Computer Roundoff Research Articles

Graph-Based Deep Decomposition for Overlapping Large-Scale Optimization Problems

Decomposition methods play a critical role in cooperative co-evolutionary algorithms (CCEAs) for solving large-scale optimization problems. Although some well-performing decomposition methods have been designed based on the interactions among variables (IaV), their grouping accuracy is still limited due to the poor performance on the overlapping problems and the computational roundoff errors of IaV in the implementation. To deal with these limitations, a graph-based deep decomposition (GDD) method is proposed to obtain more accurate grouping results, especially for the overlapping problems. On the one hand, the GDD mines the IaV information and obtains the minimum vertex separator of the interaction graph of variables, so as to group variables deeply and recursively. On the other hand, the GDD has the ability of fault tolerance to deal with the computational roundoff errors of IaV and can improve the grouping accuracy. For better experimental studies of overlapping problems, a novel overlapping function generator is designed with the random and complicate overlap type, and two new metrics are proposed to evaluate the grouping accuracy. Comprehensive experiments show that GDD can greatly improve the grouping accuracy and help CCEAs perform better than other existing algorithms, especially on the overlapping problems. In addition, the GDD is highly fault tolerant and can divide problems accurately even on the inaccurate IaV.

Equivalence of Two Solutions of Wahba’s Problem

Many attitude estimation methods are based on an optimization problem posed in 1965 by Grace Wahba. All these methods yield the same optimal estimate, except for inevitable computer roundoff errors. This note shows shows that Shuster’s Quaternion Estimator (QUEST) and Mortari’s Estimator of the Optimal Quaternion (ESOQ) are essentially identical even in the presence of roundoff errors. It also shows some connections between two other algorithms for solving Wahba’s problem: Davenport’s q method and the Singular Value Decomposition (SVD) method.

An algorithm for computing a Padé approximant with minimal degree denominator

In this paper, a new definition of a reduced Padé approximant and an algorithm for its computation are proposed. Our approach is based on the investigation of the kernel structure of the Toeplitz matrix. It is shown that the reduced Padé approximant always has nice properties which the classical Padé approximant possesses only in the normal case. The new algorithm allows us to avoid the appearance of Froissart doublets induced by computer roundoff in the non-normal Padé table.

A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing

An overview of a comprehensive framework is given for estimating the predictive uncertainty of scientific computing applications. The framework is comprehensive in the sense that it treats both types of uncertainty (aleatory and epistemic), incorporates uncertainty due to the mathematical form of the model, and it provides a procedure for including estimates of numerical error in the predictive uncertainty. Aleatory (random) uncertainties in model inputs are treated as random variables, while epistemic (lack of knowledge) uncertainties are treated as intervals with no assumed probability distributions. Approaches for propagating both types of uncertainties through the model to the system response quantities of interest are briefly discussed. Numerical approximation errors (due to discretization, iteration, and computer round off) are estimated using verification techniques, and the conversion of these errors into epistemic uncertainties is discussed. Model form uncertainty is quantified using (a) model validation procedures, i.e., statistical comparisons of model predictions to available experimental data, and (b) extrapolation of this uncertainty structure to points in the application domain where experimental data do not exist. Finally, methods for conveying the total predictive uncertainty to decision makers are presented. The different steps in the predictive uncertainty framework are illustrated using a simple example in computational fluid dynamics applied to a hypersonic wind tunnel.

Noise influence on pole solutions of the Sivashinsky equation for planar and outward propagating flames

The dynamics of planar and outward propagating cylindrical flames has been studied in terms of exact solutions of the Sivashinsky equation with a random force term. The force term models the computational roundoff errors or a variety of perturbations of physical origins. In contrast to noiseless conditions, the number of poles in the system does not conserve and new poles appear due to the external forcing. It was found that modification of the pole solutions taking into account the appearance of new poles captures the features typical for the hydrodynamically unstable flames, which cannot be detected by the pole solutions with a fixed number of poles. Investigations based on the pole solutions make it possible to exclude the uncontrolled numerical noise that is always present in direct computations of the Sivashinsky equation, and to examine the interplay between noises and hydrodynamic instability. The study clearly demonstrates that the presence of noises is a necessary condition for flame acceleration.

