Discrete Integration Research Articles

On Sharp Rate of Convergence for Discretization of Integrals Driven by Fractional Brownian Motions and Related Processes with Discontinuous Integrands

We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order H>12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H>\\frac{1}{2}$$\\end{document} with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to n1-2H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{1-2H}$$\\end{document}, which is twice as good as the best known results in the case of discontinuous integrands and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply a change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.

Continuum limits for rational, breather solutions, and gauge equivalent model of the generalized discrete nonlinear Schrödinger equation

In this paper, we focus on an integrable higher‐order discrete nonlinear Schrödinger (dNLS) equation, which yields the Lakshmanan–Porsezian–Daniel (LPD) equation in the continuum limit. The breather solutions of the higher‐order dNLS are obtained for the first time. We show that the rational solutions and breather solutions of the higher‐order dNLS equation yield the counterparts of the integrable fourth‐order NLS equation under proper continuous limits. Most importantly, we also explore the gauge equivalent equation of the higher‐order dNLS equation from the point of view of the continuous limit theory. We make integrable discretization for the linear spectrum problem of the integrable generalized Heisenberg ferromagnetic equation that is just the gauge equivalent structure of the LPD equation. Compatibility condition of the discrete linear spectrum problem leads to a discrete integrable generalized Heisenberg Ferromagnetic equation. It is revealed that the discrete linear spectrum problem of the discrete integrable generalized ferromagnetic equation converges to the corresponding spectrum problem of the integrable generalized Feisenberg spin model in the continuum limit.In addition, with the help of the method of linear combinations for the conserved fluxes and densities, the conservation laws of the integrable higher‐order dNLS equation converge to the homologous results of the LPD equation.

DEVELOPMENT OF INTEGRATED DISCRETE MATHEMATICS TEACHINGBOOK WITH OPS TRANSFORMATION

The Ops Transformation has been analytically proven to be able to solve discrete math problems related to ordinary generating functions. The purpose of this research is to develop a discrete mathematics textbook which has included ops transformation theory as one of its chapters. The research method used is the 4D model research and development method. The results of expert validation on discrete mathematics textbooks equipped with the Ops transformation show that the books are very valid.

Sampling discretization error of integral norms for function classes with small smoothness

We consider infinitely dimensional classes of functions and instead of the relative error setting, which was used in previous papers on the integral norm discretization, we consider the absolute error setting. We demonstrate how known results from two areas of research – supervised learning theory and numerical integration – can be used in sampling discretization of the square norm on different function classes. We prove a general result, which shows that the sequence of entropy numbers of a function class in the uniform norm dominates, in a certain sense, the sequence of errors of sampling discretization of the square norm of this class. Then we use this result for establishing new error bounds for sampling discretization of the square norm on classes of multivariate functions with mixed smoothness.

Discrete integration for measuring aerodynamic loads on trains in crosswinds − realizable strategies of discretization and discrete integration

Discrete integration occupies an important place in train aerodynamic tests in crosswinds. Improved delayed detached eddy simulations based on shear stress transport k-ω turbulence models were carried out to calculate the side force, lift, rolling moment around the lee rail, and surface pressure on bluff and streamlined vehicles. The changepoints and piecewise linearities of the pressure coefficients were evaluated, and the maximum coefficient of determination in the elements was 0.9973. A realizable strategy of the discretization based on the Lagrange rectangular elements was suggested, including the largest lengths and numbers of the elements. From this, a strategy of the discrete integration was presented to measure the aerodynamic loads, considering the real orientation of the elements. The maximum errors of the mean aerodynamic load coefficients of the bluff and streamlined vehicles were 4.1% and 2.2% (except the mean lift coefficient of the bluff vehicle), respectively. The errors were less than those in the previous studies, especially for the streamlined vehicle, which reduced by up to 8.7%. The unsteady aerodynamic loads with no delay obtained by the strategies were near to natural ones in the frequency range that people would be concerned about in crosswinds (at the Strouhal number of less than 0.4). Some suggestions were made for using the strategies in the full-scale tests and model tests, which provided a foundation for further studies of the running safety in natural crosswinds.

