Full Newton Method Research Articles

Frequency prediction from exact or self-consistent mean flows

A number of approximations have been proposed to estimate basic hydrodynamic quantities, in particular the frequency of a limit cycle. One of these, RZIF (for Real Zero Imaginary Frequency), calls for linearizing the governing equations about the mean flow and estimating the frequency as the imaginary part of the leading eigenvalue. A further reduction, the SCM (for Self-Consistent Model), approximates the mean flow as well, as resulting only from the nonlinear interaction of the leading eigenmode with itself. Both RZIF and SCM have proven dramatically successful for the archetypal case of the wake of a circular cylinder. Here, the SCM is applied to thermosolutal convection, for which a supercritical Hopf bifurcation gives rise to branches of standing waves and traveling waves. The SCM is solved by means of a full Newton method coupling the approximate mean flow and leading eigenmode. Although the RZIF property is verified for the traveling waves, the SCM reproduces the nonlinear frequency only very near the onset of the bifurcation and for another isolated parameter value. Thus, the nonlinear interaction arising from the leading mode is insufficient to reproduce the nonlinear mean field and frequency.

Implicit and coupled fluid plasma solver with adaptive Cartesian mesh and its applications to non-equilibrium gas discharges

We present a new fluid plasma solver with adaptive Cartesian mesh (ACM) based on a full-Newton (nonlinear, implicit) scheme for non-equilibrium gas discharge plasma. The electrons and ions are described using drift–diffusion approximation coupled to Poisson equation for the electric field. The electron-energy transport equation is solved to account for electron thermal conductivity, Joule heating, and energy loss of electrons in collisions with neutral species. The rate of electron-induced ionization is a function of electron temperature and could also depend on electron density (important for plasma stratification). The ion and gas temperature are kept constant. The transport equations are discretized using a non-isothermal Scharfetter–Gummel scheme to resolve possible large temperature gradients in the sheaths. We demonstrate the new solver for simulations of direct current (DC) and radiofrequency (RF) discharges. The implicit treatment of the coupled equations allows using large time steps. The full-Newton method (FNM) enables fast nonlinear convergence at each time step, offering significantly improved simulation efficiency. We discuss the selection of time steps for solving different plasma problems. The new solver enables solving several problems we could not solve before with existing software: two- and three-dimensional structures of the entire DC discharges including cathode and anode regions, electric field reversals and double-layer formation, the normal cathode spot and an anode ring, moving striations in diffuse and constricted DC discharges, and standing striations in RF discharges. The developed FNM-ACM technique offers many benefits for tackling the disparity of gas discharge plasma systems' time scales and nonlinearity.

A Newton solver for micromorphic computational homogenization enabling multiscale buckling analysis of pattern-transforming metamaterials

Mechanical metamaterials feature engineered microstructures designed to exhibit exotic, and often counter-intuitive, effective behaviour such as negative Poisson’s ratio or negative compressibility. Such a specific response is often achieved through instability-induced transformations of the underlying periodic microstructure into one or multiple patterning modes. Due to a strong kinematic coupling of individual repeating microstructural cells, non-local behaviour and size effects emerge, which cannot easily be captured by classical homogenization schemes. In addition, the individual patterning modes can mutually interact in space as well as in time, while at the engineering scale the entire structure can buckle globally. For efficient numerical predictions of macroscale engineering applications, a micromorphic computational homogenization scheme has recently been developed (Rokoš et al., J. Mech. Phys. Solids 123, 119–137, 2019). Although this framework is in principle capable of accounting for spatial and temporal interactions between individual patterning modes, its implementation relied on a gradient-based quasi-Newton solution technique. This solver is suboptimal because (i) it has sub-quadratic convergence, and (ii) the absence of Hessians does not allow for proper bifurcation analyses. Given that mechanical metamaterials often rely on controlled instabilities, these limitations are serious. Addressing them will reduce the dependency of the solution on the initial guess by perturbing the system towards the correct deformation when a bifurcation point is encountered. Eventually, this enables more accurate and reliable modelling and design of metamaterials. To achieve this goal, a full Newton method, entailing all derivations and definitions of the tangent operators, is provided in detail in this paper. The construction of the macroscopic tangent operator is not straightforward due to specific model assumptions on the decomposition of the underlying displacement field pertinent to the micromorphic framework, involving orthogonality constraints. Analytical expressions for the first and second variation of the total potential energy are given, and the complete algorithm is listed. The developed methodology is demonstrated with two examples in which a competition between local and global buckling exists and where multiple patterning modes emerge. The numerical results indicate that local to global buckling transition can be predicted within a relative error of 6% in terms of the applied strains. The expected pattern combinations are triggered even for the case of multiple patterns.

