Explicit Ordinary Differential Equations Research Articles

On the area-preserving Willmore flow of small bubbles sliding on a domain’s boundary

Abstract We consider the area-preserving Willmore evolution of surfaces ϕ that are close to a half-sphere with a small radius, sliding on the boundary S of a domain Ω while meeting it orthogonally. We prove that the flow exists for all times and keeps a “half-spherical” shape. Additionally, we investigate the asymptotic behavior of the flow and prove that for large times the barycenter of the surfaces approximately follows an explicit ordinary differential equation. Imposing additional conditions on the mean curvature of S, we then establish convergence of the flow.

Normal form analysis in the presence of SVC and STATCOM and the effect of their siting on damping of rotor swings

Normal form analysis is used to assess the nonlinear behavior; it can also be used for siting controllers. Normal form analysis is straight-forward for systems governed by explicit ordinary differential equations. However, the analysis becomes complicated if the governing equations are differential–algebraic. The presence of SVC/STATCOM results in the governing equations for which the normal form analysis is not as straight-forward as that for explicit differential equations. In this paper, it is shown how semi-explicit differential–algebraic equations of index 1 are amenable to normal form analysis. In particular, methods are proposed for normal form analysis in the presence of SVC and STATCOM. Normal form analysis can be used to find the location of SVC and STATCOM that results in the best damping of rotor swings; the results are analyzed by case studies and also compared with the performance obtained by locating SVC and STATCOM using linearization methods.

Error estimates for Runge–Kutta schemes of optimal control problems with index 1 DAEs

In this paper we derive error estimates for Runge–Kutta schemes of optimal control problems subject to index one differential–algebraic equations (DAEs). Usually, Runge–Kutta methods applied to DAEs approximate the differential and algebraic state in an analogous manner. These schemes can be considered as discretizations of the index reduced system where the algebraic equation is solved for the algebraic variable to get an explicit ordinary differential equation. However, in optimal control this approach yields discrete necessary conditions that are not consistent with the continuous necessary conditions which are essential for deriving error estimates. Therefore, we suggest to treat the algebraic variable like a control, obtaining a new type of Runge–Kutta scheme. For this method we derive consistent necessary conditions and compare the discrete and continuous systems to get error estimates up to order three for the states and control as well as the multipliers.

Лучевой поиск для расширения пространства обыкновенных дифференциальных уравнений, разрешенных относительно производной

Расширение пространства для явных ОДУ заключается во введении новых уравнений в набор уравнений, где новые неизвестные функционально зависят от исходных неизвестных. Цель состоит в том, чтобы преобразовать набор ОДУ в форму, которая имеет чисто полиномиальные правые части второй степени. Это необходимый шаг предварительной обработки для ряда некоторых методов решения. Полиномиальные ОДУ могут быть преобразованы в ОДУ с членами чисто второй степени посредством пространственного расширения. В предыдущей работе показано, что расширение пространства с наименьшим числом новых неизвестных может быть найдено полным перебором. Однако полный поиск неэффективен в вычислительном отношении. В этой статье используется эффективный в вычислительном отношении поиск (поиск луча), но оптимальность (наименьшее количество новых неизвестных) не гарантируется. Численные эксперименты показывают, что поиск луча помогает найти полезное расширение пространства даже для многочленов с относительно более высокими степенями Space extension for explicit ODEs is considering introduction of new equations to the equation set where the new unknowns are functionally dependent on the original unknowns. The purpose is to convert the ODE set into a form that has purely second degree multinomial right hand side functions. This is a necessary preprocessing step for certain series solution methods. Multinomial ODEs can be converted to ODEs with purely second degree terms through space extension. In a previous work, it is shown that the space extension with the smallest number of new unknowns can be found by a complete search. However, the complete search is not computationally efficient. In this paper, a computationally efficient search (beam search) is utilized but optimality (smallest number of new unknowns) is not guaranteed. The numerical experiments show that beam search is powerful in finding a useful space extension even for multinomials with relatively higher degrees.

Revealing hidden dynamics from time-series data by ODENet

To derive the hidden dynamics from observed data is one of the fundamental but also challenging problems in many different fields. In this study, we propose a new type of interpretable network called the ordinary differential equation network (ODENet), in which the numerical integration of explicit ordinary differential equations (ODEs) is embedded into the machine learning scheme to build a general framework for revealing the hidden dynamics buried in massive time-series data efficiently and reliably. ODENet takes full advantage of both machine learning algorithms and ODE modeling. On one hand, the embedding of ODEs makes the framework more interpretable benefiting from the mature theories of ODEs. On the other hand, the schemes of machine learning enable data handling, paralleling, and optimization to be easily and efficiently implemented. From classical Lotka-Volterra equations to chaotic Lorenz equations, the ODENet exhibits its remarkable capability in handling time-series data even in the presence of large noise. We further apply the ODENet to real actin aggregation data, which shows an impressive performance as well. These results demonstrate the superiority of ODENet in dealing with noisy data, data with either non-equal spacing or large sampling time steps over other traditional machine learning algorithms.

