Low-order Polynomial Approximation Research Articles

Identification of cable tension through physical models and non-contact measurements

Determining the static and dynamic response of structural cables is a mechanical problem of theoretical and practical interest in many aspects of civil engineering. In particular, the cable tension is a fundamental variable to be assigned in direct design problems, as well as a fundamental unknown to be assessed in inverse identification problems. The present paper outlines an original analytical strategy for identifying the axial tension of inclined sagged cables in cable-stayed structures, based on static measurements. Starting from the classic formulation of an inextensible, perfectly flexible model of catenary suspended cables, a perturbation-based low-order polynomial approximation of the static configuration under self-weight is achieved (direct problem). The quadratic and cubic coefficients of the configurational function are tension-dependent quantities, whose analytical expressions can be mathematically inverted to define consistent parametric formulas for tension identification (inverse problem). The experimental assessment of the static cable configuration, which serves as input for the inverse problem, is provided by the geometric data acquired by modern and noninvasive techniques of non-contact measurement. The methodological advantage is that the laser scanner acquisition of point cloud models can efficiently and economically provide precise and redundant three-dimensional geometric descriptions of the cable configuration, which is suitable for a highly accurate and statistically robust data treatment. Lastly, the identification strategy is successfully applied to a real-scale structure, for tension identification in the stays of a cable-stayed footbridge. The results are discussed from a qualitative and quantitative viewpoint, and finally validated through a comparison with the outcomes of a dynamic identification procedure based on high-speed camera acquisitions.

SigML++: Supervised Log Anomaly with Probabilistic Polynomial Approximation

Security log collection and storage are essential for organizations worldwide. Log analysis can help recognize probable security breaches and is often required by law. However, many organizations commission log management to Cloud Service Providers (CSPs), where the logs are collected, processed, and stored. Existing methods for log anomaly detection rely on unencrypted (plaintext) data, which can be a security risk. Logs often contain sensitive information about an organization or its customers. A more secure approach is always to keep logs encrypted (ciphertext). This paper presents “SigML++”, an extension of “SigML” for supervised log anomaly detection on encrypted data. SigML++ uses Fully Homomorphic Encryption (FHE) according to the Cheon–Kim–Kim–Song (CKKS) scheme to encrypt the logs and then uses an Artificial Neural Network (ANN) to approximate the sigmoid (σ(x)) activation function probabilistically for the intervals [−10,10] and [−50,50]. This allows SigML++ to perform log anomaly detection without decrypting the logs. Experiments show that SigML++ can achieve better low-order polynomial approximations for Logistic Regression (LR) and Support Vector Machine (SVM) than existing methods. This makes SigML++ a promising new approach for secure log anomaly detection.

Learning High-Order Graph Convolutional Networks via Adaptive Layerwise Aggregation Combination.

Graph convolutional networks have attracted wide attention for their expressiveness and empirical success on graph-structured data. However, deeper graph convolutional networks with access to more information can often perform worse because their low-order Chebyshev polynomial approximation cannot learn adaptive and structure-aware representations. To solve this problem, many high-order graph convolution schemes have been proposed. In this article, we study the reason why high-order schemes have the ability to learn structure-aware representations. We first prove that these high-order schemes are generalized Weisfeiler-Lehman (WL) algorithm and conduct spectral analysis on these schemes to show that they correspond to polynomial filters in the graph spectral domain. Based on our analysis, we point out twofold limitations of existing high-order models: 1) lack mechanisms to generate individual feature combinations for each node and 2) fail to properly model the relationship between information from different distances. To enable a node-specific combination scheme and capture this interdistance relationship for each node efficiently, we propose a new adaptive feature combination method inspired by the squeeze-and-excitation module that can recalibrate features from different distances by explicitly modeling interdependencies between them. Theoretical analysis shows that models with our new approach can effectively learn structure-aware representations, and extensive experimental results show that our new approach can achieve significant performance gain compared with other high-order schemes.