A new full-Newton step interior-point method for $$P_{*}(\kappa )$$-LCP based on a positive-asymptotic kernel function

A new full-Newton step interior-point method for $$P_{*}(\kappa )$$-LCP based on a positive-asymptotic kernel function

Seismic full-waveform inversion using minimization of virtual scattering sources

Full-waveform inversion (FWI) is a powerful tool for imaging underground structures with high resolution; however, this approach commonly suffers from the cycle-skipping issue. Recently, various FWI methods have been suggested to address this problem. Such methods are mainly classified into either data-space manipulation or model-space extension. We developed an alternative FWI method that belongs to the latter class. First, we define the virtual scattering source based on perturbation theory. The virtual scattering source is estimated by minimizing the differences between observed and simulated data with a regularization term penalizing the weighted virtual scattering source. The inverse problem for obtaining the virtual scattering source can be solved by the linear conjugate gradient method. The inverted virtual scattering source is used to update the wavefields; thus, it helps FWI to better approximate the nonlinearity of the inverse scattering problem. As the second step, the virtual scattering source is minimized to invert the velocity model. By assuming that the variation of the reconstructed wavefield is negligible, we can apply an approximated full Newton method to the velocity inversion with reasonable cost comparable to the Gauss-Newton method. From the numerical examples using synthetic data, we confirm that the proposed method performs better and more robust than the simple gradient-based FWI method. In addition, we show that our objective function has fewer local minima, which helps to mitigate the cycle-skipping problem.

A generalized Newton iteration for computing the solution of the inverse Henderson problem

ABSTRACT We develop a generalized Newton scheme called IHNC (inverse hypernetted-chain iteration) for the construction of effective pair potentials for systems of interacting point-like particles. The construction is realized in such a way that the distribution of the particles matches a given radial distribution function. The IHNC iteration uses the hypernetted-chain integral equation for an approximate evaluation of the inverse of the Jacobian of the forward operator. In contrast to the full Newton method realized in the Inverse Monte Carlo (IMC) scheme, the IHNC algorithm requires only a single molecular dynamics computation of the radial distribution function per iteration step and no further expensive cross-correlations. Numerical experiments are shown to demonstrate that the method is as efficient as the IMC scheme, and that it easily allows to incorporate thermodynamical constraints.

Improved Multiple-Mixing-Cell Method for Accelerating Minimum Miscibility Pressure Calculations

Summary An improved algorithm for accelerating minimum miscibility pressure (MMP) computation using the multiple-mixing-cell (MMC) methods is presented. The MMC method is widely used to accurately calculate the MMP. In this study, we proposed an acceleration algorithm toward original MMC method to directly locate the shortest key tie-line (TL) after a certain amount of contacts through the adjustment of the gas/oil mixing ratio during the calculation process. The algorithm contains the following key components: (1) mixing cell cutoff strategy to avoid unnecessary flash calculations; (2) gas/oil mixing ratio adjustment to prevent lost information on the shortest key TL during the cell cutoff process; (3) a search algorithm for pressure to improve the next step pressure estimate; (4) the fast and reliable two-phase flash implementation by combining full Newton method with recently proposed iteration variables and conventional successive substitution method. The improved MMC model is shown to be faster than the original MMC method in computing MMP.

Adaptive full newton-step infeasible interior-point method for sufficient horizontal LCP

ABSTRACTAn adaptive full Newton-step infeasible-interior-point method for solving sufficient horizontal linear complementarity problems is analysed and sufficient conditions are given for the superlinear convergence of the sequence of iterates. The main feature of the method is that the parameter defining the Newton-step is adaptively chosen at each iteration, in contrast with previous full-Newton step methods where this parameter is kept fixed at all iterations. We mention that no superlinear convergence results are known for the latter methods. The theoretical complexity of our method matches the best known results in the literature. In the first algorithm, we assume that an upper bound for the handicap of the problem is known. The second algorithm does not depend on the handicap of the problem, so that it can readily be applied to any horizontal linear complementarity problem.