Evolution of Nonlinear Reduced-Order Solutions for PDEs with Conserved Quantities

Reduced-order models of time-dependent partial differential equations (PDEs) where the solution is assumed as a linear combination of prescribed modes are rooted in a well-developed theory. However, more general models where the reduced solutions depend nonlinearly on time varying parameters have thus far been derived in an ad hoc manner. Here, we introduce reduced-order nonlinear solutions (RONS): a unified framework for deriving reduced-order models that depend nonlinearly on a set of time-dependent parameters. The set of all possible reduced-order solutions are viewed as a manifold immersed in the function space of the PDE. The parameters are evolved such that the instantaneous discrepancy between reduced dynamics and the full PDE dynamics is minimized. This results in a set of explicit ordinary differential equations on the tangent bundle of the manifold. In the special case of linear parameter dependence, our reduced equations coincide with the standard Galerkin projection. Furthermore, any number of conserved quantities of the PDE can readily be enforced in our framework. Since RONS does not assume an underlying variational formulation for the PDE, it is applicable to a broad class of problems. We demonstrate the efficacy of RONS on three examples: an advection-diffusion equation, the nonlinear Schrödinger equation, and Euler's equation for ideal fluids.

Antiperiodic boundary value problem for an implicit ordinary differential equation

The paper is devoted to the investigation of the antiperiodic boundary value problem for an implicit nonlinear ordinary differential equation $$f(t,x,\dot x)=0, \quad x(0)+x(\tau)=0.$$ We assume that the mapping $f:\mathbb{R}\times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^k$ defining the equation under consideration is smooth and satisfies the condition of uniform nondegeneracy of the first derivative $$ \inf \bigl\{ {\rm cov} f'_v (t,x,v):\, (t,x,v)\in \mathbb{R}\times \mathbb{R}^n \times \mathbb{R}^n \bigr\}>0. $$ Here ${\rm cov} A$ is the Banach constant of the linear operator $A.$ The assumption of uniform non-degeneracy holds, in particular, for the mapping $f$ defining an explicit ordinary differential equation. For implicit equations, sufficient conditions for the existence of a solution to an antiperiodic boundary value problem are obtained, and estimates for solutions are found. Co\-ro\-l\-la\-ries for normal ordinary differential equations are formulated. To prove the main result, the original implicit equation is reduced to an explicit differential equation by applying a nonlocal implicit function theorem. Then we prove an auxiliary assertion on the solvability of the equation $x+\psi(x)=0,$ which is an analog of Brouwer's fixed point theorem. It is shown that the mapping $\psi,$ that assigns the value of the solution of the Cauchy problem at the point $\tau$ to an arbitrary initial point $x_0,$ is well defined and satisfies the assumptions of the auxiliary statement. This reasoning completes the proof of the existence of a solution to the boundary value problem.

Design of a stigmatic lens implementing a required ray mapping.

We consider a method for designing stigmatic lenses implementing required ray mappings. A ray mapping relates the angular coordinates of the rays at the image points with the angular coordinates of the rays coming from the object. The calculation of a stigmatic lens with a required ray mapping is reduced to solving an explicit ordinary differential equation. As examples, we design stigmatic lenses satisfying the Abbe sine condition, the Herschel's condition, and the constant angular magnification condition.

Design of a stigmatic lens with minimal Fresnel losses

A method for designing double aspheric lenses enabling minimal Fresnel losses in the class of stigmatic lenses is considered. Minimization of the Fresnel losses is provided by ensuring equal ray-deviation angles on both aspheric surfaces of the lens. The design of the lens is reduced to the integration of an explicit ordinary differential equation. Simple analytical approximations for the lens profiles are also presented.

Designing stigmatic lenses with minimal Fresnel losses.

We present a method for designing lenses with two aspherical surfaces having minimal Fresnel losses among the class of stigmatic lenses. Minimization of Fresnel losses is achieved by ensuring equal ray deviation angles on the lens surfaces. Calculation of lenses with minimal Fresnel losses is reduced to solving an explicit ordinary differential equation. Simple analytical approximations are also obtained for the lens profiles.

