Truncation Error Estimates Research Articles

Where is my footprint located? Estimating the geographical variance of hybrid LCA footprints

AbstractCurrent implementations of hybrid life cycle assessment (LCA) mostly do not fully exploit valuable information from multi‐regional input–output databases by aggregating regional supply chains to the lower geographical resolution of process LCA databases. We propose a method for sampling the various individual regions within the aggregated regional scope of LCA processes. This sampling maximizes the information content of hybrid LCA footprint results by preserving the regional variance, and it allows for regional price distributions from BACI/UN‐COMTRADE international trade statistics to be used to simultaneously improve the accuracy of the hybrid model. This work makes the previously hidden regional and price variance explicit and analyzes uncertainty of the hybrid carbon (global warming potential 100, GWP100) and land use footprints arising from these variances, both separately and in combination. We find that the median process footprint intensity increases by for the GWP100 due to hybridization, and for the land use footprint. Results show that the magnitude of the footprint uncertainty strongly depends on the product sector of the LCA process and environmental impact considered. In a case study of Swiss household consumption, we find truncation error estimates of for the GWP100 and for the land use footprint. Our results highlight the importance of regionalization of process LCA databases, as it has the potential to significantly improve both the precision and accuracy of derived hybrid LCA models. This article met the requirements for a gold/gold JIE data openness badge described at http://jie.click/badges.

Adaptive mesh refinement in binary black holes simulations

We discuss refinement criteria for the Berger–Rigoutsos (block-based) refinement algorithm in our numerical relativity code GR-Athena++ in the context of binary black hole (BBH) merger simulations. We compare three different strategies: the ‘box-in-box’ approach, the ‘sphere-in-sphere’ approach and a local criterion for refinement based on the estimation of truncation error of the finite difference scheme. We extract and compare gravitational waveforms using the three different mesh refinement methods and compare their accuracy against a calibration waveform and demonstrate that the sphere-in-sphere approach provides the best strategy overall when considering computational cost and the waveform accuracy. Ultimately, we demonstrate the capability of each mesh refinement method in accurately simulating gravitational waves from BBH systems—a crucial aspect for their application in next-generation detectors. We quantify the mismatch achievable with the different strategies by extrapolating the gravitational wave mismatch to higher resolution.

A second order numerical method for singularly perturbed Volterra integro-differential equations with delay

Abstract This study deals with singularly perturbed Volterra integro-differential equations with delay. Based on the properties of the exact solution, a hybrid difference scheme with appropriate quadrature rules on a Shishkin-type mesh is constructed. By using the truncation error estimate techniques and a discrete analogue of Grönwall’s inequality it is proved that the hybrid finite difference scheme is almost second order accurate in the discrete maximum norm. Numerical experiments support these theoretical results and indicate that the estimates are sharp.

Fast Ewald summation for Stokes flow with arbitrary periodicity

A fast and spectrally accurate Ewald summation method for the evaluation of stokeslet, stresslet and rotlet potentials of three-dimensional Stokes flow is presented. This work extends the previously developed Spectral Ewald method for Stokes flow to periodic boundary conditions in any number (three, two, one, or none) of the spatial directions, in a unified framework. The periodic potential is split into a short-range and a long-range part, where the latter is treated in Fourier space using the fast Fourier transform. A crucial component of the method is the modified kernels used to treat singular integration. We derive new modified kernels, and new improved truncation error estimates for the stokeslet and stresslet. An automated procedure for selecting parameters based on a given error tolerance is designed and tested. Analytical formulas for validation in the doubly and singly periodic cases are presented. We show that the computational time of the method scales like O(Nlog⁡N) for N sources and targets, and investigate how the time depends on the error tolerance and window function, i.e. the function used to smoothly spread irregular point data to a uniform grid. The method is fastest in the fully periodic case, while the run time in the free-space case is around three times as large. Furthermore, the highest efficiency is reached when applying the method to a uniform source distribution in a primary cell with low aspect ratio. The work presented in this paper enables efficient and accurate simulations of three-dimensional Stokes flow with arbitrary periodicity using e.g. boundary integral and potential methods.

