Numerical Ill-conditioning Research Articles

Advancing the temporal direct deconvolution method with spatial regularization

This study continues the exploration of temporal large-eddy simulation, particularly the extension of the temporal direct deconvolution method (TDDM) with a regularization term based on spatial dissipation. Furthermore, we aim to put insight stemming from previous work to test. Specifically, the hypothesis is that the temporal residual-stress leads to a reduction of the required artificial dissipation in under-resolved simulations. Moreover, this work seeks corroborate earlier discoveries with a posteriori results. We perform a numerical examination of two different spatial regularization terms in conjunction with TDDM: a spatial variant of selective frequency damping, functioning as a relaxation term that gradually drifts the velocity toward the filtered velocity, and the dynamic Smagorinsky model incorporating a prefactor. We test various cases, including the Taylor–Green vortex flow with a Reynolds number of Re = 3000, forced homogeneous isotropic turbulence with Reλ=200, turbulent channel flow at Reτ=590, and the flow over a periodic hill with Re = 10 935. Additionally, we also analyze the various dissipation contributions in TDDM as well as their interrelations. We also discuss grid artifacts and energy budget errors using these to compare the different models. Our results confirm the hypothesis that residual-stress dissipation reduces the necessary artificial dissipation. Because of the numerical ill-conditioning of deconvolution, whether temporal or spatial, there are practical limitations in the size of the filter width. Due to these limitations, the impact remains relatively minor. The a posteriori results of the new spatial regularization term show it to be effective in eliminating energy from the high wavenumber range.

A Globalized and Preconditioned Newton-CG Solver for Metric-Aware Curved High-Order Mesh Optimization

We present a specific-purpose globalized and preconditioned Newton-CG solver to minimize a metric-aware curved high-order mesh distortion. The solver is specially devised to optimize curved high-order meshes for high polynomial degrees with a target metric featuring non-uniform sizing, high stretching ratios, and curved alignment — exactly the features that stiffen the optimization problem. To this end, we consider two ingredients: a specific-purpose globalization and a specific-purpose Jacobi-iLDLT(0) preconditioning with varying accuracy and curvature tolerances (dynamic forcing terms) for the CG method. These improvements are critical in stiff problems because, without them, the large number of non-linear and linear iterations makes curved optimization impractical. First, to enhance the global convergence of the non-linear solver, the globalization strategy modifies Newton’s direction to a feasible step. In particular, our specific-purpose globalization strategy memorizes the length of the feasible step (step-length continuation) between the optimization iterations while ensuring sufficient decrease and progress. Second, to compute Newton’s direction in second-order optimization problems, we consider a conjugate-gradient iterative solver with specific-purpose preconditioning and dynamic forcing terms. To account for the metric stretching and alignment, the preconditioner uses specific orderings for the mesh nodes and the degrees of freedom. We also present a preconditioner switch between Jacobi and iLDLT(0) preconditioners to control the numerical ill-conditioning of the preconditioner. In addition, the dynamic forcing terms determine the required accuracy for the Newton direction approximation. Specifically, they control the residual tolerance and enforce sufficient positive curvature for the conjugate-gradients method. Finally, to analyze the performance of our method, the results compare the specific-purpose solver with standard optimization methods. For this, we measure the matrix–vector products indicating the solver computational cost and the line-search iterations indicating the total amount of objective function evaluations. When we combine the globalization and the linear solver ingredients, we conclude that the specific-purpose Newton-CG solver reduces the total number of matrix–vector products by one order of magnitude. Moreover, the number of non-linear and line-search iterations is mainly smaller but of similar magnitude.

A new approach for direct discretization of fractional order operator in delta domain

The fractional order system (FOS) comprises fractional order operator. In order to obtain the discretized version of the fractional order system, the first step is to discretize the fractional order operator, commonly expressed as s?m, 0 < m < 1. The fractional order operator can be used as fractional order differentiator or integrator, depending upon the values of . In general, there are two approaches for discretization of fractional order operator, one is indirect method of discretization and another is direct method of discretization. The direct discretization method capitalizes the method of formation of generating function where fractional order operator s?m is expressed as a function of Z in the shift operator parameterization and continued fraction expansion (CFE) method is then utilized to get the corresponding discrete domain rational transfer function. There is an inherent problem with this discretization method using shift operator parameterization (discrete Z-domain). At fast sampling time, the discretized version of the continuous time operator or system should resemble that of the continuous time counterpart if the sampling theorem is satisfied. At very high sampling rate, the shift operator parameterized system fails to provide meaningful information due to its numerical ill conditioning. To overcome this problem, Delta operator parameterization for discretization is considered in this paper, where at fast sampling rate, the continuous time results can be obtained from the discrete time experiments and therefore a unified framework can be developed to get the discrete time results and continuous time results hand to hand. In this paper a new generating function is proposed to discretize the fractional order operator using the Gauss-Legendre 2-point quadrature rule. Additionally, the function has been expanded using the CFE in order to obtain rational approximation of the fractional order operator. The detailed mathematical formulations along with the simulation results in MATLAB, with different fractional order systems are considered, in order to prove the newness of this formulation for discretization of the FOS in complex Delta domain.

