Quasi-linear Methods Research Articles

A Quasi‐Newton Reformulated Geostatistical Approach on Reduced Dimensions for Large‐Dimensional Inverse Problems

AbstractFor classic geostatistical approaches relying on Gauss‐Newton or quasilinear methods, iterative evaluations of forward models to compute the Jacobians is a major computational challenge for large‐dimensional, underdetermined nonlinear inverse problems. Recent developments in dimension reduction based on principal component analysis have lower the number of forward model implementations in each iteration to the number of retained principal components given prior covariance or structural parameters. The present research develops a quasi‐Newton approach on reduced dimensions within the reformulated geostatistical approach framework to further lower the number of forward model calls and thus decrease the overall computational cost. The proposed algorithm uses Broyden's method to update the approximate Jacobian of the forward model with respect to the projections on the retained principal axes and the drift coefficients. When the secant condition is satisfied, no additional forward model runs or adjoint solver runs are needed. The efficiency and validity of the developed approach are tested by numerical experiments of large‐dimensional hydraulic tomography with true or biased prior information. Compared with Newton method using the full‐rank update of the Jacobians, the quasi‐Newton reformulated geostatistical approach provides similarly high‐quality inverse estimates with a 60–70% reduction of required forward model simulations. For inverse problems with biased prior information, the iterative correction of the principal axes or the update of the prior covariance may be influenced by the rank‐one update of the Jacobian, yielding slightly lower map accuracies, but still providing satisfactory inverse results with substantially reduced computational costs.

Big-data-driven pre-stack seismic intelligent inversion

Seismic inversion is an image formation process for the spatial structures and physical properties of underground rock strata according to the seismic data observed on the surface or in wells under the constraints of known geological laws and drilling and logging data. It is an important component of geophysical inversion. The inversion of reservoir parameters using seismic data is a non-linear problem. Therefore, the use of linear or quasi-linear methods to solve this problem makes it easy to fall into local optimal solutions. In contrast, intelligent optimisation algorithms based on the global optimum have strong local and global optimisation abilities and good convergence. Thus, they can improve the calculation efficiency and are suitable for geophysical inversion problems with multiple parameters and multiple extremums. In this study, the pre-stack seismic inversion problem is considered as the research object. According to the non-linear characteristics of the problem, the pre-stack seismic intelligent inversion method with a hybrid genetic algorithm is proposed, which solves the problems for which the standard genetic algorithm may easily fall into local optimums, resulting in an indistinct inversion effect, especially an unsatisfactory optimisation effect for density parameters. The experimental results show that the inversion parameters fit well with the theoretical model logging curve. In view of the high data volume in inversion, a new parallel strategy is proposed and combined with the MapReduce framework. This strategy can reduce the operating time while maintaining the diversity of the population. The strategy is implemented on the Hadoop platform and its effectiveness is verified.

Inversion for magnetotelluric data using the particle swarm optimization and regularized least squares

The major disadvantage of using traditional linear or quasi-linear methods to perform magnetotelluric data inversion is that these methods heavily depend on the initial models. If a selected initial model is not suitable, these inversion algorithms may fall into a local minimum or diverge during the iterations. Meanwhile, in most geophysical inversions, it is difficult to obtain a proper initial model because of the lack of sufficient prior information. To solve this problem, a hybrid inversion algorithm based on the combined particle swarm optimization and regularized least squares is proposed to invert magnetotelluric data. Particle swarm optimization algorithms have good performances in global searching. More importantly, these algorithms are not sensitive to the initial models. Therefore, this approach can be used to preliminary search the model parameter space. Then, the searched results are used as an initial model for iterations of inversion based on the least square regularization. Both synthetic data and field data tests demonstrated that the hybrid inversion has a good performance. The numerical experiments show that by utilizing the reliable initial model generated from the particle swarm optimization, the search results of hybrid inversion are greatly refined. The hybrid algorithm offers a significant improvement in both inversion efficiency and accuracy. In particular, this algorithm is suitable for the situations that lack initial information while inverting the field magnetotelluric data.

A new method of estimating derived cetane number for hydrocarbon fuels

A new spectroscopic predictor for estimating ignition characteristics including derived cetane number is proposed for hydrocarbon fuels. This spectroscopic predictor is the ratio of room temperature absorbance of unreacted fuel vapor at 3.41 and 3.39 μm, termed here as the “3.41/3.39 absorption ratio.” Its wide availability and applicability are demonstrated for a range of pure hydrocarbons, mixtures of pure hydrocarbons, and distillate and synthetic jet fuels. Quasi-linear calculation methods are provided for practical use. Spectroscopic and kinetic interpretations are provided based on the fraction of –CH2– hydrogen relative to all carbon-bonded hydrogen. In addition, the correlations between the proposed predictor and ignition delay time and C2H4 yield are presented and discussed to exhibit the predictor’s potential as a fuel screening tool.

