optimal control problems

<abstract><p>In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results.</p></abstract>

Математичні моделі визначення температурних полів у теплоактивних елементах цифрових пристроїв з локальним внутрішнім нагріванням та із урахуванням термочутливості

Linear and non-linear mathematical models for the determination of the temperature field, and subsequently for the analysis of temperature regimes in isotropic spatial heat-active media subjected to internal local heat load, have been developed. In the case of a nonlinear boundary-value problem, the Kirchhoff transformation is applied, using which the original nonlinear heat conduction equation and nonlinear boundary conditions are linearized, and as a result, a linearized second-order differential equation with partial derivatives and a discontinuous right-hand side and partially linearized boundary conditions is obtained. For the final linearization of the partially linearized boundary conditions, the approximation of the temperature by the radial spatial coordinate on the boundary surface of the thermosensitive medium was performed by a piecewise constant function, as a result of which the boundary value problem was obtained completely linearized. To solve the linear boundary value problem, as well as the obtained linearized boundary value problem with respect to the Kirchhoff transformation, the Henkel integral transformation method was used, as a result of which analytical solutions of these problems were obtained. For a heat-sensitive environment, as an example, a linear dependence of the coefficient of thermal conductivity of the structural material of the structure on temperature, which is often used in many practical problems, was chosen. As a result, an analytical relationship was obtained for determining the temperature distribution in this medium. Numerical analysis of temperature behavior as a function of spatial coordinates for given values of geometric and thermophysical parameters was performed. The influence of the power of internal heat sources and environmental materials on the temperature distribution was studied. To determine the numerical values of the temperature in the given structure, as well as to analyze the heat exchange processes in the middle of these structures, caused by the internal heat load, software tools were developed, using which a geometric image of the temperature distribution depending on the spatial coordinates was made. The developed linear and nonlinear mathematical models for determining the temperature field in spatial heat-active environments with internal heating testify to their adequacy to a real physical process. They make it possible to analyze such environments for their thermal stability. As a result, it becomes possible to increase it and protect it from overheating, which can cause the destruction of not only individual nodes and their elements, but also the entire structure.

Immersed finite element methods for convection diffusion equations

<abstract><p>In this work, we develop two IFEMs for convection-diffusion equations with interfaces. We first define bilinear forms by adding judiciously defined convection-related line integrals. By establishing Gårding's inequality, we prove the optimal error estimates both in $ L^2 $ and $ H^1 $-norms. The second method is devoted to the convection-dominated case, where test functions are piecewise constant functions on vertex-associated control volumes. We accompany the so-called upwinding concepts to make the control-volume based IFEM robust to the magnitude of convection terms. The $ H^1 $ optimal error estimate is proven for control-volume based IFEM. We document numerical experiments which confirm the analysis.</p></abstract>

Control parameterization approach to time-delay optimal control problems: A survey

<p style='text-indent:20px;'>Control parameterization technique is an effective method to solve optimal control problems. It works by approximating the control function by piece-wise constant (or linear) function. In this way, the optimal control problems are approximated by optimal parameter selection problems, which can be regarded as finite-dimensional optimization problems, where the control heights and switching times of the piece-wise constant function are taken as the decision variables. They can be solved by using gradient-based optimization methods. For this, it requires the gradient formulas of the objective and constraint functions with respect to the decision variables. There are two methods for deriving the required gradient formulas. The aim of this paper is to survey various numerical methods developed based on control parameterization for solving optimal control problems governed by time-delay systems. Particular focus is on those numerical methods for optimizing switching time points. This is because the process of optimizing the switching time points is much more complicated for optimal control problems with time delays. In addition, we shall also review the computational methods developed based on control parameterization technique for solving two optimal control problems involving more complicated time delays.</p>

Generalized Integral Equations for Timing Games

We consider timing games, the payoff functions of which have additional break lines outside the unit square diagonal. A special case of such games are games with piecewise constant payoff functions. Solving these games is reduced to solving a pair of integral equations for the distribution functions of equalizing strategies. The proposed solution methods can be used in the study of random walks on a segment in a variable environment.