An Approximation Method for Variational Inequality with Uncertain Variables

In this paper, a Stieltjes integral approximation method for uncertain variational inequality problem (UVIP) is studied. Firstly, uncertain variables are introduced on the basis of variational inequality. Since the uncertain variables are based on nonadditive measures, there is usually no density function. Secondly, the expected value model of UVIP is established after the expected value is discretized by the Stieltjes integral. Furthermore, a gap function is constructed to transform UVIP into an uncertain constraint optimization problem, and the optimal value of the constraint problem is proved to be the solution of UVIP. Finally, the convergence of solutions of the Stieltjes integral discretization approximation problem is proved.

Integrable discretization and numerical simulation for the Ito equation

An integrable time-discretization of the Ito equation is presented. By use of Hirota’s bilinear method, the Bäcklund transformation, Lax pair and soliton solutions to the semi-discrete system are also derived. Since the integrable time-discrete system converges to the continuous Ito equation when the step size tends to zero and does not destroy the conserved quantities, we design a numerical scheme to calculate the solutions to the Ito equation based on the integrable time-discretization and pseudo-spectral space-discretization. Numerical results show that our new scheme has obvious advantage in preserving the conserved quantities.

Opposing effects of KDM6A and JDP2 on glucocorticoid sensitivity in T-ALL.

Glucocorticoids (GCs) are the cornerstone of acute lymphoblastic leukemia (ALL) therapy. Although mutations in NR3C1, which encodes the GC receptor (GR), and other genes involved in GC signaling occur at relapse, additional mechanisms of adaptive GC resistance are uncertain. We transplanted and treated 10 primary mouse T-lineage acute lymphoblastic leukemias (T-ALLs) initiated by retroviral insertional mutagenesis with GC dexamethasone (DEX). Multiple distinct relapsed clones from 1 such leukemia (T-ALL 8633) exhibited discrete retroviral integrations that upregulated Jdp2 expression. This leukemia harbored a Kdm6a mutation. In the human T-ALL cell line CCRF-CEM, enforced JDP2 overexpression conferred GC resistance, whereas KDM6A inactivation unexpectedly enhanced GC sensitivity. In the context of KDM6A knockout, JDP2 overexpression induced profound GC resistance, counteracting the sensitization conferred by KDM6A loss. These resistant "double mutant" cells with combined KDM6A loss and JDP2 overexpression exhibited decreased NR3C1 mRNA and GR protein upregulation upon DEX exposure. Analysis of paired samples from 2 patients with KDM6A-mutant T-ALL in a relapsed pediatric ALL cohort revealed a somatic NR3C1 mutation at relapse in 1 patient and a markedly elevated JDP2 expression in the other. Together, these data implicate JDP2 overexpression as a mechanism of adaptive GC resistance in T-ALL, which functionally interacts with KDM6A inactivation.

Real-Time FPGA Simulation of High-Voltage Silicon Carbide MOSFETs

This article presents a real-time field-programmable gate array (FPGA)-based dynamic model of high-voltage and high-current silicon carbide (SiC) metal-oxide-semiconductor field-effect transistor ( <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mosfet</small> ) half-bridge power modules. The dynamic switching model utilizes the Shichman and Hodges equations using voltage-dependent nonlinear device capacitances and module electrical parameters to obtain an accurate dynamic model of the device switching transients. The key device states gate-source voltage, drain current, and drain-source voltage are modeled and discretized using forward Euler discrete integration method. Analysis of synthesizing the discrete-time model into real-time FPGA-based system with real-time data output from the on-board digital-to-analog converter is presented in detail. The model is utilized in modeling a 320-kW, medium-voltage dc/dc dual-active bridge converter and verified using dynamic experimental results from a 3.3-kV SiC <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mosfet</small> half-bridge power module. It has been shown that the presented discrete-time dynamic switching model accurately describes the turn- <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> and turn- <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> switching transients of the SiC power module at various voltage and current levels. Such models are useful for rapid and cost effective design and prototyping of SiC-based power electronic systems by defining key design and operating parameters, such as deadtime, switching frequency, and switching losses.

Low-rank Parareal: a low-rank parallel-in-time integrator

In this work, the Parareal algorithm is applied to evolution problems that admit good low-rank approximations and for which the dynamical low-rank approximation (DLRA) can be used as time stepper. Many discrete integrators for DLRA have recently been proposed, based on splitting the projected vector field or by applying projected Runge–Kutta methods. The cost and accuracy of these methods are mostly governed by the rank chosen for the approximation. These properties are used in a new method, called low-rank Parareal, in order to obtain a time-parallel DLRA solver for evolution problems. The algorithm is analyzed on affine linear problems and the results are illustrated numerically.