CosmoMHD: A Cosmological Magnetohydrodynamics Code

In this era of precision cosmology, a detailed physical understanding on the evolution of cosmic baryons is required. Cosmic magnetic fields, though still poorly understood, may represent an important component in the global cosmic energy flow that affects the baryon dynamics. We have developed an Eulerian-based cosmological magnetohydrodynamics code (CosmoMHD) with modern shock capturing schemes to study the formation and evolution of cosmic structures in the presence of magnetic fields. The code solves the ideal MHD equations as well as the non-equilibrium rate equations for multiple species, the Vlasov equation for dynamics of collisionless particles, the Poisson's equation for the gravitational potential field and the equation for the evolution of the intergalactic ionizing radiation field. In addition, a detailed star formation prescription and feedback processes are implemented. Several methods for solving the MHD by high-resolution schemes with finite-volume and finite-difference methods are implemented. The divergence-free condition of the magnetic fields is preserved at a level of computer roundoff error via the constraint transport method. We have also implemented a high-resolution method via dual-equation formulations to track the thermal energy accurately in very high Mach number or high Alfven-Mach number regions. Several numerical tests have demonstrated the efficacy of the proposed schemes.

Global Convergence of GHA Learning Algorithm With Nonzero-Approaching Adaptive Learning Rates

The generalized Hebbian algorithm (GHA) is one of the most widely used principal component analysis (PCA) neural network (NN) learning algorithms. Learning rates of GHA play important roles in convergence of the algorithm for applications. Traditionally, the learning rates of GHA are required to converge to zero so that its convergence can be analyzed by studying the corresponding deterministic continuous-time (DCT) equations. However, the requirement for learning rates to approach zero is not a practical one in applications due to computational roundoff limitations and tracking requirements. In this paper, nonzero-approaching adaptive learning rates are proposed to overcome this problem. These proposed adaptive learning rates converge to some positive constants, which not only speed up the algorithm evolution considerably, but also guarantee global convergence of the GHA algorithm. The convergence is studied in detail by analyzing the corresponding deterministic discrete-time (DDT) equations. Extensive simulations are carried out to illustrate the theory.

Robust stability analysis for discrete-time LQG system under finite wordlength effects, noise uncertainties and time-varying structured parameter perturbations

In this paper, a sufficient condition is proposed to analyze the robust stability of the discrete-time LQG system under linear time-varying structured parameter perturbations, noise uncertainties and finite wordlength effects. Digital controllers are realized with floating-point arithmetic. The robust stability condition presented in this paper can be used to analyze the stability of the discrete-time LQG systems with a linear time-varying perturbation matrix due to the combination of structured parameter perturbations, feedback gain quantization error, estimator gain quantization error and computational roundoff errors. An example is given to illustrate the application of the proposed robust stability criterion.

Substructure Analysis of a Flexible System Contact-Impact Event

In this paper, the application of the substructure methodology to contact-impact analysis of flexible multibody systems is validated. Various impact model parameters that affect the model’s accuracy are presented. A contact-impact system is used that consists of a flexible cantilever bar longitudinally struck at its free end by a rigid body moving at a finite velocity. First, a dynamic model using the substructure method is established. Second, the initial conditions of the system’s dynamic responses during contact-impact are derived. Finally, a numeric contact-impact simulation is performed. The excellent agreement between the numeric solutions to both the substructure model and the analytical solutions demonstrates that the substructure model can successfully describe stress wave propagation within flexible bodies during contact-impact. The method can also clearly display the contact force time history and deformation distribution along the bar during contact-impact time and correctly predict the displacement of the contact surface of the flexible bar and the contact duration of the two bodies. It is shown that a larger substructure number will improve the accuracy of the numerical solutions, but an excessive number will lower the model’s accuracy since increasingly fine substructures increase the number of modal coordinates and lead to more serious computational round off errors and longer computational time.

Robust stability measure for discrete‐time linear regulators with computational delays under finite wordlength effects and time‐varying parameter perturbations

A non‐Lyapunov approach is introduced to analyze the robust stability of discrete‐time linear regulators with computational delays under both finite wordlength effects and linear time‐varying parameter perturbations. A robust stability criterion (sufficient condition) presented in this paper can be used to obtain the stability bound on a linear time‐varying uncertainty matrix due to the combination of linear time‐varying parameter perturbations and computational roundoff errors. This bound provides a quantitative measure concerning robustness for control engineers to design actual discrete‐time linear regulators with computational delays. An example is given to illustrate the application of the proposed robust stability criterion.

