Accurate Integrator Research Articles

AMFR-W-methods for parabolic problems with mixed derivates. Applications to the Heston model

The combination of Approximate Matrix Factorization (AMF), W-methods and iterative refinement in the solution of linear systems leads to the definition of AMFR-W-methods. This method class provides stable and accurate time integrators for parabolic PDEs with mixed derivatives discretized in space by means of Finite Differences (or Finite Volumes) in an arbitrary number of spatial dimensions. When the coefficients of the PDE actually depend on the spatial variables, the approximation of the pure diffusion coefficients by its respective maximum value produces simplified AMFR-W-methods requiring only a reduced number of LU decompositions of banded matrices with small bandwidth. The new class of methods is shown to be unconditionally stable regardless of the spatial dimension on a linear test problem relevant for homogeneous or periodic boundary conditions. Furthermore, high orders of convergence in PDE sense are observed when homogeneous boundary conditions are assumed. For general Robin boundary conditions, a simple algorithm is provided to convert a PDE problem into one where such conditions are homogeneous. Numerical experiments with the new simplified AMFR-W-methods on a linear parabolic problem with variable coefficients and the Heston problem from financial option pricing are presented.

A robust nonsmooth generalized-$\alpha $ scheme for flexible systems with impacts

The aim of this work is the development of a robust and accurate time integrator for the simulation of the dynamics of multibody systems composed of rigid and/or flexible bodies subject to frictionless contacts and impacts. The integrator is built upon a previously developed nonsmooth generalized-$\alpha $ scheme time integrator which was able to deal well with nonsmooth dynamical problems avoiding any constraint drift phenomena and capturing vibration effects without introducing too much numerical dissipation. However, when dealing with problems involving nonlinear bilateral constraints and/or flexible elements, it is necessary to adopt small time-step sizes to ensure the convergence of the numerical scheme. In order to tackle these problems more efficiently, a fully decoupled version of the nonsmooth generalized-$\alpha $ method is proposed in this work, avoiding these inconveniences. Several examples are considered to assess its accuracy and robustness.

Who should make decision on this pull request? Analyzing time-decaying relationships and file similarities for integrator prediction

In pull-based development model, integrators are responsible for making decisions about whether to accept pull requests and integrate code contributions. Ideally, pull requests are assigned to integrators and evaluated within a short time after their submissions. However, the volume of incoming pull requests is large in popular projects, and integrators often encounter difficulties in processing pull requests in a timely fashion. Therefore, an automatic integrator prediction approach is required to assign appropriate pull requests to integrators. In this paper, we propose an approach TRFPre which analyzes Time-decaying Relationships and File similarities to predict integrators. We evaluate the effectiveness of TRFPre on 24 projects containing 138,373 pull requests. Experimental results show that TRFPre makes accurate integrator predictions in terms of accuracies and Mean Reciprocal Rank. Less than 2 predictions are needed to find correct integrator in 91.67% of projects. In comparison with state-of-the-art approaches cHRev, WRC, TIE, CoreDevRec and ACRec, TRFPre improves top-1 accuracy by 68.2%, 73.9%, 49.3%, 14.3% and 46.4% on average across 24 projects.

Pushing the frontiers of modeling excited electronic states and dynamics to accelerate materials engineering and design

Electronic excitations and their dynamics are oftentimes at the foundation of how we use and probe materials. While recent experimental advances allow us to do so with unprecedented accuracy and time resolution, their interpretation relies on solid theoretical understanding. This can be provided by cutting-edge, first-principles theoretical-spectroscopy based on many-body perturbation theory (MBPT) and time-dependent density functional theory (TDDFT). In this work we review some of our recent results as successful examples for how electronic-structure methods lead to interesting insight into electronic excitations and deep understanding of modern materials. In many cases these techniques are accurate and even predictive, yet they rely on approximations to be computationally feasible. We illustrate the need for further theoretical understanding, using dielectric screening as an example in MBPT and faster, more accurate numerical integrators as a challenge for real-time TDDFT. Finally, we describe how incorporating online databases into computational materials research on excited electronic states can side-step the problem of high computational cost to facilitate materials design.