Density-based phase envelope construction

A new density-based phase envelope construction procedure is proposed. The equations for two-phase saturation molar density and temperature computations are established by equating to zero the tangent plane distance (TPD) function in terms of component molar densities and temperature to find a non-trivial set of component molar densities at various specifications: mixture molar density, temperature and the modified equilibrium constants. The latter are defined as the ratios of feed to incipient phase component molar density. The nonlinear system of equations is solved by the full Newton method. The sequence of calculations is started at some conditions where convergence is easily obtained and an extrapolation procedure provides high quality initial guesses for subsequent calculations. The phase envelope is traced in the molar density-temperature plane, where a unique saturation temperature exists at specified mixture molar density; the representation in the pressure-temperature plane is obtained by calculating explicitly the pressure from the equation of state at given temperature and component molar densities on the saturation curve. The equation of state (EoS) must not be solved for volume and the elements of the Jacobian matrix have simpler expressions than their conventional (pressure-based) counterparts. The proposed method is successfully tested for a variety of mixtures, ranging from binary and ternary mixtures to reservoir fluids, exhibiting usual (closed) and unusual phase envelopes, such as open-shaped envelopes characterized by a bubble point branch extending to infinity, branches with negative pressures constructed in a single run (unlike in the conventional case), several branches, double retrograde behavior and swallowtail pattern corresponding to liquid demixing at low temperatures. The proposed method is not dependent of the thermodynamic model and any pressure-explicit equation of state can be used, provided the required partial derivatives of fugacity and pressure with respect to mole numbers, temperature and volume are available.

FEM×DEM multiscale modeling: Model performance enhancement from Newton strategy to element loop parallelization

SummaryThis paper presents a multiscale model based on a FEM×DEM approach, a method that couples discrete elements at the microscale and finite elements at the macroscale. FEM×DEM has proven to be an effective way to treat real‐scale engineering problems by embedding constitutive laws numerically obtained using discrete elements into a standard finite element framework. This proposed paper focuses on some numerical open issues of the method. Given the nonlinearity of the problem, Newton's method is required. The standard full Newton method is modified by adopting operators different from the consistent tangent matrix and by developing ad‐hoc solution strategies. The efficiency of several existing operators is compared, and a new and original strategy is proposed, which is shown to be numerically more efficient than the existing propositions. Furthermore, a shared memory parallelization framework using OpenMP directives is introduced. The combination of these enhancements allows to overcome the FEM×DEM computational limitations, thus making the approach competitive with classical FEM in terms of stability and computational cost.

Estimation of elastic constants for HTI media using Gauss-Newton and full-Newton multiparameter full-waveform inversion

In seismic full-waveform inversion (FWI), subsurface parameters are estimated by iteratively minimizing the difference between the modeled and the observed data. We have considered the problem of estimating the elastic constants of a fractured medium using multiparameter FWI and modeling naturally fractured reservoirs as equivalent anisotropic media. Multiparameter FWI, although promising, remains exposed to a range of challenges, one being the parameter crosstalk problem resulting from the overlap of Fréchet derivative wavefields. Parameter crosstalk is strongly influenced by the form of the scattering pattern for each parameter. We have derived 3D radiation patterns associated with scattering from a range of elastic constants in general anisotropic media. Then, we developed scattering patterns specific to a horizontal transverse isotropic (HTI) medium to draw conclusions about parameter crosstalk in FWI. Bare gradients exhibit crosstalk, as well as artifacts caused by doubly scattered energy in the data residuals. The role of the multiparameter Gauss-Newton (GN) Hessian in suppressing parameter crosstalk is revealed. We have found that the second-order term in the multiparameter Hessian, which is associated with multiparameter second-order scattering effects, can be constructed with the adjoint-state technique. We have examined the analytic scattering patterns for HTI media with a 2D numerical example. We have examined the roles played by the first- and second-order terms in multiparameter Hessian to suppress parameter crosstalk and second-order scattering artifacts numerically. We have also compared the multiparameter GN and full-Newton methods as methods for determining the elastic constants in HTI media with a two-block-layer model.