Fast Computation of MD-DCT-IV/MD-DST-IV by MD-DWT or MD-DCT-II

This paper reveals relationships between the type-IV multidimensional discrete cosine transform (MD-DCT-IV) and the type-II multidimensional discrete cosine transform (MD-DCT-II) or the multidimensional discrete W transform (MD-DWT). Based on the relationships, MD-DCT-IV can be efficiently computed by using the fast algorithms for MD-DCT-II or MD-DWT-III or MD-DWT-III. Efficient method for implementation of the post-addition or pre-addition process is also presented. Compared to the widely used row-column method, considerable savings on the number of arithmetic operations are achieved. For example, the number of multiplications for computing an r-dimensional DCT-IV is only about $\frac{1}{r}$ times that needed by the row-column method, and the number of additions is also reduced. Numerical experiments are made to test the computational time and roundoff errors. They show that the proposed fast algorithms save time.

Two New Mixture Models: Living with Collinearity but Removing its Influence

Fitting equations to mixture data collected from highly constrained regions has challenged modelers for the past 35 years. Computer roundoff error, low precision of the estimates, and collinearity among the terms in the models are just a few of the problems encountered. It is well known that, due to collinearity, fitted models can contain coefficient estimates with magnitudes many hundreds or even a thousand times greater that the magnitude of the data values themselves. This can be very problematic to the modeler trying to convince a client the model is adequate. Fitting models in pseudocomponents with and without centering the terms has been one of the strategies used in an effort to reduce the effect of collinearity, but often to no avail. We introduce two additional model forms where the terms are scaled, and we illustrate the benefits of fitting the new models using two numerical examples. The equivalence among models of four different, but related, component systems is shown for the first time.

Sensitivity of variable angle spectroscopic ellipsometry to isotropic thin film properties

Ellipsometry involves the solution of complicated mathematical relations between the measured data and the desired physical parameters. Dealing with these equations remains a central issue in the method, especially in light of the increased measurement speed and data set size. Here the sensitivity of ellipsometry measurements to parameters of interest for isotropic thin films is investigated as a first step to developing an ellipsometry expert system. A general approach to selecting incidence angles and light wavelengths is demonstrated for the particular case of ZrO 2 films sputter deposited onto fused silica substrates. At first the measurements must be sensitive to changes in the parameter of interest and, beyond that, the system of equations must be sufficiently well posed considering measurement error and computer truncation and roundoff. Simulations of the ZrO 2 film show that the incidence angles for best sensitivity do not correspond to those for the best equation conditions. This method can be applied to any reflecting surface to assist in the selection of angles and wavelengths. It is planned that a very large number of such simulations covering an increasing number of reflecting surface materials and configurations will be performed and stored in a high performance database for easy access. Such progress in the development of expert systems to make the equations work in the background for all but the most inquisitive ellipsometrist is described.

A new trigonometric functions set for predicting plate bending, using the hierarchical finite element method

This communication proposes a new hierarchical functions set to predict flexural motion of platelike structures in the medium-frequency range. This functions set is built from trigonometric functions instead of polynomials as classically encountered in the literature. It is shown that such a trigonometric set presents all the advantages of a classical hierarchical polynomials set and additional ones which are of interest if very high-order functions are desired to be used. It is stated that this new trigonometric set can be used at very high orders, up to 2048, without taking care of computer roundoff errors, while the polynomials set fails at order 46 because of the limited numerical dynamic of computers. This trigonometric set can be easily implemented on a computer. It does not require quadruple precision precomputed arrays. Only a very low number (which do not depend on the function order) of basic operations is needed when calling such functions. Moreover, it is shown that this trigonometric set presents a better convergence rate than polynomials when predicting high-order natural flexural modes of rectangular plates with any boundary conditions.

PROCESSING OF SIMULTANEOUS MECHANICAL RANDOM RESPONSE SIGNALS: INTEGRATION, DIFFERENTIATION AND PHASE SHIFTS CORRECTION

Several experimental system identification methods require simultaneous sampling of mechanical quantities. The paper discusses problems connected with the use of multisensor set-ups and problems arising when integration or differentiation processes are employed to obtain unmeasured time histories and reduce the number of sensors. Three orders of problems related to integration or differentiation procedures must be taken into account: intrinsic problems, numerical algorithm problems and computer round off problems. Experimental and numerical details to obtain reliable data are given in the paper. The estimated data are compared with the exact experimental data using two different experimental set-ups. To avoid phase shift among measured time histories and to obtain real-time acquisition, no filtering of transducer output signals was carried out; in spite of this, phase shifts among measured time histories have been detected. A complete analysis of the frequency response of the transducers showed constant phase delays between different channels; consequently the data have been time shifted in order to obtain phase coherence.