On the implementation of explicit two-step peer methods with Runge–Kutta stability

Explicit general linear method of two-step peer-type of order 1 to 4 suitable for solving non-stiff ordinary differential equations are derived. Estimates of the local truncation error are derived with which a variable stepsize and variable order scheme is implemented. The new methods are compared in numerical experiments with ode45 from Matlab.

Chiral uncertainties in ab initio elastic nucleon-nucleus scattering

The effective interaction between a nucleon and a nucleus is one of the most important ingredients for reaction theories. Theoretical formulations were introduced early by Feshbach and Watson, and efforts of deriving and computing those ‘optical potentials’ in a microscopic fashion have a long tradition. However, only recently the leading order term in the Watson multiple scattering approach could be calculated fully ab initio, meaning that the same nucleon-nucleon (NN) interaction enters both the structure as well as the reaction pieces on equal footing. This allows the uncertainties from the underlying chiral effective NN interaction to be systematically explored in nucleon-nucleus elastic scattering observables. In this contribution the main ingredients for arriving at the ab initio leading order of the effective nucleon-nucleus interaction in the Watson approach will be reviewed. Concentrating on one specific chiral NN interaction from the LENPIC collaboration and light nuclei with a 0+ ground state, the leading order nucleon-nucleus interaction is calculated using up to the third chiral order (N2LO) in the nucleon-nucleon potential, and elastic scattering observables are extracted. Then pointwise as well as correlated uncertainty quantification is used for the estimation of the chiral truncation error. Elastic scattering observables for 4He, 12C, and 16O for between 65 and 200 MeV projectile energy will be analyzed.

An Optimal Adaptive Grid Method Based on L1 Scheme for a Nonlinear Caputo Fractional Differential Equation

A nonlinear fractional differential equation with a Caputo derivative of order α is studied. This problem is discretized by using the L1 scheme on an arbitrary nonuniform mesh. By utilizing the Taylor expansion with integral remainder term, an optimal local truncation error estimation of L1 scheme is proved. Based on this truncation error estimation and the mesh equidistribution principle, a new monitor function is constructed to construct an adaptive grid generation algorithm. Numerical experiments are performed to confirm the accuracy of our new adaptive grid algorithm.

Estimation of Truncation Error in Statistical Description of Communication Signals over mm-Wave Channels

This paper presents an illustration of how knowledge from the field of special functions, orthogonal polynomials and numerical series can be applied to solve a very important problem in the field of modern wireless communications. We present the formulas for the probability density function (PDF) and cumulative distribution function (CDF) of the composite signal envelope over an mm-Wave channel. The formulas for the PDF and CDF are expressed in the convergent infinity series form. The main contribution of the paper is in estimating the upper bounds for absolute truncation error in evaluating PDF and CDF of the signal envelope. We also derive the formulas for the required number of terms in the summation under the condition of achieving a given accuracy for typical values of channel parameters. In deriving these formulas, we use the alternating series estimation theorem, as well as some properties of orthogonal polynomials in order to derive upper bounds for hypergeometric functions. Based on the newly derived formulas, numerical results are presented and commented upon. The analytical results are verified by Monte Carlo simulations. The results are essential in the designing and performance estimating of the fifth-generation (5G) and beyond wireless networks.

Spectral Collocation Algorithm for Solving Fractional Volterra-Fredholm Integro-Differential Equations via Generalized Fibonacci Polynomials

In this research article, we build and implement an efficient spectral algorithm for handling linear/nonlinear mixed Volterra-Fredholm integro-differential equations. First, we expand the exact solution as a truncated series of the generalized Fibonacci polynomials, and then we discretize the equation via Simpson's quadrature formula. Finally, we collocate the resulted residual at the roots of the shifted first-kind Chebyshev polynomials. Also, the rate of convergence is studied and the truncation error estimate is reported. Some numerical examples are exhibited to prove the applicability and accuracy of the algorithm.