Complete Description of Bounded Solutions for a Duffing-Type Equation with a Periodic Piecewise Constant Coefficient

We consider the equation $u_{xx}^{}-u+W(x)u^3=0$ where $W(x)$ is a periodic alternating piecewise constant function. It is proved that under certain conditions for $W(x)$ solutions of this equation, which are bounded on $\mathbb{R}$, $|u(x)|<\xi$, can be put in one-to-one correspondence with bi-infinite sequences of numbers $n\in \{-N,\,\ldots,\,N\}$ (called ``codes'' of the solutions). The number $N$ depends on the bounding constant $\xi$ and the characteristics of the function $W(x)$. The proof makes use of the fact that, if $W(x)$ changes sign, then a ``great part'' of the solutions are singular, i.e., they tend to infinity at a finite point of the real axis. The nonsingular solutions correspond to a fractal set of initial data for the Cauchy problem in the plane $(u,\,u_x^{})$. They can be described in terms of symbolic dynamics conjugated with the map-over-period (monodromy operator) for this equation. Finally, we describe an algorithm that allows one to sketch plots of solutions by its codes.

Joint Optimization of Transmit Beamforming and Base Station Cache Allocation in Multi-Cell C-RAN

This paper studies a downlink cloud radio access network (C-RAN) consisting of a centralized processor (CP) and multiple cells each one of which has one cache-enabled base station (BS) connected to the CP through wireless backhaul links. In such a network, we aim to determine sizes of the cache allocated to the BSs such that the long-term delivery time of the available files is minimized. Considering the two-stage nature of this problem, we formulate a beamforming optimization problem and a cache allocation problem. The purpose of the first one is to deliver un-cached portions of files to the BSs in the shortest possible time during the beamforming stage. The second problem aims to minimize long-term expectation of delivery time through optimal cache allocation during the cache allocation stage. We rigorously prove that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">time-varying beamformers</i> can be assumed to be piece-wise constant functions of time. Based on this finding, the beamforming optimization problem does not appear to be amenable to a computationally efficient solution. Hence, we resort to the zero-forcing (ZF) beamforming approach to tackle the beamforming optimization problem. Using the results of the beamforming optimization problem, we prove that the cache allocation problem is a convex optimization problem; hence it can be solved by any convex optimization solvers. Simulation results show that the loss of optimality due to adopting ZF beamforming is negligible. Moreover, the superiority of our proposed cache allocation scheme over several other heuristic schemes including proportional cache size allocation is shown.

Existence, Uniqueness, and Stability of Solutions to Variable Fractional Order Boundary Value Problems

This paper investigates the sufficient conditions for the existence and uniqueness of a class of Riemann-Liouville fractional differential equations of variable order with fractional boundary conditions. The problem is converted into differential equations of constant orders by combining the concepts of generalized intervals and piecewise constant functions. We derive the required conditions for ensuring the uniqueness of the problem in order to utilize the Banach fixed point theorem. The stability of the obtained solution in the Ulam-Hyers-Rassias (UHR) sense is also investigated, and we finally provide an illustrative example.

Central concentration of asymmetric features in post-starburst galaxies at z ∼ 0.8

ABSTRACT We present morphological analyses of post-starburst galaxies (PSBs) at 0.7 &amp;lt; z &amp;lt; 0.9 in the COSMOS field. We fitted ultraviolet to mid-infrared multiband photometry of objects with i &amp;lt; 24 from COSMOS2020 catalogue with population synthesis models assuming non-parametric, piece-wise constant function of star formation history, and selected 94 of those galaxies that have high specific star formation rates (SSFRs) of more than 10−9.5 yr−1 in 321–1000 Myr before observation and an order-of-magnitude lower SSFRs within recent 321 Myr. We devised a new non-parametric morphological index which quantifies concentration of asymmetric features, CA, and measured it as well as concentration C and asymmetry A on the Hubble Space Telescope/Advanced Camera for Surveys IF814W-band images. While relatively high C and low A values of PSBs are similar to those of quiescent galaxies rather than star-forming galaxies, we found that PSBs show systematically higher values of CA than both quiescent and star-forming galaxies; 36 per cent of PSBs have log CA &amp;gt; 0.8, while only 16 per cent (2 per cent) of quiescent (star-forming) galaxies show such high CA values. Those PSBs with high CA have relatively low overall asymmetry of A ∼ 0.1, but show remarkable asymmetric features near the centre. The fraction of those PSBs with high CA increases with increasing SSFR in 321–1000 Myr before observation rather than residual ongoing star formation. These results and their high surface stellar mass densities suggest that those galaxies experienced a nuclear starburst in the recent past, and processes that cause such starbursts could lead to the quenching of star formation through rapid gas consumption, supernova/active galactic nucleus feedback, and so on.