A Method for Long-Term Target Anti-Interference Tracking Combining Deep Learning and CKF for LARS Tracking and Capturing

Autonomous underwater vehicles (AUV) recycling in an underwater environment is particularly challenging due to the continuous exploitation of marine resources. AUV recycling via visual technology is the primary method. However, the current visual technology is limited by harsh sea conditions and has problems, such as poor tracking and detection. To solve these problems, we propose a long-term target anti-interference tracking (LTAT) method, which integrates Siamese networks, You Only Look Once (YOLO) networks and online learning ideas. Meanwhile, we propose using the cubature Kalman filter (CKF) for optimization and prediction of the position. We constructed a launch and recovery system (LARS) tracking and capturing the AUV. The system consists of the following parts: First, images are acquired via binocular cameras. Next, the relative position between the AUV and the end of the LARS was estimated based on the pixel positions of the tracking AUV feature points and binocular camera data. Finally, using a discrete proportion integration differentiation (PID) method, the LARS is controlled to capture the moving AUV via a CKF-optimized position. To verify the feasibility of our proposed system, we used the robot operating system (ROS) platform and Gazebo software to simulate the system for experiments and visualization. The experiment demonstrates that in the tracking process when the AUV makes a sinusoidal motion with an amplitude of 0.2 m in the three-dimensional space and the relative distance between the AUV and LARS is no more than 1 m, the estimated position error of the AUV does not exceed 0.03 m. In the capturing process, the final capturing error is about 28 mm. Our results verify that our proposed system has high robustness and accuracy, providing the foundation for future AUV recycling research.

A new characterization of congruences and the discrete Sugeno integral on LB-valued general fuzzy automata

The present study aimed at exploring the idempotent and compatible aggregation functions on a complete infinitely distributive lattice B of propositions about the LB-valued general fuzzy automata (for simplicity, LB-valued GFA), where the compatibility is related to the congruences on B. Further, we have shown that all aggregation functions on B can be obtained as usual composition of lattice operations ∨,∧ and certain unary and binary aggregation functions. Moreover, our results yield a new characterization of discrete Sugeno integrals on B.

Full-deautonomisation of a class of second-order mappings in ancillary form

We present an application of the full-deautonomisation method to a class of secondorder mappings which, using an ancillary variable, can be cast into a form that greatly facilitates the study of their singularities. The ancillary approach was originally introduced to make it possible to construct discrete Painlevé equations associated with the affine Weyl group E (1) 8 by deautonomising a QRT mapping. The full-deautonomisation method has been shown to offer a practical technique for calculating the exact dynamical degree of a mapping, whereby allowing the detection of discrete integrability using only singularity analysis. We study the confinement property for a given singularity, for a wide class of mappings that includes the autonomous limit of the standard additive Painlevé equation with E (1) 8 symmetry. This leads to a class of non-autonomous mappings, which can be integrable or not, for which we obtain their exact dynamical degrees. The case of a non-confining singularity is also analysed and again we obtain the corresponding dynamical degrees.

A linearly implicit energy-preserving exponential time differencing scheme for the fractional nonlinear Schrödinger equation

&lt;abstract&gt;&lt;p&gt;In this paper, we present a new method to solve the fractional nonlinear Schrödinger equation. Our approach combines the invariant energy quadratization method with the exponential time differencing method, resulting in a linearly-implicit energy-preserving scheme. To achieve this, we introduce an auxiliary variable to derive an equivalent system with a modified energy conservation law. The proposed scheme uses stabilized exponential time differencing approximations for time integration and Fourier pseudo-spectral discretization in space to obtain a linearly-implicit, fully-discrete scheme. Compared to the original energy-preserving exponential integrator scheme, our approach is more efficient as it does not require nonlinear iterations. Numerical experiments confirm the effectiveness of our scheme in conserving energy and its efficiency in long-time computations.&lt;/p&gt;&lt;/abstract&gt;

On integrability of the deformed Ruijsenaars-Schneider system

We find integrals of motion for the recently introduced deformed Ruijsenaars-Schneider many-body system, which is the dynamical system for poles of elliptic solutions to the Toda lattice with constraint of type B. Our method is based on the fact that the equations of motion for this system coincide with those for pairs of Ruijsenaars-Schneider particles which stick together preserving a special fixed distance between the particles. We also obtain Bäcklund transformations and integrable time discretization of the deformed Ruijsenaars-Schneider system, which is shown to be the dynamical system for poles of elliptic solutions to the fully discrete Kadomtsev-Petviashvili equation of type B. In additon, we propose a field analogue of the deformed Ruijsenaars-Schneider system on a space-time lattice. Bibliography: 35 titles.