On the solution calculation of nonlinear ordinary differential equations via exact quadratization

We show a general method allowing the solution calculation, in the form of a power series, for a very large class of nonlinear Ordinary Differential Equations (ODEs), namely the real analytic σπ-ODEs (and, more in general, the real analytic σπ-reducible ODEs) in many indeterminates, characterized by an ODE-function given by generalized polynomials of the indeterminates and their derivatives, i.e. functions formally polynomial with exponents, though the exponent can be any real number, and whose coefficients are analytic time functions. The solution method consists in reducing the ODE to a certain canonical homogeneous quadratic ODE, named driver-type Riccati equation with a larger number of indeterminates, whose solutions include, as sub-solutions, the original solutions, and for which a recursive formula is shown, giving all coefficients of the solution power series directly from the ODE parameters. The reduction method is named exact quadratization and was formerly introduced in another our article, where we considered explicit ODEs only. In the present paper, which is self-contained to a large extent, we review and complete the theory of exact quadratization by solving issues, such as for instance the existence of a piecewise-quadratization, that had remained open, and also extend it to the more general case of an implicitly defined ODE. Finally, we argue that the result can be seen as a partial solution of a differential version of the 22nd Hilbert's problem.

Parallel manipulator dynamics embedded in singularity free domains of functionality

Ordinary differential equations of motion for a broad spectrum of parallel manipulators are derived and implemented in manipulator input coordinates for system simulation and controller design. Equations of motion are systematically embedded in singularity free domains of manipulator configuration space, assuring controllability and existence of a unique solution of the equations of motion. The kinematic foundation for the formulation is a mechanics-based model of the underlying mechanism with forward and inverse kinematic mappings that are evaluated on singularity free domains of manipulator configuration space. The kinetic foundation is d’Alembert’s variational formulation of multibody dynamics that, with forward kinematic mappings, yields explicit ordinary differential equations of motion. No ad-hoc derivations of Lagrange’s equations, Newton–Euler equations, or Lagrange multiplier-based differential-algebraic equations are required. Differential equations of motion for a planar front-end loader and a spatial Stewart platform manipulator are implemented to illustrate the formulation.

Stochastic rounding and reduced-precision fixed-point arithmetic for solving neural ordinary differential equations.

Although double-precision floating-point arithmetic currently dominates high-performance computing, there is increasing interest in smaller and simpler arithmetic types. The main reasons are potential improvements in energy efficiency and memory footprint and bandwidth. However, simply switching to lower-precision types typically results in increased numerical errors. We investigate approaches to improving the accuracy of reduced-precision fixed-point arithmetic types, using examples in an important domain for numerical computation in neuroscience: the solution of ordinary differential equations (ODEs). The Izhikevich neuron model is used to demonstrate that rounding has an important role in producing accurate spike timings from explicit ODE solution algorithms. In particular, fixed-point arithmetic with stochastic rounding consistently results in smaller errors compared to single-precision floating-point and fixed-point arithmetic with round-to-nearest across a range of neuron behaviours and ODE solvers. A computationally much cheaper alternative is also investigated, inspired by the concept of dither that is a widely understood mechanism for providing resolution below the least significant bit in digital signal processing. These results will have implications for the solution of ODEs in other subject areas, and should also be directly relevant to the huge range of practical problems that are represented by partial differential equations.This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

Design of axisymmetric double-surface refractive optical elements generating required illuminance distributions

We propose a novel technique for designing double-surface axisymmetric refractive optical elements with minimized Fresnel losses and maximized lighting efficiency. The design of the optical element is reduced to the integration of three explicit ordinary differential equations. Minimization of the Fresnel losses is provided by ensuring equal refraction angles on both “working” surfaces of the optical element. To demonstrate high performance of the method, we designed axisymmetric double-surface optical elements generating uniform illuminance distributions both in the far (angular size of 50°) and in the near (angular size of 90°) field. The lighting efficiency exceeds 87.8% for both point and extended light sources. Besides, we demonstrated that optical surfaces designed using the proposed method can serve as an excellent initial approximation in the case of a large extended light source. In the considered example with the ratio of the element height to the source diameter of only 2.2, the lighting efficiency of the optimized element exceeds 89.5%, whereas the uniformity of the resulting illuminance distribution exceeds 96%.

Novel Approach to Dynamic Soaring Modeling and Simulation

This paper revisits dynamic soaring on the basis of a nonlinear point-mass flight dynamics model previously used for scale-model aircraft to design path-following autopilots endowed with theoretically and experimentally demonstrated stability and convergence properties. The energy-harvesting process associated with specific maneuvers of a glider subjected to horizontal wind, and on which dynamic soaring relies, is explained in the light of this model. Expressions for the estimates of various variables involved in dynamic soaring along inclined circular paths crossing a thin wind shear layer, as experienced by model glider pilots over the world, are derived via approximate integration of the model equations. Given a glider’s path and a wind profile, this model also presents the asset of yielding an explicit ordinary differential equation that entirely characterizes the time evolution of the modeled glider’s state along the path, thus allowing for an easy simulation of dynamic soaring over a large variety of operating conditions. This simulation facility is viewed as a tool that usefully complements other studies of dynamic soaring that focus on trajectory optimization via dynamic programming. Its usefulness is here illustrated by first validating the aforementioned estimates in the case of circular trajectories crossing a thin wind shear layer, and then by showing how it applies to other examples of trajectories and ocean wind profile models commonly considered in studies about the dynamic soaring abilities of albatrosses.