Uniformly Accurate Oscillatory Integrators for the Klein--Gordon--Zakharov System from Low- to High-Plasma Frequency Regimes

We present a novel class of oscillatory integrators for the Klein--Gordon--Zakharov system which are uniformly accurate with respect to the plasma frequency $c$. Convergence holds from the slowly v...

Accurate dissipative forces in optimization integrators

We propose a method for accurately simulating dissipative forces in deformable bodies when using optimization-based integrators. We represent such forces using dissipation functions which may be nonlinear in both positions and velocities, enabling us to model a range of dissipative effects including Coulomb friction, Rayleigh damping, and power-law dissipation. We propose a general method for incorporating dissipative forces into optimization-based time integration schemes, which hitherto have been applied almost exclusively to systems with only conservative forces. To improve accuracy and minimize artificial damping, we provide an optimization-based version of the second-order accurate TR-BDF2 integrator. Finally, we present a method for modifying arbitrary dissipation functions to conserve linear and angular momentum, allowing us to eliminate the artificial angular momentum loss caused by Rayleigh damping.

Linear instability of the Peregrine breather: Numerical and analytical investigations

We study the linear stability of the Peregrine breather both numerically and with analytical arguments based on its derivation as the singular limit of a single-mode spatially periodic breather as the spatial period becomes infinite. By constructing solutions of the linearization of the nonlinear Schrödinger equation in terms of quadratic products of components of the eigenfunctions of the Zakharov–Shabat system, we show that the Peregrine breather is linearly unstable. A numerical study employing a highly accurate Chebychev pseudo-spectral integrator confirms exponential growth of random initial perturbations of the Peregrine breather.

A uniformly and optimally accurate multiscale time integrator method for the Klein–Gordon–Zakharov system in the subsonic limit regime

We present a uniformly and optimally accurate numerical method for discretizing the Klein–Gordon–Zakharov system (KGZ) with a dimensionless parameter 0<ε≤1, which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., 0<ε≪1, the solution of KGZ system propagates waves with O(ε)- and O(1)-wavelength in time and space, respectively, and rapid outspreading initial layers with speed O(1∕ε) in space due to the singular perturbation of the wave operator in KGZ and/or the incompatibility of the initial data. Based on a multiscale decomposition by frequency and amplitude, we propose a multiscale time integrator Fourier pseudospectral method by applying the Fourier spectral discretization for spatial derivatives followed by using the exponential wave integrator in phase space for integrating the decomposed system at each time step. The method is explicit and easy to be implemented. Extensive numerical results show that the MTI-FP method converges optimally in both space and time, with exponential and quadratic convergence rate, respectively, which is uniformly for ε∈(0,1]. Finally, the method is applied to study the convergence rates of the KGZ system to its limiting models in the subsonic limit and wave dynamics and interactions of the KGZ system in 2D.

A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime

A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and rigorously analyzed for the nonlinear Dirac equation (NLDE), which involves a dimensionless parameter ε ∈ (0, 1] inversely proportional to the speed of light. The solution to the NLDE propagates waves with wavelength O (ε2) and O (1) in time and space, respectively. In the nonrelativistic regime,i.e., 0 &lt; ε ≪ 1, the rapid temporal oscillation causes significantly numerical burdens, making it quite challenging for designing and analyzing numerical methods with uniform error bounds inε ∈ (0, 1]. The key idea for designing the MTI-FP method is based on adopting a proper multiscale decomposition of the solution to the NLDE and applying the exponential wave integrator with appropriate numerical quadratures. Two independent error estimates are established for the proposed MTI-FP method as hm0+τ2/ε2andhm0 + τ2 + ε2, where his the mesh size, τis the time step and m0depends on the regularity of the solution. These two error bounds immediately suggest that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O (τ) for all ε ∈ (0, 1] and optimally with quadratic convergence rate at O (τ2) in the regimes when either ε = O (1) or 0 &lt; ε ≲ τ. Numerical results are reported to demonstrate that our error estimates are optimal and sharp.