Dynamics of zonal flows: failure of wave-kinetic theory, and new geometrical optics approximations

The self-organisation of turbulence into regular zonal flows can be fruitfully investigated with quasi-linear methods and statistical descriptions. A wave-kinetic equation that assumes asymptotically large-scale zonal flows leads to ultraviolet divergence. From an exact description of quasi-linear dynamics emerges two better geometrical optics approximations. These involve not only the mean flow shear but also the second and third derivative of the mean flow. One approximation takes the form of a new wave-kinetic equation, but is only valid when the zonal flow is quasi-static and wave action is conserved.

COHERENT NONHELICAL SHEAR DYNAMOS DRIVEN BY MAGNETIC FLUCTUATIONS AT LOW REYNOLDS NUMBERS

Nonhelical shear dynamos are studied with a particular focus on the possibility of coherent dynamo action. The primary results -- serving as a follow up to the results of Squire & Bhattacharjee [arXiv:1506.04109 (2015)] -- pertain to the "magnetic shear-current effect" as a viable mechanism to drive large-scale magnetic field generation. This effect raises the interesting possibility that the saturated state of the small-scale dynamo could drive large-scale dynamo action, and is likely to be important in the unstratified regions of accretion disk turbulence. In this paper, the effect is studied at low Reynolds numbers, removing the complications of small-scale dynamo excitation and aiding analysis by enabling the use of quasi-linear statistical simulation methods. In addition to the magnetically driven dynamo, new results on the kinematic nonhelical shear dynamo are presented. These illustrate the relationship between coherent and incoherent driving in such dynamos, demonstrating the importance of rotation in determining the relative dominance of each mechanism.

Linear and non-linear properties of feature selectivity in V4 neurons.

Extrastriate area V4 is a critical component of visual form processing in both humans and non-human primates. Previous studies have shown that the tuning properties of V4 neurons demonstrate an intermediate level of complexity that lies between the narrow band orientation and spatial frequency tuning of neurons in primary visual cortex and the highly complex object selectivity seen in inferotemporal neurons. However, the nature of feature selectivity within this cortical area is not well understood, especially in the context of natural stimuli. Specifically, little is known about how the tuning properties of V4 neurons, measured in isolation, translate to feature selectivity within natural scenes. In this study, we assessed the degree to which preferences for natural image components can readily be inferred from classical orientation and spatial frequency tuning functions. Using a psychophysically-inspired method we isolated and identified the specific visual “driving features” occurring in natural scene photographs that reliably elicited spiking activity from single V4 neurons. We then compared the measured driving features to those predicted based on the spectral receptive field (SRF), estimated from responses to narrowband sinusoidal grating stimuli. This approach provided a quantitative framework for assessing the degree to which linear feature selectivity was preserved during natural vision. First, we found evidence of both spectrally and spatially tuned suppression within the receptive field, neither of which were present in the linear SRF. Second, we found driving features that were stable during translation of the image across the receptive field (due to small fixational eye movements). The degree of translation invariance fell along a continuum, with some cells showing nearly complete invariance across the receptive field and others exhibiting little to no position invariance. This form of limited translation invariance could indicate that a subset of V4 neurons are insensitive to small fixational eye movements, supporting perceptual stability during natural vision.

Statistical simulation of the magnetorotational dynamo.

Turbulence and dynamo induced by the magnetorotational instability (MRI) are analyzed using quasilinear statistical simulation methods. It is found that homogenous turbulence is unstable to a large-scale dynamo instability, which saturates to an inhomogenous equilibrium with a strong dependence on the magnetic Prandtl number (Pm). Despite its enormously reduced nonlinearity, the dependence of the angular momentum transport on Pm in the quasilinear model is qualitatively similar to that of nonlinear MRI turbulence. This demonstrates the importance of the large-scale dynamo and suggests how dramatically simplified models may be used to gain insight into the astrophysically relevant regimes of very low or high Pm.