Discrete Geometric Control of Planar Flexible Link Manipulators

Discrete control laws for flexible beam models obtained using structure-preserving discretization procedures are few in the existing literature so far and many control laws, including distributed ones, exist for continuous time models. Much of the effort has focussed on the discretization of the flexible structure and the fidelity of such models. The work in this paper examines the feasibility of a discretized version of a continuous time distributed control strategy, with a few approximations, to the discrete variational integrator of a flexible beam, in our case, a flexible single-link manipulator (FLM). The nonlinear control strategy is inspired by a potential shaping method in the literature based on the Cosserat continuum framework. We provide stability proof for this continuous-time control and approximate it to a discrete-time geometric framework. We use a geometrically exact beam model in which the manipulator's configuration space lies on an infinite-dimensional Lie group. The model-based discrete-time control uses a Lie Group Variational Integrator obtained by applying the variational principle on the Lagrangian of the FLM. Results show that the discrete closed-loop system stabilizes the FLM at setpoints during point-to-point rotational maneuvers. The controller demands fine time-sampling and we acknowledge the associated challenges for real-time implementation of the control scheme and intend to pursue it as our future work.

Fully discrete low-regularity integrator for the Korteweg–de Vries equation

Fully discrete low-regularity integrator for the Korteweg–de Vries equation

On the preservation of second integrals by Runge-Kutta methods

One can elucidate integrability properties of ordinary differential equations (ODEs) by knowing the existence of second integrals (also known as weak integrals or Darboux polynomials for polynomial ODEs). However, little is known about how they are preserved, if at all, under numerical methods. Here, we leverage the recently discovered theory of discrete second integrals to show novel results about Runge-Kutta methods. In particular, we show that any Runge-Kutta method preserves all affine second integrals but cannot preserve all quadratic second integrals of an ODE. A number of interesting corollaries are also discussed, such as the preservation of certain rational integrals by Runge-Kutta methods. The special case of affine second integrals with constant cofactor are also discussed as well the preservation of third and higher integrals.

Study on Single-Loop FCS-MPC for DC-Based DFIG System

Doubly fed induction generator (DFIG) system suffers from complex control structure, slow dynamic response speed and cumbersome parameter design due to its traditional cascaded dual-loop control strategy. A single-loop finite control set model predictive control (SLMPC) is proposed in the paper for DC-based DFIG in DC grid, which simplifies its control structure and parameter design, and enhances system dynamic response. There are three improved aspects. First, the single-loop control structure is proposed to eliminate intermediate link in dual-loop cascaded structure, and enhance system dynamic response. Second, system reduced-order discretization algorithm is proposed by differential and integral discretization method to reduce finite control set model predictive control strategy (FCS-MPC) design difficulty. Third, cost function with nonlinear additional current limiting function is designed to protect system from overcurrent effectively. Finally, the feasibility of proposed strategy is verified by simulations and experiments.

High-order unfitted characteristic finite element methods for moving interface problem of Oseen equations

Moving interface problems have broad applications in multi-phase flows and fluid–structure interactions. The design of high-order numerical methods for such problems becomes challenging, in particular, when the interface is severely deformed. We propose kth-order unfitted characteristic finite element methods (k=2,3,4) for solving two-dimensional Oseen equations with a time-varying interface. In each time step, the interface is not specified with an explicit function, but constructed numerically with a (k+1)th-order interface-tracking algorithm. We obtain optimal convergence orders of both the discrete velocity and the discrete pressure under the H1-norm. Our convergence analysis includes errors of the approximate interface and the time integration and spatial discretization of the governing equations. Numerical experiments are performed on a rotating object and a severely deformed interface and show optimal convergence orders of the method.