A staggered cell‐centered finite element method for Stokes problems with variable viscosity on general meshes

AbstractIn this paper, we propose to extend the staggered cell‐centered finite element method (SCFEM) on general meshes for the Stokes problems with variable viscosity (possibly discontinuous). The scheme is cell‐centered in the sense that the solution can be computed by cell unknowns of the primal mesh and the dual mesh for the velocity and the pressure, respectively, where the velocity is approximated by piecewise linear functions () on the triangular dual submesh, and the pressure is approximated by piecewise constant functions () on the dual mesh. In order to get the local continuity of numerical stresses across the interfaces, the scheme gives the auxiliary edge unknowns interpolated by the multipoint stress approximation technique. Its stability and convergence properties are presented in the rigorous theoretical framework. Numerical results are carried out to highlight accuracy and computational cost.

Approximations of the set of trajectories and integral funnel of the non-linear control systems with Lp norm constraints on the control functions

Abstract In this paper, approximations of the set of trajectories and integral funnel of the control system described by non-linear ordinary differential equation with integral constraint on the control functions are considered. The set of admissible control functions is replaced by a set, consisting of a finite number of piecewise-constant control functions. It is shown that the set of trajectories generated by a finite number of piecewise-constant control functions is an internal approximation of the set of trajectories. Further, each trajectory generated by a piecewise-constant control function is substituted by appropriate Euler’s broken line and it is proved that the set consisting of a finite number of Euler’s broken lines is an approximation of the set of trajectories of given control system. An approximation of the system’s integral funnel by a set consisting of a finite number of points is obtained.

Investigation of Gradient Platinum Loading and Porosity Distribution for Anion Exchange Membrane Fuel Cells

Abstract Anion exchange membrane fuel cells (AEMFCs) are in development as a low-cost alternative to proton exchange membrane fuel cells (PEMFCs). AEMFCs produce water at the anode side and consume it at the cathode side, resulting in no cathode water flooding like in PEMFCs. However, it brings complexity to water transportation behavior and requires appropriate water balance to avoid membrane drying out. In this study, a two-dimensional two-phase multi-physics model has been developed to investigate the impacts of three key electrode parameters (porosity, catalyst loading, and ionomer content) that are responsible for water production and transport as well as the performance of an AEMFC. A piecewise constant function along the x-direction (reactant diffusion direction) is used to apply the gradient on the porosity and platinum loading. The present results show that a larger porosity gradient near the cathode gas diffusion layer (GDL)/flow channel interface and lower near the GDL/microporous layer (MPL) interface can enhance mass transport and water removal, which is benefited the AEMFC performance. However, anode GDL porosity gradients show a lower AEMFC performance compared to the cathode porosity gradients. Moreover, it was confirmed that for both electrodes, the performance of AEMFC was significantly dependent on each electrode parameter.

How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms

In this paper we consider two topological transforms that are popular in applied topology: the Persistent Homology Transform and the Euler Characteristic Transform. Both of these transforms are of interest for their mathematical properties as well as their applications to science and engineering, because they provide a way of summarizing shapes in a topological, yet quantitative, way. Both transforms take a shape, viewed as a tame subset M M of R d \mathbb { R}^d , and associates to each direction v ∈ S d − 1 v\in S^{d-1} a shape summary obtained by scanning M M in the direction v v . These shape summaries are either persistence diagrams or piecewise constant integer-valued functions called Euler curves. By using an inversion theorem of Schapira, we show that both transforms are injective on the space of shapes, i.e. each shape has a unique transform. Moreover, we prove that these transforms determine continuous maps from the sphere to the space of persistence diagrams, equipped with any Wasserstein p p -distance, or the space of Euler curves, equipped with certain L p L^p norms. By making use of a stratified space structure on the sphere, induced by hyperplane divisions, we prove additional uniqueness results in terms of distributions on the space of Euler curves. Finally, our main result proves that any shape in a certain uncountable space of PL embedded shapes with plausible geometric bounds can be uniquely determined using only finitely many directions.