Transport bifurcation induced by sheared toroidal flow in tokamak plasmas

First-principles numerical simulations are used to describe a transport bifurcation in a differentially rotating tokamak plasma. Such a bifurcation is more probable in a region of zero magnetic shear than one of finite magnetic shear, because in the former case the component of the sheared toroidal flow that is perpendicular to the magnetic field has the strongest suppressing effect on the turbulence. In the zero-magnetic-shear regime, there are no growing linear eigenmodes at any finite value of flow shear. However, subcritical turbulence can be sustained, owing to the existence of modes, driven by the ion temperature gradient and the parallel velocity gradient, which grow transiently. Nonetheless, in a parameter space containing a wide range of temperature gradients and velocity shears, there is a sizeable window where all turbulence is suppressed. Combined with the relatively low transport of momentum by collisional (neoclassical) mechanisms, this produces the conditions for a bifurcation from low to high temperature and velocity gradients. A parametric model is constructed which accurately describes the combined effect of the temperature gradient and the flow gradient over a wide range of their values. Using this parametric model, it is shown that in the reduced-transport state, heat is transported almost neoclassically, while momentum transport is dominated by subcritical parallel-velocity-gradient-driven turbulence. It is further shown that for any given input of torque, there is an optimum input of heat which maximises the temperature gradient. The parametric model describes both the behaviour of the subcritical turbulence (which cannot be modelled by the quasi-linear methods used in current transport codes) and the complicated effect of the flow shear on the transport stiffness. It may prove useful for transport modelling of tokamaks with sheared flows.

Linear and quasi-linear iterative splitting methods: Theory and applications

Linear and quasi-linear iterative splitting methods: Theory and applications

Piecewise-quasilinearization techniques for singularly perturbed Volterra integro-differential equations

Three families of iterative quasilinear methods for the numerical solution of singularly perturbed Volterra integro-differential equations are presented. Two of these methods are based on the piecewise integration of the differential equations and the use of quadrature rules, while the third one employs a locally frozen approximation for the coefficients and right-hand side of the differential equation. It is shown that these quasilinear methods are first-order accurate for step sizes equal to or larger than the initial layer thickness, and are less accurate than exponentially fitted techniques whose coefficients are selected so that the finite difference equations have truncation errors of second- and third-order, even though they exhibit only first-order accuracy for initial layer thicknesses equal to or smaller than the step size. It is also shown that the quasilinear methods presented in this paper are of exponential type and provide more accurate results than the backward Euler and trapezoidal methods for step sizes larger than the initial layer thickness, and the quasilinear frozen method is less accurate than quasilinear techniques that make use of an integrating factor and numerical quadratures.

A New Approximation for 3D Electromagnetic Scattering in the Presence of Anisotropic Conductive Media

Accurate and efficient modeling of three-dimensional (3D) electromagnetic (EM) scattering remains an open challenge in the presence of anisotropic conductive media. Numerical algorithms used to simulate the response of dipping and anisotropic rock formations can easily exceed standard computer resources as EM fields become fully coupled in general. In the past, several scattering approximations have been developed to efficiently simulate complex EM problems arising in the probing of subsurface rock formations. These approximations include the Born, Rytov, Extended Born (ExBorn) and quasi-linear (QL) methods, among others. However, so far none of these approximations have been adapted to simulate scattering in the presence of anisotropic conductive media. In this paper, we describe and benchmark a novel EM scattering approximation that remains accurate and efficient in the presence of 3D anisotropic conductive media. The approximation is based on the integral formulation of EM scattering and takes advantage of the spatial smoothness and general vectorial properties of EM fields internal to scatterers. A general vectorial formulation is used to properly account for complex EM coupling due to anisotropy. Several numerical examples borrowed from borehole induction logging are used to describe and assess the accuracy and efficiency of the new EM scattering approximation. The approximation allows one to accurately simulate the EM response of more than 1 million cells within a few minutes of CPU time on a serial computer with standard memory and speed resources.

The Anisotropic Lagrangian Averaged Euler and Navier-Stokes Equations

The purpose of this paper is twofold. First, we give a derivation of the Lagrangian averaged Euler (LAE-α) and Navier-Stokes (LANS-α) equations. This theory involves a spatial scale α and the equations are designed to accurately capture the dynamics of the Euler and Navier-Stokes equations at length scales larger than α, while averaging the motion at scales smaller than α. The derivation involves an averaging procedure that combines ideas from both the material (Lagrangian) and spatial (Eulerian) viewpoints. This framework allows the use of a variant of G. I. Taylor's turbulence hypothesis as the foundation for the model equations; more precisely, the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow. In this article, we use this hypothesis to derive the averaged Lagrangian for the theory, and all the terms up to and including order α^2 are accounted for. The equations come in both an isotropic and anisotropic version. The anisotropic equations are a coupled system of PDEs (partial differential equations) for the mean velocity field and the Lagrangian covariance tensor. In earlier works by Foias, Holm & Titi [10], and ourselves [16], an analysis of the isotropic equations has been given. In the second part of this paper, we establish local in time well-posedness of the anisotropic LANS-α equations using quasilinear PDE type methods.