A Variational Integrator for the distance-based formation control of multi-agent systems

Abstract Distance-based formation control of second order agents can be seen as a physical system of particles linked by springs, whose evolution can be described by a Lagrangian function. An interesting family of geometric integrators, called variational integrators, is defined by using discretizations of the Hamilton’s principle of critical action. The variational integrators preserve some geometric features such as the symplectic structure, they preserve the momentum map, and the evolution of the system’s energy presents a good (bounded) behavior. We derive variational integrators that can be employed in the context of distance-based formation control algorithms. In particular, we provide an accurate numerical integrator with a lower computational cost than traditional solutions. Consequently, we can provide a faster identification of regions of attraction for desired distance-based shapes, and more computationally efficient estimation algorithms like Kalman filters that employ distance-based controllers as prediction models. We use a formation consisting of four autonomous planar agents as an example and benchmark to test and compare the performances of the proposed variational integrator.

Uniformly accurate exponential-type integrators for Klein-Gordon equations with asymptotic convergence to the classical NLS splitting

We introduce efficient and robust exponential-type integrators for Klein-Gordon equations which resolve the solution in the relativistic regime as well as in the highly-oscillatory non-relativistic regime without any step-size restriction, and under the same regularity assumptions on the initial data required for the integration of the corresponding limit system. In contrast to previous works we do not employ any asymptotic/multiscale expansion of the solution. This allows us derive uniform convergent schemes under far weaker regularity assumptions on the exact solution. In particular, the newly derived exponential-type integrators of first-, respectively, second-order converge in the non-relativistic limit to the classical Lie, respectively, Strang splitting in the nonlinear Schrodinger limit.

Example-based damping design

To date, material modeling in physically based computer animation has largely focused on mass and stiffness material properties. However, deformation dynamics is largely affected also by the damping properties. In this paper, we propose an interactive design method for nonlinear isotropic and anisotropic damping of complex three-dimensional solids simulated using the Finite Element Method (FEM). We first give a damping design method and interface whereby the user can set the damping properties so that motion aligned with each of a few chosen example deformations is damped by an independently prescribed amount, whereas the rest of the deformation space follows standard Rayleigh damping, or any viscous damping. Next, we demonstrate how to design nonlinear damping that depends on the magnitude of the deformation along each example deformation, by editing a single spline curve for each example deformation. Our user interface enables an art-directed and intuitive approach to controlling damping in solid simulations. We mathematically prove that our nonlinear anisotropic damping generalizes the frequency-dependent Caughey damping model, when starting from the Rayleigh damping. Finally, we give an inverse design method whereby the damping curve parameters can be inferred automatically from high-level user input, such as the amount of amplitude loss in one oscillation cycle along each of the chosen example deformations. To minimize numerical damping for implicit integration, we introduce an accurate and stable implicit integrator, which removes spurious high-frequency oscillations while only introducing a minimal amount of numerical damping. Our damping can generate effects not possible with previous methods, such as controllable nonlinear decaying envelopes whereby large deformations are damped faster or slower than small deformations, and damping anisotropic effects. We also fit our damping to videos of real-world objects undergoing large deformations, capturing their nonlinear and anisotropic damping dynamics.

A stiffly accurate integrator for elastodynamic problems

We present a new integration algorithm for the accurate and efficient solution of stiff elastodynamic problems governed by the second-order ordinary differential equations of structural mechanics. Current methods have the shortcoming that their performance is highly dependent on the numerical stiffness of the underlying system that often leads to unrealistic behavior or a significant loss of efficiency. To overcome these limitations, we present a new integration method which is based on a mathematical reformulation of the underlying differential equations, an exponential treatment of the full nonlinear forcing operator as opposed to more standard partially implicit or exponential approaches, and the utilization of the concept of stiff accuracy which ensures that the efficiency of the simulations is significantly less sensitive to increased stiffness. As a consequence, we are able to tremendously accelerate the simulation of stiff systems compared to established integrators and significantly increase the overall accuracy. The advantageous behavior of this approach is demonstrated on a broad spectrum of complex examples like deformable bodies, textiles, bristles, and human hair. Our easily parallelizable integrator enables more complex and realistic models to be explored in visual computing without compromising efficiency.