Regression-Based Stochastic Study of Electromagnetic Fields Due to Lightning Strikes

Electromagnetic (EM) fields triggered by lightning strikes are likely to display non-negligible variability in practice, with an apparent effect on various quantities of interest, such as induced voltages on transmission lines. This paper investigates the stochastic properties of lightning-generated fields, when the latter are affected by random ground parameters and/or nondeterministic lightning base current. In essence, the EM components are represented herein by truncated series of orthogonal polynomials, according to the generalized polynomial-chaos theory, and their expansion coefficients are computed via a regression approach. The necessary samplings of the random spaces are constructed with either tensor-based or sparse grids, and their performance is validated and compared. It is shown that reliable calculations of the expectation can be conducted with low-order polynomial approximations, while at least third-order expansions are required for the standard deviation. In addition, we conclude that nontrivial variability is induced in lightning-produced pulses, when variations of the considered stochastic parameters are taken into account.

Large-scale simulation of mantle convection based on a new matrix-free approach

In this paper, we report on a two-scale approach for efficient matrix-free finite element simulations. It is an extended version of our previous conference publication [1]. The proposed method is based on surrogate element matrices constructed by low order polynomial approximations. It is applied to a Stokes-type PDE system with variable viscosity as is a key component in mantle convection models. We set the ground for a rigorous performance analysis inspired by the concept of parallel textbook multigrid efficiency and study the weak scaling behavior on SuperMUC, a peta-scale supercomputer system. For a complex geodynamical model, we achieve, on up to 47 250 compute cores, a parallel efficiency of 93% for application of the discrete operator and 83% for a complete Uzawa V-cycle including the coarse grid solve. Our largest simulation uses a trillion (O(1012)) degrees of freedom for a global mesh resolution of 1.5 km. Applicability of our new approach for geodynamical problems is demonstrated by investigating dynamic topography for classical benchmark settings as well as for high-resolution models with lateral viscosity variations.

Towards fast hemodynamic simulations in large-scale circulatory networks

In this work, an effective numerical method specifically conceived to simulate fluid flow in networks composed by tubular domains is presented. The present contribution addresses an extension of the so-called Transversally Enriched Pipe Element Method (TEPEM) recently developed by the authors in the computational hemodynamics realm to embrace the case of branching domains. The TEPEM approach relies on a slab-based partition of the network of vessels complemented with basis functions devised to render high-order polynomial enrichment for transversal flow phenomena, while keeping a low-order polynomial approximation for the fields along the axial direction. The main goal of such an approach is to provide a fast numerical tool to characterize flow-related quantities in large-scale networks of vessels, which are of relevance in the hemodynamics field, such as the wall shear stress. Through several numerical examples ranging from academic to patient-specific networks, it is demonstrated that the proposed strategy yields reasonably accurate qualitative and quantitative results and substantially reduced computational cost when compared with standard 3D finite element models. Hence, we conclude that this approach can be regarded as an appealing numerical technology placed halfway over-simplistic cheap 1D and computationally demanding accurate 3D models.

On the efficiency of a matrix-free linearly implicit time integration strategy for high-order Discontinuous Galerkin solutions of incompressible turbulent flows

The paper aims at investigating and comparing the efficiency of matrix-based and matrix-free implementations of implicit time integration strategies, here linearly implicit Rosenbrock-type Runge–Kutta schemes, in the context of a high order Discontinous Galerkin space discretization applied to DNS/ILES of turbulent flows. In particular, the effects of time step size, GMRES solution tolerance and preconditioner, mesh size, order of polynomial approximation of the solution and parallel efficiency are here considered. The performance of the solvers have been compared though the solution of the two-dimensional laminar traveling waves test case, the ILES of the Rayleigh–Bénard natural convection up to Ra=108, and the channel flow up to Reτ=950. The results highlight that while matrix-based strategies are the most adequate for low-order polynomial approximations, considerable improvements in computational efficiency can be achieved, both in terms of CPU time and memory footprint, by a proper use of the matrix-free solver for three-dimensional test cases and very high-order accurate DG space discretizations.