Application of the least-squares inversion method: Fourier series versus waveform inversion

We describe an implicit link between waveform inversion and Fourier series based on inversion methods such as gradient, Gauss–Newton, and full Newton methods. Fourier series have been widely used as a basic concept in studies on seismic data interpretation, and their coefficients are obtained in the classical Fourier analysis. We show that Fourier coefficients can also be obtained by inversion algorithms, and compare the method to seismic waveform inversion algorithms. In that case, Fourier coefficients correspond to model parameters (velocities, density or elastic constants), whereas cosine and sine functions correspond to components of the Jacobian matrix, that is, partial derivative wavefields in seismic inversion. In the classical Fourier analysis, optimal coefficients are determined by the sensitivity of a given function to sine and cosine functions. In the inversion method for Fourier series, Fourier coefficients are obtained by measuring the sensitivity of residuals between given functions and test functions (defined as the sum of weighted cosine and sine functions) to cosine and sine functions. The orthogonal property of cosine and sine functions makes the full or approximate Hessian matrix become a diagonal matrix in the inversion for Fourier series. In seismic waveform inversion, the Hessian matrix may or may not be a diagonal matrix, because partial derivative wavefields correlate with each other to some extent, making them semi-orthogonal. At the high-frequency limits, however, the Hessian matrix can be approximated by either a diagonal matrix or a diagonally-dominant matrix. Since we usually deal with relatively low frequencies in seismic waveform inversion, it is not diagonally dominant and thus it is prohibitively expensive to compute the full or approximate Hessian matrix. By interpreting Fourier series with the inversion algorithms, we note that the Fourier series can be computed at an iteration step using any inversion algorithms such as the gradient, full-Newton, and Gauss–Newton methods similar to waveform inversion.

Source wavelet estimation using common mid-point gathers

The source wavelet estimation is a crucial process for full waveform inversion because we can hardly know source information from observed data. In order to implement the full waveform inversion in the common mid-point (CMP) domain, source wavelet estimation may be performed using CMP gathers for efficiency and consistency. In this paper, we introduce the source wavelet estimation using CMP gather data. Since the CMP gather is the time–domain data, the source wavelet is transformed into the frequency domain to apply full Newton method. Finally, it is transformed back into the time domain. Through numerical examples, we show how sensitive the source wavelet inversion is to both the direct waves on CMP gathers and the uncertainty of initial velocity. We can succeed in accurately estimating source wavelet if we introduce time damping and the CMP data has direct waves in spite of an incorrect initial velocity. This is because direct waves, which are key elements for source estimation, propagate through water layer of which velocity information is precisely known. Therefore, we have to exploit direct wave for successful source wavelet estimation in full waveform inversion.

Partial Newton methods for a system of equations

We define and analyse partial Newton iterations for the solutions of a system of algebraic equations. Firstly we focus on a linear system of equations which does not require a line search. To apply a partial Newton method to a system of nonlinear equations we need a line search to ensure that the linearized equations are valid approximations of the nonlinear equations. We also focus on the use of one or two components of the displacement vector to generate a convergent sequence. This approach is inspired by the Simplex Algorithm in Linear Programming. As expected the partial Newton iterations are found not to have the fast convergence properties of the full Newton method. But the proposed partial Newton iteration makes it significantly simpler and faster to compute in each iteration for a system of equations with many variables. This is because it uses only one or two variables instead of all the search variables in each iteration.

Source estimation and direct wave reconstruction in Laplace-domain waveform inversion for deep-sea seismic data

SUMMARY We propose a strategy to overcome the high sensitivity to early-time noise of the Laplace-domain waveform inversion. In deep-sea seismic data, this problem is particularly crucial to obtaining accurate velocity structures. To this end, rather than simply filtering or muting early-time data, we propose replacing the original, noise-contaminated direct waves with analytically computed noise-free waves. To reconstruct the noise-free direct waves, we compute Green's functions for half-space media, estimate the source wavelet from the original direct waves using the full Newton method in the frequency domain and then convolve the Green's functions with the estimated source wavelet. The data obtained by merging the reconstructed direct waves with the original late-time data set can then be used for Laplace- and Laplace-Fourier-domain waveform inversions. To verify the source estimation and direct wave reconstruction strategy, we applied it to field data acquired in a deep-sea environment and obtained a realistic 2-D velocity model. The source estimation and direct wave reconstruction methods can also be applied to 3-D Laplace-domain waveform inversion.