CCII and RC fractance based fractional order current integrator

Integrators are an important functional module of several filters, PID controllers and automated systems. Designing these integrators in fractional domain enhance their operations and make the responses highly precise and accurate. This paper presents a Fractional Order Current Integrator using second generation current conveyor as the active block, and fractional capacitor as a grounded fractance. Analog realization of this fractance comprises of resistances and capacitances arranged in parallel RC ladder topology. The motivation behind this work is that more accurate and stable fractional integrator with fewer passive elements, for lower bias voltage and high dynamic range, can be designed and implemented. Integrators of fractional orders from 0.1 to 0.9 are simulated using TSMC 0.25µm technology parameters. The transient and frequency responses obtained in Mentor Graphics are in close conformity with the theoretical values of magnitude and phase. Robustness of the proposed model is verified by performing Monte Carlo analysis in time and in frequency domain. Comparisons of fractional order current integrators with existing analog fractance models have also been included to further validate the work presented in the paper.

Uniformly accurate multiscale time integrators for second order oscillatory differential equations with large initial data

We apply the modulated Fourier expansion to a class of second order differential equations which consists of an oscillatory linear part and a nonoscillatory nonlinear part, with the total energy of the system possibly unbounded when the oscillation frequency grows. We comment on the difference between this model problem and the classical energy bounded oscillatory equations. Based on the expansion, we propose the multiscale time integrators to solve the ODEs under two cases: the nonlinearity is a polynomial or the frequencies in the linear part are integer multiples of a single generic frequency. The proposed schemes are explicit and efficient. The schemes have been shown from both theoretical and numerical sides to converge with a uniform second order rate for all frequencies. Comparisons with popular exponential integrators in the literature are done.

Empirical manipulation of the thermoinactivation kinetics of Bacillus amyloliquefaciens and Bacillus licheniformis α-amylases for thermal process evaluations

Abstract In an attempt to obtain more accurate and stable time and temperature integrators (TTIs) for the evaluation of commercial pasteurisation process, the thermoinactivation kinetics (D and z value) of amylases contained in the TTIs from two sources, Bacillus amyloliquefaciens (BAA) and Bacillus licheniformis (BLA) were determined under different TTI conditions ( i.e ., pH, Ca 2+ , and enzyme concentration). Under 21 different TTI conditions tested for BAA, D 70 value is found to vary from 5.9 to 159.9 min, D 85 value vary from 4.5 to 40.5 min, and z value vary from 6.2 to 30.4°C, respectively; Under 14 different TTI conditions tested for BLA, D 90 value is found to vary from 2.5 to 9.9 min, and z value vary from 9.3 to 17.4°C, respectively. The thermoinactivation kinetics of TTI has been successfully modified to match the thermoinactivation kinetics of target microorganisms in commercial mild pasteurisation processes through these manipulations. We believe this empirical approach of manipulation has provided practical data and a better understanding of how to obtain desired thermoinactivation kinetics. This makes the enzymes and therefore TTIs better candidates for quantifying the thermal impact on both safety and quality parameters in commercial pasteurisation process. Industrial relevance Quantification of the thermal process is of paramount importance to ensure the safety and quality of the heat-preserved food. Bacillus amylase-based time and temperature integrators (TTIs) have been developed as an alternative tool for thermal process evaluations in food industries, where conventional thermal sensors can not apply. Based on the results of this study, it is possible to manipulate the thermoinactivation kinetics of both Bacillus amyloliquefaciens and Bacillus licheniformis amylases to match the thermoinactivation kinetics of the target microorganisms in commercial pasteurisation process. Manipulation of the D and z value may be required when the supplied enzyme batch changes. The ability to undertake this manipulation contributes significantly to producing more accurate and consistent time and temperature integrators for its commercial use ( e.g ., thermal process validation). Moreover, being able to manipulate amylase activity to achieve target D and z values may also make it a potential candidate for the development of TTIs to monitor quality and nutritional markers. Consequently, TTIs can be developed to quantify the thermal impact on quality and nutritional parameters in pasteurised products.