Estimation of Basis Line-Integrals in a Spectral Distortion-Modeled Photon Counting Detector Using Low-Order Polynomial Approximation of X-ray Transmittance.

Photon counting detector (PCD)-based computed tomography exploits spectral information from a transmitted x-ray spectrum to estimate basis line-integrals. The recorded spectrum, however, is distorted and deviates from the transmitted spectrum due to spectral response effect (SRE). Therefore, the SRE needs to be compensated for when estimating basis line-integrals. One approach is to incorporate the SRE model with an incident spectrum into the PCD measurement model and the other approach is to perform a calibration process that inherently includes both the SRE and the incident spectrum. A maximum likelihood estimator can be used to the former approach, which guarantees asymptotic optimality; however, a heavy computational burden is a concern. Calibration-based estimators are a form of the latter approach. They can be very efficient; however, a heuristic calibration process needs to be addressed. In this paper, we propose a computationally efficient three-step estimator for the former approach using a low-order polynomial approximation of x-ray transmittance. The low-order polynomial approximation can change the original non-linear estimation method to a two-step linearized approach followed by an iterative bias correction step. We show that the calibration process is required only for the bias correction step and prove that it converges to the unbiased solution under practical assumptions. Extensive simulation studies validate the proposed method and show that the estimation results are comparable to those of the ML estimator while the computational time is reduced substantially.

Transmission Efficient Channel Estimation for OFDM System Under Multi-path Fast Fading Channel

This paper considers the channel estimation method in the case of a high mobility communication in orthogonal frequency-division multiplexing (OFDM) systems. In this paper, several successive OFDM symbols experienced channel are treated as a window. The polynomial approximation of the channel parameters in the time domain is considered. We show that the low order polynomials are good enough to model the channel within a window. Hence the interpolation performing over a whole window profits pilot reduction. Moreover, the method has an advantage of computational complexity because of the low order polynomial approximation and without the requirement of inter-carrier interference mitigation. The method is derived with the least-square approach for polynomial interpolation of arbitrary order. Simulation results show that the proposed method is robust to high Doppler frequency and enhancing the transmission efficiency.

The finite cell method for bone simulations: verification and validation

Standard methods for predicting bone's mechanical response from quantitative computer tomography (qCT) scans are mainly based on classical h-version finite element methods (FEMs). Due to the low-order polynomial approximation, the need for segmentation and the simplified approach to assign a constant material property to each element in h-FE models, these often compromise the accuracy and efficiency of h-FE solutions. Herein, a non-standard method, the finite cell method (FCM), is proposed for predicting the mechanical response of the human femur. The FCM is free of the above limitations associated with h-FEMs and is orders of magnitude more efficient, allowing its use in the setting of computational steering. This non-standard method applies a fictitious domain approach to simplify the modeling of a complex bone geometry obtained directly from a qCT scan and takes into consideration easily the heterogeneous material distribution of the various bone regions of the femur. The fundamental principles and properties of the FCM are briefly described in relation to bone analysis, providing a theoretical basis for the comparison with the p-FEM as a reference analysis and simulation method of high quality. Both p-FEM and FCM results are validated by comparison with an in vitro experiment on a fresh-frozen femur.

Time-dependent generalized polynomial chaos

Generalized polynomial chaos (gPC) has non-uniform convergence and tends to break down for long-time integration. The reason is that the probability density distribution (PDF) of the solution evolves as a function of time. The set of orthogonal polynomials associated with the initial distribution will therefore not be optimal at later times, thus causing the reduced efficiency of the method for long-time integration. Adaptation of the set of orthogonal polynomials with respect to the changing PDF removes the error with respect to long-time integration. In this method new stochastic variables and orthogonal polynomials are constructed as time progresses. In the new stochastic variable the solution can be represented exactly by linear functions. This allows the method to use only low order polynomial approximations with high accuracy. The method is illustrated with a simple decay model for which an analytic solution is available and subsequently applied to the three mode Kraichnan–Orszag problem with favorable results.

