Nonlinear Situations Research Articles

Global Well-Posedness for the Generalized Large-Scale Semigeostrophic Equations

We prove existence and uniqueness of global classical solutions to the generalized large-scale semigeostrophic equations with periodic boundary conditions. This family of Hamiltonian balance models for rapidly rotating shallow water includes the L1 model derived by R. Salmon in 1985 and its 2006 generalization by the second author. The results are, under the physical restriction that the initial potential vorticity is positive, as strong as those available for the Euler equations of ideal fluid flow in two dimensions. Moreover, we identify a special case in which the velocity field is two derivatives smoother in Sobolev space as compared to the general case. Our results are based on careful estimates which show that, although the potential vorticity inversion is nonlinear, bounds on the potential vorticity inversion operator remain linear in derivatives of the potential vorticity. This permits the adaptation of an argument based on elliptic Lp theory, proposed by Yudovich in 1963 for proving existence and uniqueness of weak solutions for the two-dimensional Euler equations, to our particular nonlinear situation.

On the Smoothed Minimum Error Entropy Criterion

Recent studies suggest that the minimum error entropy (MEE) criterion can outperform the traditional mean square error criterion in supervised machine learning, especially in nonlinear and non-Gaussian situations. In practice, however, one has to estimate the error entropy from the samples since in general the analytical evaluation of error entropy is not possible. By the Parzen windowing approach, the estimated error entropy converges asymptotically to the entropy of the error plus an independent random variable whose probability density function (PDF) corresponds to the kernel function in the Parzen method. This quantity of entropy is called the smoothed error entropy, and the corresponding optimality criterion is named the smoothed MEE (SMEE) criterion. In this paper, we study theoretically the SMEE criterion in supervised machine learning where the learning machine is assumed to be nonparametric and universal. Some basic properties are presented. In particular, we show that when the smoothing factor is very small, the smoothed error entropy equals approximately the true error entropy plus a scaled version of the Fisher information of error. We also investigate how the smoothing factor affects the optimal solution. In some special situations, the optimal solution under the SMEE criterion does not change with increasing smoothing factor. In general cases, when the smoothing factor tends to infinity, minimizing the smoothed error entropy will be approximately equivalent to minimizing error variance, regardless of the conditional PDF and the kernel.

Impact of nonlinearity and discontinuity on the spatial scaling effects of the leaf area index retrieved from remotely sensed data

This article addresses the major scaling problems in leaf area index (LAI) retrieval for a heterogeneous surface associated with (1) the nonlinearity in the relationships between remotely sensed reflectances and LAI products, (2) the discontinuity caused by the mixture of contrasting cover types that is categorized as the dominating type within a large-scale pixel, and (3) the algorithm for the dominant cover type being used for the retrieval of the LAI in that large-scale mixed pixel. Through mathematical analysis, two scaling models (a component-based model and a pixel-based model) are proposed on the basis of the Taylor series expansion with the corresponding textural and contextural parameters (i.e. variance–covariance matrices and component fractions) to correct for the scaling effects among LAI products at different scales. These models express the magnitude of the scaling effects for the nonlinear and discontinuous situations as a function of (1) the degree of nonlinearity quantified by the second derivative of the retrieval function, (2) the spatial heterogeneity quantified by variance–covariance matrices, and (3) the component fractions in the large-scale mixed pixel. To evaluate the proposed scaling models, a scaling correction test is performed and analysed on a SPOT (Système Pour l'Observation de la Terre) image for two vegetation types. The component fractions have proven to be the main reason for the scaling effects in a mixed pixel. Compared to the results before scaling, using either of the two proposed models greatly reduces the retrieval errors that the scaling effects cause. The relative scaling effects of the LAI may be up to 55% in an uncorrected, large-scale mixed pixel. However, the relative scaling errors can be as low as 2% with the intra-component textural parameters and about 13% with the intra-pixel textural parameters. Because the scaling effects can be corrected for the spatial heterogeneity caused either by density changes within the same cover or by cover type changes, our work indicates that the proposed scaling models are promising and feasible.

Nonlinear Sierpiński and Riesel numbers

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k ⋅ 2 n + 1 is composite for all positive integers n . Such values of k are known as Sierpiński numbers . Extending the ideas of Sierpiński to a nonlinear situation, Chen showed that there exist infinitely many positive integers k such that k r ⋅ 2 n + d is composite for all positive integers n , where d ∈ { − 1 , 1 } , provided that r is a positive integer with r ≢ 0 , 4 , 6 , 8 ( mod 12 ) . Filaseta, Finch and Kozek improved Chenʼs result by completely lifting the restrictions on r when d = 1 , and they asked if a similar result exists if k r is replaced by f ( k ) , where f ( x ) is an arbitrary nonconstant polynomial in Z [ x ] . In this article, we address this question when f ( x ) = a x r + b x + c ∈ Z [ x ] . In particular, we show, for various values of a , b , c , d and r , that there exist infinitely many positive integers k such that f ( k ) ⋅ 2 n + d is composite for all integers n ⩾ 1 . When d = 1 or −1, we refer to such values of k as nonlinear Sierpiński or nonlinear Riesel numbers, respectively.