A Novel Second-Order Adaptive Grid Method for Singularly Perturbed Convection–Diffusion Equations

In this paper, a singularly perturbed convection–diffusion equation is studied. At first, the original problem is transformed into a parameterized singularly perturbed Volterra integro-differential equation by using an integral transform. Then, a second-order finite difference method on an arbitrary mesh is given. The stability and local truncation error estimates of the discrete schemes are analyzed. Based on the mesh equidistribution principle and local truncation error estimation, an adaptive grid algorithm is given. In addition, in order to calculate the parameters of the transformation equation, a nonlinear unconstrained optimization problem is constructed. Numerical experiments are given to illustrate the effectiveness of our presented adaptive grid algorithm.

Average Sampling Theorems on Multidimensional Random Signals

In this paper, a lower bounded of sequences on classical sampling theorem for random signals is given and is used to extend the applicable range of frame. The convergence property of sampling series, the estimate of truncation error both in mean square sense and for almost sure results on sampling theorem for multidimensional random signals from asymmetrical average sampling are analyzed. Using the new results on frames recently, the famous Shannon sampling theorem on multidimensional random signals has been extended with a new idea.

Approximation of radiating waves in the near-field: Error estimates and application to a class of inverse problems

Numerical approximation of radiating waves, with a priori truncation parameter and error estimates, is crucial for efficient simulation of forward and inverse scattering models. Convergence of a series ansatz for the wave field using the classical radiating wave functions is known only when the field is evaluated exterior to a ball circumscribing the configuration. If the configuration comprises non-convex and/or elongated scatterers, evaluation of the scattered field in the interior region of the ball is important for applications including for the far-field-data based inverse problem of identifying the scatterer boundary. In this article we develop a new error estimate for the series ansatz that facilitates identification of the truncation-parameter dependent interior convergence region. This in turn facilitates an estimate-based approach for solving the boundary identification inverse problem. We demonstrate, through numerical experiments, excellent agreement of the theoretical error estimate with respect to the truncation parameter, and the efficiency of the approach to identify scatterer shapes.

Numerical integration of an elasto-plastic critical state model for soils under unsaturated conditions

This paper presents the complete set of incremental equations for the numerical integration of the Glasgow Coupled Model (GCM) and a comprehensive algorithm for its numerical integration. The incremental formulation proposed is expressed in terms of strain and suction increments (i.e. strain-driven) and defines an initial value problem (IVP) that can be solved once the initial state and the pair of increments of the driven variables are known. The numerical integration of this IVP is carried out by extending to unsaturated condition, the well-known explicit substepping formulation with automatic error control widely used for saturated soils. A notable feature of the substepping integration scheme presented is that it integrates simultaneously the model equations for both mechanical and water retention responses. Hence, the estimate of the local truncation error to automatically adjust the size of the integration step is not only affected by the local error in stresses and mechanical hardening parameter (as in a saturated soil model) but, additionally, by the local error incurred in the integration of the water retention relations (i.e. degree of saturation and water retention hardening parameter). The correctness of the integration scheme is then verified by comparison of computational outcomes against analytical/reference solutions.

Finite differences for higher order derivatives of low resolution data

The viability of finite difference stencils for higher-order derivatives of low-resolution data, encountered in various applications, is examined. This study’s illustrative example involves the evaluation of the fourth derivative of the digitized free surface radius obtained from pixelated images. Procedures to obtain an optimal approximation to the derivative are made from estimates of truncation and roundoff error, with emphasis on the analysis of roundoff error. A method of successive approximations to evaluate the optimal grid spacing is presented. Higher-order stencils allow for larger optimal grid spacing, thereby reducing the roundoff error that would otherwise dominate. Hence, higher-order stencils are effective for low precision data.

Implementation of Local Chiral Interactions in the Hyperspherical Harmonics Formalism

With the goal of using chiral interactions at various orders to explore the properties of the few-body nuclear systems, we write the recently developed local chiral interactions as spherical irreducible tensors and implement them in the hyperspherical harmonics expansion method. We devote particular attention to three-body forces at next-to-next-to leading order, which play an important role in reproducing experimental data. We check our implementation by benchmarking the ground-state properties of 3H, 3He, and 4He against the available Monte Carlo calculations. We then confirm their order-by-order truncation error estimates and further investigate uncertainties in the charge radii obtained by using the precise muonic atom data for single-nucleon radii. Having local chiral Hamiltonians at various orders implemented in our hyperspherical harmonics suites of codes opens up the possibility to test such interactions on other light-nuclei properties, such as electromagnetic reactions.