Conservation properties of higher order time stepping schemes based on variational integrators

AbstractThis paper deals with higher order accurate variational integrators for finite element systems. The variational integrator (VI) is based on higher order LAGRANGE polynomials as shape functions and a higher order GAUSSIAN quadrature rule. The goals of this paper are to implement a discrete gradient to preserve the balance of total energy and fulfill the constraints with the LAGRANGE multiplier method and a NEWTON‐COTÊS quadrature rule. We show the calculation of bearing forces from the LAGRANGE multipliers, which are essential for the balance of total linear momentum and the balance of total angular momentum. (© 2016 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Variational integration in endochronic theory for small strain elastoplastodynamics

AbstractThis paper presents a basic endochronic plasticity model with isotropic hardening for small strain theory according to Valanis. The key point of this model is a convolution integral over an intrinsic time scale involving past values of the strain measure and a so‐called memory kernel which leads to a smooth evolution equation for the internal variable. For the temporal discretization of the underlying constitutive equations and the resulting evolution equation we use higher order accurate variational integrators (VI). The remarkable feature is the fact that we approximate the position vector in terms of velocities according to partitioned Runge‐Kutta methods (pRK). As a representative model problem serves a quasistatic, uniaxial tensile testing as well as a dynamic, elastoplastic cantilever beam with a smooth plasticity model according to the Valanis framework. (© 2016 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

A uniformly accurate multiscale time integrator spectral method for the Klein–Gordon–Zakharov system in the high-plasma-frequency limit regime

A multiscale time integrator sine pseudospectral (MTI-SP) method is presented for discretizing the Klein–Gordon–Zakharov (KGZ) system with a dimensionless parameter 0<ε≤1, which is inversely proportional to the plasma frequency. In the high-plasma-frequency limit regime, i.e. 0<ε≪1, the solution of the KGZ system propagates waves with amplitude at O(1) and wavelength at O(ε2) in time and O(1) in space, which causes significantly numerical burdens due to the high oscillation in time. The main idea of the numerical method is to carry out a multiscale decomposition by frequency (MDF) to the electric field component of the solution at each time step and then apply the sine pseudospectral discretization for spatial derivatives followed by using the exponential wave integrator in phase space for integrating the MDF and the equation of the ion density component. The method is explicit and easy to be implemented. Extensive numerical results show that the MTI-SP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O(τ) for ε∈(0,1] with τ time step size and optimally with quadratic convergence rate at O(τ2) in the regime when either ε=O(1) or 0<ε≤τ. Thus the meshing strategy requirement (or ε-scalability) of the MTI-SP for the KGZ system in the high-plasma-frequency limit regime is τ=O(1) and h=O(1) for 0<ε≪1, which is significantly better than classical methods in the literatures. Finally, we apply the MTI-SP method to study the convergence rates of the KGZ system to its limiting models in the high-plasma-frequency limit and the interactions of bright solitons of the KGZ system, and to identify certain parameter regimes that the solution of the KGZ system will be blow-up in one dimension.

A uniformly accurate (UA) multiscale time integrator Fourier pseudospectral method for the Klein–Gordon–Schrödinger equations in the nonrelativistic limit regime

A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and analyzed for solving the Klein-Gordon-Schr\"{o}dinger (KGS) equations in the nonrelativistic limit regime with a dimensionless parameter $0<\varepsilon\le1$ which is inversely proportional to the speed of light. In fact, the solution to the KGS equations propagates waves with wavelength at $O(\varepsilon^2)$ and $O(1)$ in time and space, respectively, when $0<\varepsilon\ll 1$, which brings significantly numerical burdens in practical computation. The MTI-FP method is designed by adapting a multiscale decomposition by frequency to the solution at each time step and applying the Fourier pseudospectral discretization and exponential wave integrators for spatial and temporal derivatives, respectively. We rigorously establish two independent error bounds for the MTI-FP at $O(\tau^2/\varepsilon^2+h^{m_0})$ and $O(\varepsilon^2+h^{m_0})$ for $\varepsilon\in(0,1]$ with $\tau$ time step size, $h$ mesh size and $m_0\ge 4$ an integer depending on the regularity of the solution, which imply that the MTI-FP converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at $O(\tau)$ for $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regime when either $\varepsilon=O(1)$ or $0<\varepsilon\le \tau$. Thus the meshing strategy requirement (or $\varepsilon$-scalability) of the MTI-FP is $\tau=O(1)$ and $h=O(1)$ for $0<\varepsilon\ll 1$, which is significantly better than classical methods. Numerical results demonstrate that our error bounds are optimal and sharp. Finally, the MTI-FP method is applied to study numerically convergence rates of the KGS equations to the limiting models in the nonrelativistic limit regime.