Numerical issues affecting vortex asymmetries computed with the conical Navier-Stokes equations

An implicit upwind symmetric factorization finite volume solver for the conical Navier-Stokes equations was used to compute asymmetric vortices about a 5 ° cone at a 20 ° angle of attack. Asymmetric vortices were computed with no asymmetric disturbances other than computational roundoff errors. Variations in grid resolution, flux limiters and grid symmetry were studied. A brief proof of algorithm symmetry is included as an appendix. The results indicate that numerical dissipation in the form of flux limiters or poor grid resolution can prevent the formation of asymmetric vortices in a manner similar to that shown by Thomas [15] [J. Thomas, Reynolds number effects on supersonic asymmetrical flows over a cone. J. Aircraft 30, 4 (1993)] for viscous dissipation.

Robust Stability Bound on Linear Time-Varying Uncertainties for Linear Digital Control Systems under Finite Wordlength Effects

An approach is introduced to analyze the robust stability of linear digital control systems under both finite wordlength effects and linear time-varying parameter perturbations. Digital controllers are realized with floating-point arithmetic. The robust stability criterion (sufficient condition) presented in this paper can be used to obtain the stability bound on a linear time-varying uncertainty matrix due to the combination of parameter perturbations, feedback gain quantization error and computational roundoff errors. This bound provides a quantitative measure for control engineers to design an actual closed-loop digital control system with robustness. An example is given to illustrate the application of the proposed robust stability criterion.

Padé approximations in noise filtering

We first recall the results on the distribution of poles and zeros of Padé approximations to noisy Taylor series around a point. We then report the numerical experiment on noisy data coming from inelastic electron-scattering cross-sections on rare gases, as functions of the momentum transfer at fixed energy. We apply the Thiele algorithm (Padé approximations of type II) to those data, which are known to belong to an analytic function in a cut plane from −∞ to 0. In this case, the poles and zeros split into three families: (i) Doublet pole-zero at a distance of the order of the computer roundoff. (ii) Doublet pole-zero at a distance of the order of the noise in the experimental data. (iii) Remaining poles and zeros compatible with the analytic structure of the experimentally measured function. By throwing away the first two families of poles and zeros (noisy ones), we were able to obtain an excellent filtering of the data compatible with the analytic structure of the analyzed function.

On the Hamiltonian interpolation of near-to-the identity symplectic mappings with application to symplectic integration algorithms

We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Moser's scheme, we prove that for any mapping ψe, analytic and e-close to the identity, there exists an analytic autonomous Hamiltonian system, He such that its time-one mapping ΦHe differs from ψe by a quantity exponentially small in 1/e. This result is applied, in particular, to the problem of numerical integration of Hamiltonian systems by symplectic algorithms; it turns out that, when using an analytic symplectic algorithm of orders to integrate a Hamiltonian systemK, one actually follows “exactly,” namely within the computer roundoff error, the trajectories of the interpolating Hamiltonian He, or equivalently of the rescaled Hamiltonian Ke=e-1He, which differs fromK, but turns out to be e5 close to it. Special attention is devoted to numerical integration for scattering problems.

Likelihood and Bayesian Prediction of Chaotic Systems

Abstract There has recently been considerable interest in both applied disciplines and in mathematics, as well as in the popular science literature, in the areas of nonlinear dynamical systems and chaotic processes. By a nonlinear, deterministic dynamical system, we mean a time series in which, starting at some initial condition, the values of the series are some fixed, nonlinear function of the previous states. One of the more intriguing aspects of these models is their propensity for displaying very complex, apparently random behavior, even when simple models are analyzed. A consequence of such chaotic behavior is that it is difficult to predict the exact behavior of a chaotic system. The difficulty in prediction stems from the fact that even the tiniest of errors, including computer roundoff, in either the specification of the function or the initial condition, can lead to huge errors in prediction. After a brief review of dynamical systems and the role of probability in dealing with uncertainty, a common example of a simple dynamical system, the logistic map, is introduced. In Section 2 statistical prediction for a dynamical system, based on observing a portion of a realization of the system with error, is considered. The emphasis is on likelihood and Bayesian approaches to the problem of prediction. Example computations, based on simulated data, suggest some novel characteristics of chaotic data analysis. Section 3 is an excursus into the relationships between ergodic theory and chaos, with specific application to Bayesian forecasting. In Section 4 a common numerical approach, which generates discrete-time dynamical systems, to the solution of differential equations is described briefly. An example involving the so-called Duffing equation is presented.