A gradient aggregate asymptotical smoothing algorithm for training max–min fuzzy neural networks

A gradient aggregate asymptotical smoothing algorithm is proposed for training fuzzy neural networks (FNNs), which develops the smoothing algorithm (SA) described in Li et al. (2017). By introducing the framework of asymptotical approximation to the algorithm, any degree of approximation for the error function of max–min FNNs can be obtained by an aggregate smoothing function with a variable precision parameter. The algorithm minimizes a sequence of asymptotically approximate functions using the steepest descent algorithm for solving a nondifferentiable max–min optimization problem of max–min FNNs. The proposed update rule on the precision parameter reconciles the conflict between the high-accuracy approximation and the numerical ill-conditioning. The algorithm is globally convergent under Armijo line search. As shown in the simulation results for three artificial examples and a real-world problem in fault diagnosis, compared with SA, the proposed algorithm can efficiently deal with the numerical oscillations and has better performance.

The comparison of the estimators of banded toeplitz covariance structure under the high-dimensional multivariate model

In this paper the model of experiment in which several characteristics are measured or the measurements are repeated in time, is considered. It is assumed that the covariance matrix of the characteristics/time points is unknown, but has the banded Toeplitz structure. The problem of estimating such a covariance structure was considered by e.g. Cui et al. (2016), Filipiak et al. (2018). We discuss two methods that aim to deal with singularity and numerical ill-conditioning of the estimators of covariance matrix under high-dimensional regime. The method of estimation of banded Toeplitz covariance matrix based on a shrinking, described by Ledoit and Wolf (2004) for unstructured covariance matrices is proposed. The estimators obtained with the use of the proposed method are then compared with the estimators obtained by the projection of sample covariance matrix onto the asymptotic cone of the nonnegative definite Toeplitz matrices (Filipiak et al. 2018). For this purpose simulation studies concerning bias and risk are conducted for several sets of parameters.

On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries

Radial basis function generated finite difference (RBF-FD) approximations generalize grid-based regular finite differences to scattered node sets. These become particularly effective when they are based on polyharmonic splines (PHS) augmented with multi-variate polynomials (PHS+poly). One key feature is that high orders of accuracy can be achieved without having to choose an optimal shape parameter and without having to deal with issues related to numerical ill-conditioning. The strengths of this approach were previously shown to be especially striking for approximations near domain boundaries, where the stencils become highly one-sided. Due to the polynomial Runge phenomenon, regular FD approximations of high accuracy will in such cases have very large weights well into the domain. The inclusion of PHS-type RBFs in the process of generating weights makes it possible to avoid this adverse effect. With that as motivation, this study aims at gaining a better understanding of the behavior of PHS+poly generated RBF-FD approximations near boundaries, illustrating it in 1-D, 2-D and 3-D.

Comparison of approximate methods for computation of the concerted adiabatic electronic and nuclear fluxes in aligned [formula omitted

Approximate methods of computing the concerted electronic and nuclear fluxes associated with electronically adiabatic processes are developed and applied to the prototypal system, namely aligned H2+ vibrating in its electronic ground state (2Σg+), the only realistic system for which highly accurate (exact) electronic (EPD) and nuclear (NPD) probability densities, electronic (EFD) and nuclear (NFD) flux densities, as well as corresponding fluxes, are available. Alternative formulas for the electronic flux, Fe,EPD and Fe,EFD, based on either the EPD or the EFD, are derived from the continuity equation. The results of Born-Oppenheimer approximation (BOA) and of an ordered sequence of Born-Huang expansions (BHE) are presented. The BOA and first-order BHE are in excellent agreement with the exact for both the NPD and NFD, as well as for the EPD and Fe,EPD up to about 1ps. Higher-order BHE are necessary to achieve similar accuracy at longer times. In contrast, the BOA and first-order BHE yield zero EFD and therefore also zero Fe,EFD. Although the higher-order BHE give non-zero values for these properties, they disagree flagrantly with their exact correlates. The error is traceable to numerical ill-conditioning of the working expression for the EFD. In summary, the BOA is adequate to compute accurate NPD, NFD, EPD and Fe,EPD for times corresponding to several dozens of vibrational periods; the higher-order BHE is required for longer times. But neither the BOA nor the BHE can provide reliable estimates of the EFD and Fe,EFD.