A Hybrid Wavelet Multiresolution Particle Filter(HWMPF) Solution for GPS/INS Navigation

It is well known that Kalman filtering is an optimal state estimation method for GPS/INS integration navigation. However, it can perform perfectly only in the condition that transition models are well defined. The Extended Kalman filter (EKF), as its nonlinear vision, gives a better accuracy which is considered as a common method for practical integrated navigation application for error estimation and compensation. While it still has difficulties to be implemented and achieve stability, adaptability and observability. To solve this problem, particle filter (PF) based on Bayesian Optimal Estimation (BOE) has shown its advantages, even so, it has been bounded by the complex calculation for the application. Meanwhile, it is found that wavelet transform featured with multi-resolution which can effectively reduce the undesirable noise of sensors and improve the Signal-to-Noise Ratio (SNR), which broadly used in denoising the unwanted data of the sensor signal. Therefore a Hybrid Wavelet Multiresolution Particle Filter(HWMPF) solution is proposed: firstly, wavelet filter is employed to handle the inertial sensor noise, and then velocity and acceleration detection are applied to judge whether the system is in high speed and dynamic condition or not. High dynamic system is implemented in using wavelet–based particle filter to deal with strong nonlinear situation with less computational complexity, while EKF is applied for low velocity and dynamic system. In addition, Zero Velocity Updates (ZUPT) carried out in zero velocity situations. The hybrid filter solution can deal with the most part of cases in practical navigation, and simulation and experimental results have confirmed feasibility and rationality of the proposed solution.

Influence of an electric field on the capillary analogue of the dead-water effect

It is shown that the capillary analogue of the dead-water effect strongly depends on an external electrostatic field: in the presence of the field, the number of nonlinear resonance situations increases sharply. The field increases the wave amplitudes and excites both degenerate and secondary combination resonances, which are not observed without the field.

Focused waves and wave–structure interaction in a numerical wave tank

Sustainable and efficient design solutions are the aim for any engineer. In offshore engineering forces resulting from extreme wave impact are of special interest as these challenge the structure and the crew working in this harsh environment. Theoretical models tend to be limited to linear or weakly non-linear situations and are unable to predict the violent and turbulent effects of breaking waves in combination with wave run up on structures or green water loading. The classic approach for such cases is to carry out scale model tests in a physical wave tank and measure the forces, water levels and flow velocities at some chosen locations.Another approach is to use fully non-linear calculations, such as Computational Fluid Dynamics, which have the potential to investigate the design in different conditions at full scale. This paper deals with the generation and behaviour of extreme focused wave groups in a numerical wave tank. Non-linear effects of these extreme waves are shown and the implications for a numerical wave tank are discussed. Also the forces on horizontal and vertical cylinders, which represent simple models of offshore structures, are calculated. All numerical results are compared with measured data from physical experiments.

Design of a Power Factor Measurement System for Nonlinear Load

This paper introduces and develops an efficient method for measuring power factor (PF) and its nature under nonlinear load current situations. The method is based on generating a pulse width modulated signal whose width correlates to the value of PF. This signal can then be employed as a feedback signal for controlling PF related power quantities in a system. This method has the advantages of its simple implementation, less computational complexity, and its allowable error of less than 4[%], which is justified by the computer simulation results.

Geometrical expression of excess entropy production

We derive a geometrical expression of the excess entropy production for quasistatic transitions between nonequilibrium steady states of Markovian jump processes, which can be exactly applied to nonlinear and nonequilibrium situations. The obtained expression is geometrical; the excess entropy production depends only on a trajectory in the parameter space, analogous to the Berry phase in quantum mechanics. Our results imply that vector potentials are needed to construct the thermodynamics of nonequilibrium steady states.

Nonparametric particle filtering and smoothing with quasi-Monte Carlo sampling

Sequential Monte Carlo methods (also known as particle filters and smoothers) are used for filtering and smoothing in general state-space models. These methods are based on importance sampling. In practice, it is often difficult to find a suitable proposal which allows effective importance sampling. This article develops an original particle filter and an original particle smoother which employ nonparametric importance sampling. The basic idea is to use a nonparametric estimate of the marginally optimal proposal. The proposed algorithms provide a better approximation of the filtering and smoothing distributions than standard methods. The methods’ advantage is most distinct in severely nonlinear situations. In contrast to most existing methods, they allow the use of quasi-Monte Carlo (QMC) sampling. In addition, they do not suffer from weight degeneration rendering a resampling step unnecessary. For the estimation of model parameters, an efficient on-line maximum-likelihood (ML) estimation technique is proposed which is also based on nonparametric approximations. All suggested algorithms have almost linear complexity for low-dimensional state-spaces. This is an advantage over standard smoothing and ML procedures. Particularly, all existing sequential Monte Carlo methods that incorporate QMC sampling have quadratic complexity. As an application, stochastic volatility estimation for high-frequency financial data is considered, which is of great importance in practice. The computer code is partly available as supplemental material.