Adaptive Stepsize Control for Extrapolation Semi-Implicit Multistep ODE Solvers

Developing new and efficient numerical integration techniques is of great importance in applied mathematics and computer science. Among the variety of available methods, multistep ODE solvers are broadly used in simulation software. Recently, semi-implicit integration proved to be an efficient compromise between implicit and explicit ODE solvers, and multiple high-performance semi-implicit methods were proposed. However, the computational efficiency of any ODE solver can be significantly increased through the introduction of an adaptive integration stepsize, but it requires the estimation of local truncation error. It is known that recently proposed extrapolation semi-implicit multistep methods (ESIMM) cannot operate with existing local truncation error (LTE) estimators, e.g., embedded methods approach, due to their specific right-hand side calculation algorithm. In this paper, we propose two different techniques for local truncation error estimation and study the performance of ESIMM methods with adaptive stepsize control. The first considered approach is based on two parallel semi-implicit solutions with different commutation orders. The second estimator, called the “double extrapolation” method, is a modification of the embedded method approach. The introduction of the double extrapolation LTE estimator allowed us to additionally increase the precision of the ESIMM solver. Using several known nonlinear systems, including stiff van der Pol oscillator, as the testbench, we explicitly show that ESIMM solvers can outperform both implicit and explicit linear multistep methods when implemented with an adaptive stepsize.

Truncation error estimates for two‐dimensional sampling series associated with the linear canonical transform

The linear canonical transform (LCT) provides a more general framework for many well‐known linear integral transforms, such as Fourier transform, fractional Fourier transform, Fresnel transform, and so forth, in digital signal processing and optics. There has been a great deal of research on sampling expansions of bandlimited signals in the LCT domain. However, results on error estimation and convergence analysis appear to be relatively rare, especially for multi‐dimensional sampling series associated with the LCT. In this paper, we present error estimation and convergence analysis for two‐dimensional (2D) sampling series in the LCT domain. Specifically, we first prove the absolute convergence and uniform convergence of 2D LCT sampling series. Then, we analyze truncation error estimates for uniformly sampling 2D bandlimited signals in the LCT domain, as well as deriving three truncation error bounds. Finally, our theoretical results are demonstrated by numerical examples.

NOTE ON THE PROBLEM OF MOTION OF VISCOUS FLUID AROUND A ROTATING AND TRANSLATING RIGID BODY

We consider the linearized and nonlinear systems describing the motion of incompressible flow around a rotating and translating rigid body Ɗ in the exterior domain Ω = ℝ3 \ Ɗ, where Ɗ ⊂ ℝ3 is open and bounded, with Lipschitz boundary. We derive the L∞-estimates for the pressure and investigate the leading term for the velocity and its gradient. Moreover, we show that the velocity essentially behaves near the infinity as a constant times the first column of the fundamental solution of the Oseen system. Finally, we consider the Oseen problem in a bounded domain ΩR := BR ∩ Ω under certain artificial boundary conditions on the truncating boundary ∂BR, and then we compare this solution with the solution in the exterior domain Ω to get the truncation error estimate.

Artificial boundary conditions for linearized stationary incompressible viscous flow around rotating and translating body

AbstractWe consider the linearized and nonlinear stationary incompressible flow around rotating and translating body in the exterior domain , where is open and bounded, with Lipschitz boundary. We derive the pointwise estimates for the pressure in both cases. Moreover, we consider the linearized problem in a truncation domain of the exterior domain under certain artificial boundary conditions on the truncating boundary , and then compare this solution with the solution in the exterior domain to get the truncation error estimate.

Refactorization of the midpoint rule

An alternative formulation of the midpoint method is employed to analyze its advantages as an implicit second-order absolutely stable timestepping method. Legacy codes originally using the backward Euler method can be upgraded to this method by inserting a single line of new code. We show that the midpoint method, and a theta-like generalization, are B-stable. We outline two estimates of local truncation error that allow adaptive time-stepping.