Error Estimates for Adaptive Spectral Decompositions

Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield small subspaces where the characteristic functions comprising u are well approximated. When combined with Newton-like optimization methods for the solution of inverse medium problems, AS decompositions have proved remarkably efficient in providing at each nonlinear iteration a low-dimensional search space. Here, we derive L^2-error estimates for the AS decomposition of u, truncated after K terms, when u is piecewise constant and consists of K characteristic functions over Lipschitz domains and a background. Our estimates apply both to the continuous and the discrete Galerkin finite element setting. Numerical examples illustrate the accuracy of the AS decomposition for media that either do, or do not, satisfy the assumptions of the theory.

Multilevel Selective Harmonic Modulation via Optimal Control

We consider the Selective Harmonic Modulation (SHM) problem, consisting in the design of a staircase control signal with some prescribed frequency components. In this work, we propose a novel methodology to address SHM as an optimal control problem in which the admissible controls are piecewise constant functions, taking values only in a given finite set. In order to fulfill this constraint, we introduce a cost functional with piecewise affine penalization for the control, which, by means of Pontryagin’s maximum principle, makes the optimal control have the desired staircase form. Moreover, the addition of the penalization term for the control provides uniqueness and continuity of the solution with respect to the target frequencies. Another advantage of our approach is that the number of switching angles and the waveform need not be determined a priori. Indeed, the solution to the optimal control problem is the entire control signal, and therefore, it determines the waveform and the location of the switches. We also provide numerical examples in which the SHM problem is solved by means of our approach.

EXAMPLES OF INFORMATION TECHNOLOGIES FOR RECONSTRUCTION FROM THE DATA OF THE SPECTRUM OF SOME CLASSES OF RANDOM FUNCTIONS

It is known that a stationary random process is represented as a superposition of harmonic oscillations with real frequencies and uncorrelated amplitudes. In the study of nonstationary processes, it is natural to have increasing or declining oscillationсs. This raises the problem of constructing algorithms that would allow constructing broad classes of nonstationary processes from elementary nonstationary random processes. A natural generalization of the concept of the spectrum of a nonstationary random process is the transition from the real spectrum in the case of stationary to a complex or infinite multiple spectrum in the nonstationary case. There is also the problem of describing within the correlation theory of random processes in which the spectrum has no analogues in the case of stationary random processes, namely, the spectrum point is real, but it has infinite multiplicity for the operator image of the corresponding operator, and when the spectrum itself is complex. Reconstruction of the complex spectrum of a nonstationary random function is a very important problem in both theoretical and applied aspects. In the paper the procedure of reconstruction of random process, sequence, field from a spectrum for Gaussian random functions is developed. Compared to the stationary case, there are wider possibilities, for example, the construction of a nonstationary random process with a real spectrum, which has infinite multiplicity and which can be distributed over the entire finite segment of the real axis. The presence of such a spectrum leads, in contrast to the case of a stationary random process, to the appearance of new components in the spectral decomposition of random functions that correspond to the internal states of "strings", i.e. generated by solutions of systems of equations in partial derivatives of hyperbolic type. The paper deals with various cases of the spectrum of a non-self-adjoint operator A, namely, the case of a discrete spectrum and the case of a continuous spectrum, which is located on a finite segment of the real axis, which is the range of values of the real non-decreasing function a (x). The cases a (x) = 0, a (x) = const, a (x) = x and a (x) is a piecewise constant function are studied. The authors consider the recovery of nonstationary sequences for different cases of the spectrum of a non-self-adjoint operator A promising since spectral decompositions are a superposition of discrete or continuous internal states of oscillators with complex frequencies and uncorrelated amplitudes and therefore have deep physical meaning.