Robust viscous-inviscid interaction scheme for application on unstructured meshes

A coupled viscous-inviscid interaction scheme combining the continuity equation for potential flow with the three-dimensional integral boundary layer equations is presented. The inviscid problem is discretized by a finite-element approach whereas an upwind-biased finite-volume scheme is employed for the boundary layer equations. The discretization is applicable to unstructured tetrahedral-triangular meshes and results in a sparse system of non-linear equations which is solved by a Newton-type method. The mathematical reasons for the singularities commonly associated with the integral boundary layer equations in separated flow regions are analyzed and the connection between the mathematical singularities and the numerical ill-conditioning is discussed. It is shown that, by a suitable choice of closure relations, it is possible to obtain a boundary layer model free from numerical ill-conditioning in separated flow regions. The accuracy of the coupled viscous-inviscid model is investigated in a number of test cases including transitional and mildly separated flow over two different natural laminar flow airfoils and three-dimensional flow over a swept wing. It is concluded that the coupled method is able to provide reasonably accurate predictions of viscous and inviscid flow field quantities for the investigated cases.

Investigation of Spatial Control Strategies for AHWR: A Comparative Study

Large nuclear reactors such as the Advanced Heavy Water Reactor (AHWR), are susceptible to xenon-induced spatial oscillations in which, though the core average power remains constant, the power distribution may be nonuniform as well as it might experience unstable oscillations. Such oscillations influence the operation and control philosophy and could also drive safety issues. Therefore, large nuclear reactors are equipped with spatial controllers which maintain the core power distribution close to desired distribution during all the facets of operation and following disturbances. In this paper, the case of AHWR has been considered, for which a number of different types of spatial controllers have been designed during the last decade. Some of these designs are based on output feedback while the others are based on state feedback. Also, both the conventional and modern control concepts, such as linear quadratic regulator theory, sliding mode control, multirate output feedback control and fuzzy control have been investigated. The designs of these different controllers for the AHWR have been carried out using a 90th order model, which is highly stiff. Hence, direct application of design methods suffers with numerical ill-conditioning. Singular perturbation and time-scale methods have been applied whereby the design problem for the original higher order system is decoupled into two or three subproblems, each of which is solved separately. Nonlinear simulations have been carried out to obtain the transient responses of the system with different types of controllers and their performances have been compared.

Randomized Alternating Least Squares for Canonical Tensor Decompositions: Application to A PDE With Random Data

This paper introduces a randomized variation of the alternating least squares (ALS) algorithm for rank reduction of canonical tensor formats. The aim is to address the potential numerical ill-conditioning of least squares matrices at each ALS iteration. The proposed algorithm, dubbed randomized ALS, mitigates large condition numbers via projections onto random tensors, a technique inspired by well-established randomized projection methods for solving overdetermined least squares problems in a matrix setting. A probabilistic bound on the condition numbers of the randomized ALS matrices is provided, demonstrating reductions relative to their standard counterparts. Additionally, results are provided that guarantee comparable accuracy of the randomized ALS solution at each iteration. The performance of the randomized algorithm is studied with three examples, including manufactured tensors and an elliptic PDE with random inputs. In particular, for the latter, tests illustrate not only improvements in condition...

Quadrature-Based Vector Fitting for Discretized $\mathcal{H}_2$ Approximation

Vector Fitting is a popular method of constructing rational approximants designed to fit given frequency response measurements. The original method, which we refer to as VF, is based on a least-squares fit to the measurements by a rational function, using an iterative reallocation of the poles of the approximant. We show that one can improve the performance of VF significantly, by using a particular choice of frequency sampling points and properly weighting their contribution based on quadrature rules that connect the least squares objective with an H2 error measure. Our modified approach, designated here as QuadVF, helps recover the original transfer function with better global fidelity (as measured with respect to the H2 norm), than the localized least squares approximation implicit in VF. We extend the new framework also to incorporate derivative information, leading to rational approximants that minimize system error with respect to a discrete Sobolev norm. We consider the convergence behavior of both VF and QuadVF as well, and evaluate potential numerical ill-conditioning of the underlying least-squares problems. We investigate briefly VF in the case of noisy measurements and propose a new formulation for the resulting approximation problem. Several numerical examples are provided to support the theoretical discussion.