Discontinuous Galerkin finite element methods for incompressible non-linear elasticity

A discontinuous Galerkin finite element method (DGFEM) for incompressible, non-linear elasticity is derived. One of the limitations of the continuous Galerkin finite element method (CGFEM) when applied to incompressible elasticity is that the accuracy of the numerical solution may be adversely affected by the phenomenon known as locking—this may prevent the use of a low order polynomial approximation to the displacement on each element. We demonstrate using simulations that the DGFEM presented here does not suffer from this drawback. Two further advantages in this setting of this DGFEM over CGFEMs are that: (i) highly anisotropic meshes—meshes containing elements with a very high aspect ratio—may be used without significantly degrading the accuracy of the solution; and (ii) discontinuities in the elasticity parameters are handled more effectively.

Mass and heat fluxes for a binary granular mixture at low density

The Navier–Stokes order hydrodynamic equations for a low-density granular mixture obtained previously from the Chapman–Enskog solution to the Boltzmann equation are considered further. The six transport coefficients associated with mass and heat flux in a binary mixture are given as functions of the mass ratio, size ratio, composition, and coefficients of restitution. Their quantitative variation across this parameter set is demonstrated using low-order Sonine polynomial approximations to solve the exact integral equations. The results are also used to quantify the violation of the Onsager reciprocal relations for a granular mixture. Finally, the stability of the homogeneous cooling state is discussed.

Polynomial-based maximum-likelihood technique for synchronization in digital receivers

Maximum-likelihood estimation theory provides a general framework for developing near-optimum (with respect to the Cramer-Rao bound) synchronization schemes for digital communication systems. A new technique for jointly estimating the symbol timing and carrier phase in digital receivers for linear digital modulations with nonsynchronized sampling is established for both data-aided and nondata-aided systems using a block-based feedforward architecture. This technique is a practical, fully digitally implemented synchronization concept based on a low-order polynomial approximation of the likelihood functions using the Farrow-based interpolator. It is shown that low oversampling, like two samples per symbol, can be used in timing recovery without compromising the system performance. This results in efficient overall receiver implementation. It is also demonstrated that the quality of the interpolator design has a big effect on the performance of the synchronization scheme. Frequency-domain optimized polynomial interpolator designs provide significantly better performance than the well-known Lagrange interpolators.

Polynomial Response Surface Approximations for the Multidisciplinary Design Optimization of a High Speed Civil Transport

Surrogate functions have become an important tool in multidisciplinary design optimization to deal with noisy functions, high computational cost, and the practical difficulty of integrating legacy disciplinary computer codes. A combination of mathematical, statistical, and engineering techniques, well known in other contexts, have made polynomial surrogate functions viable for MDO. Despite the obvious limitations imposed by sparse high fidelity data in high dimensions and the locality of low order polynomial approximations, the success of the panoply of techniques based on polynomial response surface approximations for MDO shows that the implementation details are more important than the underlying approximation method (polynomial, spline, DACE, kernel regression, etc.). This paper selectively surveys some of the ancillary techniques—statistics, global search, parallel computing, variable complexity modeling—that augment the construction and use of polynomial surrogates.

Approximate analysis of response variability of uncertain linear systems

A probabilistic methodology is presented for obtaining the variability and statistics of the dynamic response of multi-degree-of-freedom linear structures with uncertain properties. Complex mode analysis is employed and the variability of each contributing mode is analyzed separately. Low-order polynomial approximations are first used to express modal frequencies, damping ratios and participation factors with respect to the uncertain structural parameters. Each modal response is then expanded in a series of orthogonal polynomials in these parameters. Using the weighted residual method, a system of linear ordinary differential equations for the coefficients of each series expansion is derived. A procedure is then presented to calculate the variability and statistics of the uncertain response. The technique is extended to the stochastic excitation case for obtaining the variability of the response moments due to the variability of the system parameters. The methodology can treat a variety of probability distributions assumed for the structural parameters. Compared to existing analytical techniques, the proposed method drastically reduces the computational effort and computer storage required to solve for the response variability and statistics. The performance and accuracy of the method are illustrated by examples.