Mathematical Modeling of Battery Energy Storage System Operating Modes using Volterra Integral Equations

The integration of powerful photovoltaic systems and wind farms into energy systems significantly affects the operating modes due to frequent power fluctuations, thereby justifying the use of energy storage devices. The improvement of mathematical models describing operating modes of power facilities is a consequence of structure and process complication occurring in energy systems. The article presents a method for modelling the operating parameters of battery energy storage systems, based on the Volterra integral equations. The sequence of the solution is described, as well as its applicability in energy systems research. To determine the operating modes of battery energy storage systems, a non-standard class of Volterra integral equations of the first kind is used in the research. This makes it possible to calculate the operating modes of storage batteries in terms of the inverse problem, when the generation and power imbalance functions are known. To calculate the operating modes of several types of storage devices operating simultaneously within the same power system, the form of such equations can be specified in the form of piecewise constant functions. This method has a number of advantages, since it allows one to take into account nonlinear processes in the kernels of integral equations. Thus, the processes of degradation of energy storage devices that affect the efficiency of their operation are taken into account. The use of regularizing algorithms makes it possible to reduce the error. As an example, an autonomous photovoltaic system with batteries was considered. The obtained results and their visualization show the viability of applying the approach.

Discrete approximations of continuous probability distributions obtained by minimizing Cramér-von Mises-type distances

We consider the problem of approximating a continuous random variable, characterized by a cumulative distribution function (cdf) F(x), by means of k points, x_1<x_2<dots < x_k, with probabilities p_i, i=1,dots ,k. For a given k, a criterion for determining the x_i and p_i of the approximating k-point discrete distribution can be the minimization of some distance to the original distribution. Here we consider the weighted Cramér-von Mises distance between the original cdf F(x) and the step-wise cdf {hat{F}}(x) of the approximating discrete distribution, characterized by a non-negative weighting function w(x). This problem has been already solved analytically when w(x) corresponds to the probability density function of the continuous random variable, w(x)=F'(x), and when w(x) is a piece-wise constant function, through a numerical iterative procedure based on a homotopy continuation approach. In this paper, we propose and implement a solution to the problem for different choices of the weighting function w(x), highlighting how the results are affected by w(x) itself and by the number of approximating points k, in addition to F(x); although an analytic solution is not usually available, yet the problem can be numerically solved through an iterative method, which alternately updates the two sub-sets of k unknowns, the x_i’s (or a transformation thereof) and the p_i’s, till convergence. The main apparent advantage of these discrete approximations is their universality, since they can be applied to most continuous distributions, whether they possess or not the first moments. In order to shed some light on the proposed approaches, applications to several well-known continuous distributions (among them, the normal and the exponential) and to a practical problem where discretization is a useful tool are also illustrated.

Optimal Control of a Quasilinear Parabolic Equation and its Time Discretization

In this paper we discuss the optimal control of a quasilinear parabolic state equation. Its form is leaned on the kind of problems arising for example when controlling the anisotropic Allen–Cahn equation as a model for crystal growth. Motivated by this application we consider the state equation as a result of a gradient flow of an energy functional. The quasilinear term is strongly monotone and obeys a certain growth condition and the lower order term is non-monotone. The state equation is discretized implicitly in time with piecewise constant functions. The existence of the control-to-state operator and its Lipschitz-continuity is shown for the time discretized as well as for the time continuous problem. Latter is based on the convergence proof of the discretized solutions. Finally we present for both the existence of global minimizers. Also convergence of a subsequence of time discrete optimal controls to a global minimizer of the time continuous problem can be shown. Our results hold in arbitrary space dimensions.

Log–log convexity of an optimal control problem for positive linear systems

This paper investigates the finite-time optimal control problems for positive linear systems with a time-varying control input. Cost functional of the system state is constructed. Under some assumptions on the designed parameters and cost functionals, the optimization problem with piecewise-constant matrix functions is first proved to be log–log convex and can be solved via geometric programming. Then, the log–log convex result is further extended to the optimization problem with continuous functions. Finally, an optimal control problem is investigated to verify the effectiveness of the proposed optimization framework.