Contingency Constrained VAr Planning Using Penalty Successive Conic Programming

This paper presents a new method for VAr planning under multiple operating scenarios, optimizing sizes and locations of new reactive compensation equipment to ensure that both the system voltage profile and voltage stability requirements are met. The approach is based on mbi L1-norm regularization for finding a solution with minimum VAr installation sites and on the mbi L2-norm penalty function for satisfying the multiple state constraints. The mbi L2-norm penalty function is exact in the sense that a finite penalty parameter is required to establish equivalence with the VAr planning problem, thus avoiding numerical ill-conditioning. The solution is obtained from a successive conic programming algorithm which makes use of adaptive trust-region control. The results of the new method are compared with those of an optimal power flow-based program for VAr planning.

Design of Fast Output Sampling Controller for Three-Time-Scale Systems: Application to Spatial Control of Advanced Heavy Water Reactor

This paper introduces a formulation for design of Fast Output Sampling (FOS) controllers for three-time-scale systems. It is shown that the FOS control gain for a three-time-scale system can be obtained by combining the solutions of the three subsystem problems, obtained separately. Since three smaller order subsystem problems are to be solved in lieu of one high order problem, numerical ill-conditioning is completely avoided. Techniques for block-diagonalization and composite control of a three-time-scale system are also discussed. The proposed method is applied to the problem of spatial control of advanced heavy water reactor (AHWR). The model of AHWR is decomposed into three subsystems respectively named “slow,” “fast 1,” and “fast 2,” and separate subsystem control problems are cast from which an FOS controller for the original system is derived. Efficacy of the controller thus obtained is demonstrated through dynamic simulations. The controller thus designed employs only the output information to achieve arbitrary pole placement.

Determination of a Rubber-Like Material Model as an Inverse Problem

The purpose of this paper is to describe the selection process of a rubber-like material model useful for simulation behaviour of an inflatable air cushion under multi-axial stress states. The air cushion is a part of a single segment of a pontoon bridge. The air cushion is constructed of a polyester fabric reinforced membrane such as Hypalon®. From a numerical point of view such a composite type poses a challenge since numerical ill-conditioning can occur due to stiffness differences between rubber and fabric. Due to the analysis of the large deformation dynamic response of the structure, the LS-Dyna code is used. Since LS-Dyna contains more than two-hundred constitutive models the inverse method is used to determine parameters characterizing the material on the base of results of the experimental test.

Exponentially-convergent strategies for defeating the Runge Phenomenon for the approximation of non-periodic functions, part two: Multi-interval polynomial schemes and multidomain Chebyshev interpolation

Approximating a smooth function from its values f ( x i ) at a set of evenly spaced points x i through P-point polynomial interpolation often fails because of divergence near the endpoints, the “Runge Phenomenon”. This report shows how to achieve an error that decreases exponentially fast with P by means of polynomial interpolation on N s subdomains where N s increases with P. We rigorously prove that in the limit both N s and M, the degree on each subdomain, increase simultaneously, the approximation error converges proportionally to exp ( − constant P log ( P ) ) . Thus, division into ever-shrinking, ever-more-numerous subdomains is guaranteed to defeat the Runge Phenomenon in infinite precision arithmetic. (Numerical ill-conditioning is also discussed, but is not a great difficulty in practice, though not insignificant in theory.) Although a Chebyshev grid on each subdomain is well known to be immune to the Runge Phenomenon, it is still interesting, and the same methodology can be applied as to a uniform grid. When a Chebyshev grid is used on each subdomain, there are two regimes. If c is the distance from the middle of the interval [ − 1 , 1 ] to the nearest singularity of f ( x ) in the complex plane, then when c N s ≪ 1 , the error is proportional to exp ( − c P ) , independent of the number of subdomains. When c N s ≫ 1 , the rate of convergence slows to exp ( − constant P log ( P ) ) , the same as for equispaced interpolation. However, the Chebyshev multidomain error is always smaller than the equispaced multidomain error.