Frequency-domain reverse-time migration with source estimation

Although artificially generated seismic sources such as dynamite, vibroseis, and air guns are used in seismic exploration, it is not easy to exactly recover the source wavelet in field recording or in data processing. For this reason, seismic data processing often assumes that an explosive-source wavelet can be described by a well-known function (e.g., a Ricker wavelet), a near-offset trace, or a deconvolved wavelet. In frequency-domain waveform inversion, it has been proven that a source wavelet can be estimated by an optimization method, and incorporating the source wavelet estimation into an inversion algorithm yields better inversion results. We have developed source wavelet estimation into 2D two-way frequency-domain reverse-time migration. The source wavelet is first estimated independently of reverse-time migration by an optimization method such as the full Newton method. It is then used in reverse-time migration. This source-wavelet-incorporated reverse-time migration algorithm is applied to three model data sets: a simple anticline model, the Institut Français du Petrole (IFP) Marmousi model, and the BP 2004 EAGE model. Numerical examples were used to show that better migration images can be obtained by using an estimated source wavelet than those obtained by using a designatured wavelet without source estimation. To enhance the migrated images, the Laplacian filter is applied and migrated images are scaled using the diagonal of the pseudo-Hessian matrix. It is expected that the source wavelet estimation method can be directly applied to 2D or 3D time-domain two-way reverse-time migration.

Blind Adaptive Equalization of Complex Signals based on the Constant Modulus Algorithm

The paper discuss, applicability of the second-order Newton gradient descent method for blind equalization of complex signals based on the Constant Modulus Algorithm (CMA). The Constant Modulus (CM) loss function is real with complex valued arguments, and, hence, non-analytic. The Hessian for noiseless FIR channels and rederive the known fact that the full Hessian of the CM loss function is always singular in a simpler manner. The channel model shows that the perfectly equalizing solutions are stationary points of the CM loss function and also evaluate the bit error rate. The paper also discuss of the full Newton method. Finally, to validate the proposed algorithm, simulation studies have been carried out and results are presented and compared. The simulation results show the effectiveness of the proposed algorithm. General Terms Mobile communications, FIR channels.

Use of the Newton Method for Blind Adaptive Equalization Based on the Constant Modulus Algorithm

We study the applicability of the second-order Newton gradient descent method for blind equalization of complex signals based on the constant modulus algorithm (CMA). The constant modulus (CM) loss function is real with complex valued arguments, and, hence, nonanalytic. We, therefore, use the framework of the Wirtinger calculus to derive a useful and insightful form of the Hessian for noiseless FIR channels and rederive the known fact that the full Hessian of the CM loss function is always singular in a simpler manner. For the implementation of a suboptimum version of Newton algorithm, we give the conditions under which the leading partial Hessian is nonsingular for a noiseless FIR channel model. For this channel model, we show that the perfectly equalizing solutions are stationary points of the CM loss function and also evaluate the leading partial Hessian and the full Hessian at a perfectly equalizing solution. We also discuss regularization of the full Newton method. Finally, some simulation results are given.

One-Dimensional Ablation Using a Full Newton's Method and Finite Control Volume Procedure

The development and verification of a one-dimensional constant density material thermal response code with ablation is presented. The implicit time integrator, control volume finite element spatial discretization, and Newton's method (with an analytical Jacobian) for the entire system of residual equations have been implemented and verified for variable material properties, Q* ablation, and thermochemical ablation problems. Timing studies were performed, and when accuracy is considered, the method developed in this study exhibits significant time savings over the property lagging approach. In addition, maximizing the Newton solver's convergence rate by including sensitivities to the surface recession rate reduces the overall computational time when compared to excluding recession rate sensitivities.

Full Newton method for inverse transmission line problems, utilising explicit second order derivatives

The full Newton's method is considered as an optimisation approach to the inverse transmission line problem in the frequency-domain. For the sake of accuracy and computational efficiency, the gradient and the Hessian of the cost-functional, with respect to parameter functions in the L2-space, are derived explicitly by means of the adjoint transmission line problem and the first and second order Frechét differentials of the cost-functional. The numerical implementation, when reducing to a finite dimensional parameter space, and a regularisation technique for the resulting ill-conditioned Hessian matrix are presented. For the reconstruction of one or two parameters, the algorithm is tested on synthetic reflection data contaminated with gaussian noise. The algorithm is also tested on measured reflection data to reconstruct a piecewise constant shunt-capacitance. The generalisation to the three-dimensional inverse scattering problem for bianisotropic media is presented.