Shimmy analysis of a simple aircraft nose landing gear model using different mathematical methods

Shimmy oscillations are still a problem in design and operation of aircraft landing gears, and accurate and appropriate analysis is required to master the task. Based on a nonlinear model of the mechanics of the landing gear and tire elasticity according to elastic string theory, some well known linear and nonlinear mathematical methods are applied to the shimmy analysis of a simple model of a nose gear: Computing eigenvalues, solving analytically the stability boundaries with a parameter space method, getting limit cycles by analytical formulae from describing functions, and last but not least numerical simulation of time histories. It seems that linear or quasi linear methods and analytical solutions are well suited to obtain extensive insight, respecting limitations of these methods. Numerical simulation on the other hand is a valuable tool for pointing out specific effects of a nonlinear system in large amplitude regions.

Modified equation techniques for reactive-diffusive systems. Part 1: explicit, implicit and quasilinear methods

A family of modified equation techniques for one-dimensional reaction-diffusion equations has been developed and applied to the study of a single reaction-diffusion equation which has an exact travelling wave solution. Modified equation methods have also been applied to study the propagation of a laminar flame through an ozone-oxygen mixture. It is shown that modified implicit techniques are less accurate than the standard implicit method if the time derivatives are evaluated by means of a first-order accurate formula. If the time derivatives are approximated by second-order accurate expressions, modified equation methods yield L 2- norm errors which are almost independent of the time step used in the calculations; they are also independent of time. Large accuracy improvements can be obtained by using modified equation methods. These improvements are achieved at the expense of involving five grid points and requiring iterations if quasilinear techniques are used in the calculations. A linear stability analysis of the modified equation methods presented in this paper shows that the modified explicit scheme requires smaller time steps than the standard explicit method.

Convergence of multistep methods for systems of ordinary differential equations with parameters

The author considers the convergence of quasilinear nonstationary multistep methods for systems of ordinary differential with parameters. Sufficient conditions for their convergence are given. The new numerical method is tested for two examples and it turns out to be a little better than the Hamming method.

Quasilinear Multistep Methods and Variable Step Predictor–Corrector Methods for Neutral Functional-Differential Equations

The theory of quasilinear multistep methods and predictor–corrector methods for neutral functional differential equations is developed. The theorem about the asymptotic behaviour of the global discretization error of quasilinear methods, which forms a theoretical basis for step control strategy for predictor–corrector methods for neutral equations, is given. Some problems of implementation of these methods are also addressed, and the results of numerical experiments are presented.

Convergence of multistep methods for Volterra functional differential equations

This paper deals with the convergence of nonstationary quasilinear multistep methods with varying step, used for the numerical integration of Volterra functional differential equations. A Perron ty...

Comparison of Some Optimal Control Methods for the Design of Turbine Blades

This paper attempts a comparative study of some numerical methods for the optimal control design of turbine blades whose vibration characteristics are approximated by Timoshenko beam idealizations with shear and incorporating simple boundary conditions. The blade was synthesized using the following methods (1) conjugate gradient minimization of the system Hamiltonian in function space incorporating penalty function transformations (2) projection operator methods in a function space which includes the frequencies of vibration and the control function (3) ε-technique penalty function transformation resulting in a highly nonlinear programming problem (4) finite difference discretization of the state equations again resulting in a nonlinear program (5) second variation methods with complex state differential equations to include damping effects resulting in systems of inhomogeneous matrix Riccatti equations some of which are stiff (6) quasi-linear methods based on iterative linearization of the state and adjoint equation. The paper includes a discussion of some substantial computational difficulties encountered in the implementation of these techniques together with a resume of work presently in progress using a differential dynamic programming approach.

On the transient solution of the unforced duffing equation with large damping†

The refined elliptic-function method previously developed (Soudack and Barkham 1970) is applied to the unforced Duffing equation with large damping. The theoretical development for this case, and an example, are presented. The Kryloff-Bogoliuboff solution is shown for comparison with the present result. The advantage of this approach, well illustrated by the example, is in its ability to predict the solution phase. The approach is therefore superior to quasi-linear methods, such as the Kryloff-Bogoliuboff method, which rapidly fails to give a good approximation due to phase errors.