Computation of the phase and amplitude noise in microwave oscillators and a simplified calculation method for far enough from the carrier offsets

New results regarding phase and amplitude noise analysis in microwave oscillators for moderate offset frequencies from the carrier are presented. Although the phase noise process in an oscillator is a large signal non-stationary process, it is proved that for the purpose of phase noise calculations for moderate offset frequencies, the phase noise process can be considered as a small signal stationary process and by this assumption, a valid approximation of the phase noise spectrum at these offset frequencies is obtained. By this consideration, a simplified approach for the purpose of the phase and amplitude noise spectrum calculations, at far enough from the carrier offset frequencies, by avoiding the numerical ill-conditioning of the harmonic balance equations, is presented. Owing to the presence of the common phase process in the circuit variables, for the practical resolution of the noise relations, it is necessary to introduce an extra constraint on the phase or the amplitude process. Here, it is theoretically proved that this constraint will not change the total noise spectrum of the oscillator. Another form of conversion matrix analysis is presented and the cause of error in the computation of the phase noise at offset frequencies near to the carrier is investigated. The theoretically verified characteristics of phase and amplitude noise analysis methods have been observed in a P-HEMT oscillator at 10.23 GHz.

Radial Distribution Load Flow Using Conic Programming

This paper shows that the load flow problem of a radial distribution system can be modeled as a convex optimization problem, particularly a conic program. The implications of the conic programming formulation are threefold. First, the solution of the distribution load flow problem can be obtained in polynomial time using interior-point methods. Second, numerical ill-conditioning can be automatically alleviated by the use of scaling in the interior-point algorithm. Third, the conic formulation facilitates the inclusion of the distribution power flow equations in radial system optimization problems. A state-of-the-art implementation of an interior-point method for conic programming is used to obtain the solution of nine different distribution systems. Comparisons are carried out with a previously published radial load flow program by R. Cespedes

Reconstruction of Piecewise Smooth Functions from Non-uniform Grid Point Data

Spectral series expansions of piecewise smooth functions are known to yield poor results, with spurious oscillations forming near the jump discontinuities and reduced convergence throughout the interval of approximation. The spectral reprojection method, most notably the Gegenbauer reconstruction method, can restore exponential convergence to piecewise smooth function approximations from their (pseudo-)spectral coefficients. Difficulties may arise due to numerical robustness and ill-conditioning of the reprojection basis polynomials, however. This paper considers non-classical orthogonal polynomials as reprojection bases for a general order (finite or spectral) reconstruction of piecewise smooth functions. Furthermore, when the given data are discrete grid point values, the reprojection polynomials are constructed to be orthogonal in the discrete sense, rather than by the usual continuous inner product. No calculation of optimal quadrature points is therefore needed. This adaptation suggests a method to approximate piecewise smooth functions from discrete non-uniform data, and results in a one-dimensional approximation that is accurate and numerically robust.

Utilization of collinearity in regression modeling of activated sludge processes

Identification of collinear groups among the variables of biological wastewater treatment processes is considered. The SROV (stepwise regression using orthogonalized variables) program was used to analyze the data that were collected in an experimental lane of a full-scale waste water treatment plant. The analysis has identified 10 collinear groups among the 22 measured variables of the process. It has been shown that a member of such a group (target variable) can be represented as a linear combination of the other members (predictive variables) and the associated correlation coefficient can be used for assessing the quality of the representation. The proposed technique can be beneficial in determining how many independent variables have to be measured in a process. It can be used for developing soft sensors and also for selecting the measurements that have to be used in determining parameters of mechanistic models in order to prevent numerical ill-conditioning.

Computation of three stochastic modifications of Stokes's formula for regional geoid determination

In the regional geoid studies, the modified Stokes formula is often used nowadays. This method combines local terrestrial data with an appropriate global geopotential model in a truncated form of Stokes's integral. This paper is devoted to three stochastic least-squares (LS) modifications, which were originally proposed by Sjöberg (Least squares modification of Stokes and Vening–Meinesz formulas by accounting for errors of truncation, potential coefficients and gravity data. Report No. 27, Department of Geodesy, University of Uppsala, 16pp.) in 1984 (with later developments). The main principles of the LS modifications and some spectral models of the gravity field characteristics are reviewed. Certain difficulties may be encountered when computing the modification parameters from a system of linear equations. In particular, the design matrices of the unbiased and optimum LS modifications suffer from numerical ill-conditioning. Two mathematical regularization strategies are selected in order to find a practical solution for the sought modification parameters. Typical numerical outcome of the regularization and the applicability of the obtained LS parameters are discussed. The present contribution tackles the LS modification-related problems in the context of a specially designed Matlab software package. The core quantities of the stochastic LS modifications can be computed by this software, which is made available on the Computers & Geosciences server.