Chemistry in strong laser fields: An example from methyl iodide photodissociation

Time-dependent quantum-mechanical theories and simulations provide a clear and intuitive description of molecular processes. Due to ensuing simplification of the theory and the generally employed numerical algorithms, the vast majority of these treatments are based upon perturbation theory. Especially in light of the current level of experimental sophistication, with experiments being realized which are influenced by the spectral, temporal, and spatial shape of the laser pulse, it is important to move beyond treatments limited to weak fields or idealized δ-function wave forms. Various methods to examine the results of high-field simulations are presented. All of the techniques are shown to have the familiar linear response form in the weak-field limit. In a time-dependent framework the difference between the linear and nonlinear response expressions can be seen from expectation values over stationary versus nonstationary states. The high-field photodissociation of methyl iodide illustrates this approach. Methyl iodide represents a physical system well suited for examining the effects of such exciting laser-field characteristics as strength, linewidth, and frequency upon the photodissociation dynamics. Its dissociation occurs upon coupled repulsive excited electronic potential-energy surfaces which have recently been revised to fit the most current experimental data. The effect of the surface intersection has previously been typically studied by examining the branching and the internal state distributions of the products in the two channels as a function of excitation frequency only. The collinear photodissociation dynamics is examined using a numerically exact time-dependent quantum-mechanical method. The equations of motion for the amplitudes upon the ground and two coupled excited electronic surfaces, explicitly incorporating the laser field, are integrated by a scheme which employs a low-order polynomial approximation to the evolution operator. The effects of the three field characteristics upon the branching ratio and internal state distributions of the products and the spectroscopy of the process are delineated. The course of the photodissociation dynamics is shown to be affected by these characteristics. The results demonstrate the causal connections between the pulse shape and the resulting photoprocesses. Practical manifestations of strong fields (power broadening, sub-threshold absorption, higher harmonic generation, emission shaping of the ground state, temporal development) are stressed.

Low-order polynomial approximation of propagators for the time-dependent Schrödinger equation

Polynomial approximations for propagating the time-dependent Schrödinger equation are studied. These methods are motivated by the numerical demands of systems with time-dependent Hamiltonian operators. First-order and second-order Magnus expansions are tested for approximating the time ordering operator. The polynomials considered are based on a reduced basis space which is obtained by iterating the Hamiltonian operator on an initial wavefunction. The approximate polynomials are obtained by minimizing the error in the propagation. One such approach which minimizes the residuum outside the reduced space is proved to be equivalent to the short time iterative Lanczos procedure. A second approach which uses a different optimization scheme (the residuum method) is found to be somewhat superior to the Lanczos procedure. Numerical examples are used to illustrate the convergence of these methods. The second-order Magnus approximation is found to be a significant improvement over the first-order approximation regardless of method.

A New Rapid Acquisition Technique for Direct Sequence Spread-Spectrum Communications

This paper describes a rapid acquisition technique which can be used in direct sequence spread-spectrum systems. It is sequential in nature and is based on a double threshold which defines when a decision can be made. The values of the thresholds change at every examination instant. The procedure makes use of a bound to the partial crosscorrelation function of PN sequences which ensures that only the sequence length needs to be given as far as the PN sequence particulars are concerned. A decision is made only if the probability of error associated with a decision at that point is equal to or better than a prespecified value. Finally, it is shown that most of the computational difficulties associated with the bound as given by a recursive algorithm can be eliminated when using a low-order polynomial approximation to the bound. However, either by using the bound or its polynomial approximation, a significant improvement in the acquisition time is obtained.