Design of an axisymmetrical refractive optical element generating required illuminance distribution and wavefront.

The design of an axisymmetrical refractive optical element transforming a given incident beam into an output beam with prescribed illuminance distribution and wavefront is considered. The wavefront of the output beam is represented by the eikonal function defined in a certain plane behind the optical element. The design of the optical element is reduced to the solution of two explicit ordinary differential equations of the first order. These equations can be easily integrated using conventional numerical methods. As examples, we consider the design of two optical elements transforming a spherical beam from a point Lambertian light source into the uniform-illuminance beams with a plane wavefront and with a complex wavefront providing the subsequent focusing into a line segment on the optical axis.

Transforming linear FDEs with rational orders into ODEs by modified differential operator multiplication method

This paper presents a modified differential operator multiplication (DOM) method for solving a certain class of fractional differential equations (FDEs), with emphasis on linear oscillators subjected to periodic excitation. The main idea of DOM is to transform the considered FDEs with rational order r/m into rth-order ordinary differential equations (ODEs), herein r and m are positive integers. The transformation is realized by differentiating the FDEs stepwise with fractional-order r/m, until the rth-order derivative is reached due to the accumulative property for FDs. In the modified method, differently, first-order ODEs are deduced by transforming the FDEs with order r/m into 1/m. In addition, we introduce auxiliary state space variables to represent the FDs of inhomogeneous terms such as triangle functions. Exact and explicit first-order ODEs are deduced and solved by the Runge–Kutta algorithm. To validate the modified DOM, we solve the fractional Kelvin-Voigt equation and harmonically excited oscillators with visco-elastic dampings. The presented method can provide highly accurate solution, at the expense of linearly increasing computational resources as the solution domain expands. Such high computation accuracy and efficiency would potentially enable the presented method to become a benchmark comparison for solving certain FDEs, especially harmonically excited fractional oscillators.

Accelerating explicit ODE methods on GPUs by kernel fusion

SummaryGraphics processing units (GPUs) have a promising architecture for implementing highly parallel solution methods for systems of ordinary differential equations (ODEs). However, their high performance comes at the price of caveats such as small caches or wide SIMD. For ODE methods, optimizing the memory access pattern is often crucial. In this article, instead of considering only one specific method, we generalize the description of explicit ODE methods by using data flow graphs consisting of basic operations that are suitable to cover the types of computations occurring in all common explicit methods. After showing that the straightforward approach for processing the data flow graph by calling one kernel per basic operation is memory bound, we explain how the number of memory accesses can be reduced by the kernel fusion technique, which fuses several basic operations into one kernel. Moreover, we will present enabling transformations that allow additional fusions and thus can reduce the number of memory accesses even further. We apply these optimizations to three different classes of explicit ODE methods: embedded Runge–Kutta (RK) methods, parallel iterated RK (PIRK) methods, and peer methods. A detailed experimental evaluation on three modern GPUs showed speedups between 1.86 and 3.51 compared to unfused implementations.

Design of an axisymmetric optical element generating a prescribed illuminance distribution and wavefront

Design of an axisymmetric refracting optical element to transform an incident beam into an output beam with the prescribed illuminance distribution and prescribed wavefront is considered. The wavefront of the output beam is represented by the eikonal function defined in some plane behind the optical element. The computation of the optical element surface is reduced to the solution of two explicit ordinary differential equations of the first order. As examples, we consider the design of two optical elements to transform a spherical beam from a point Lambert light source into a beam with the uniform illuminance and plane wavefront and a beam with the complex wavefront to focus into an on-axis line.

Transcritical Bifurcation without Parameters in Memristive Circuits

The transcritical bifurcation without parameters (TBWP) describes a stability change along a line of equilibria, resulting from the loss of normal hyperbolicity at a given point of such a line. Memristive circuits systematically yield manifolds of nonisolated equilibria, and in this paper we address a systematic characterization of the TBWP in circuits with a single memristor. To achieve this we develop two mathematical results of independent interest; the first is an extension of the TBWP theorem to explicit ordinary differential equations (ODEs) in arbitrary dimension; the second result drives the characterization of this phenomenon to semiexplicit differential-algebraic equations (DAEs), which provide the appropriate framework for the analysis of circuit dynamics. In the circuit context the analysis is performed in graph-theoretic terms: in this setting, our first working scenario is restricted to passive problems (an exception is made for the bifurcating memristor), and in a second step some results are presented for the analysis of nonpassive cases. The latter context is illustrated by means of a memristive neural network model.