Theoretical and Numerical Analysis on the Seismic Response of Underground Structures in Existence of Isolators

In this paper, an analytical method of the beam with springs and dampers fixed at the ends was proposed based on equivalent single degree of freedom (SDOF) system and secondary Lagrange’s dynamic equations, in order to develop a new effective method to enhance the aseismic capability of underground structures. The dynamic response of elastically supported and damply supported beams subjected to both seismic loading and static axial loading was analyzed by the proposed analytical procedure. The theoretical results were validated by the numerical simulation. In order to further investigate the effects of springs and dampers fixed at the ends of the columns in nonlinear response situation, the 3D nonlinear seismic responses of the Dakai metro station structure with and without the isolators were analyzed by ABAQUS respectively. It is demonstrated that: (1) the proposed analytical procedure can predict the dynamic response of beams with elastic and damper supports subjected to both seismic loading and axial loading. (2) Setting isolators at the supports of the column could enhance the aseismic capability of the structure effectively. (3) The axial static loading induced by the gravity of the soil and structure provide the constraint on the column, and therefore could not be neglected in the structural dynamic analysis.

Bayesian Nonparametric Modeling for Causal Inference

Researchers have long struggled to identify causal effects in nonexperimental settings. Many recently proposed strategies assume ignorability of the treatment assignment mechanism and require fitting two models—one for the assignment mechanism and one for the response surface. This article proposes a strategy that instead focuses on very flexibly modeling just the response surface using a Bayesian nonparametric modeling procedure, Bayesian Additive Regression Trees (BART). BART has several advantages: it is far simpler to use than many recent competitors, requires less guesswork in model fitting, handles a large number of predictors, yields coherent uncertainty intervals, and fluidly handles continuous treatment variables and missing data for the outcome variable. BART also naturally identifies heterogeneous treatment effects. BART produces more accurate estimates of average treatment effects compared to propensity score matching, propensity-weighted estimators, and regression adjustment in the nonlinear simulation situations examined. Further, it is highly competitive in linear settings with the “correct” model, linear regression. Supplemental materials including code and data to replicate simulations and examples from the article as well as methods for population inference are available online.

Generalized control variate methods for pricing Asian options

The conventional control variate method proposed by Kemna and Vorst (1990) to evaluate Asian options under the Black-Scholes model can be interpreted as a particular selection of linear martingale controls. We generalize the constant control parameter into a control process to gain more reduction on variance. By means of an option price approximation, we construct a martingale control variate method, which outperforms the conventional control variate method. It is straightforward to extend such linear control to a nonlinear situation such as the American Asian option problem. From the variance analysis of martingales, the performance of control variate methods depends on the distance between the approximate martingale and the optimal martingale. This measure becomes helpful for the design of control variate methods for complex problems such as Asian option under stochastic volatility models. We demonstrate multiple choices of controls and test them under MC/QMC (Monte Carlo/ Quasi Monte Carlo)- simulations. QMC methods work signicantly well after adding a control, the variance reduction ratios increase to 260 times for randomized QMC compared with 60 times for MC simulations with a control.

Bidirectional oscillations in Er-doped fiber ring cavity with polarization splitting for rotation sensing

The structure of an all-fiber Erbium-doped fiber ring cavity with bidirectional single-mode oscillations and a rotation-sensitive beat signal for gyro applications is reported in this letter. The structure is achieved by introducing a polarization splitting method to filter the bidirectional oscillations and to control the mode coupling of the counter-propagating waves. Two distinct behaviors of the beat signal are identified, namely, the linear mode coupling situation caused by the back-reflection of the optical elements and the nonlinear mode coupling situation attributed to spatially non-uniform saturation in gain medium. Experiments are conducted to show the cavity rotation rate dependence of the beat frequency shift. The results are in good agreement with the theoretical predictions.

Automated Model Generation Algorithm for High-Level Fault Modeling

High-level modeling for operational amplifiers (opamps) has been previously carried out successfully using models generated by published automated model generation approaches. Furthermore, high-level fault modeling (HLFM) has been shown to work reasonably well using manually designed fault models. However, no evidence shows that published automated model generation approaches based on opamps have been used in HLFM. This paper describes HLFM for analog circuits using an adaptive self-tuning algorithm called multiple model generation system using delta. The generation algorithms and simulation models were written in MATLAB and the hardware description language VHDL-AMS, respectively. The properties of these self-tuning algorithms were investigated by modeling complementary metal-oxide-semiconductor opamps, and comparing simulations using the HLFM against those of the original simulation program with integrated circuit emphasis circuit utilizing transient analysis. Results show that the models can handle both linear and nonlinear fault situations with better accuracy than previously published HLFMs.

An approach based on self-organizing map and fuzzy membership for decomposition of mixed pixels in hyperspectral imagery

Spectral unmixing, which decomposes the mixed pixel into typical ground signatures (endmembers) and their fractional proportions (abundances) is a meaningful job for high-accuracy ground object recognition and quantitative remote sensing analysis. In this paper, a method for decomposition of mixed pixels which combines competitive neural network and fuzzy clustering, termed self-organizing map and fuzzy membership (SOM&FM) is proposed. The proposed method only demands some data samples as prior knowledge to train the SOM neural network in a supervised way. And the unmixing is based on the fuzzy model, which satisfies the abundances non-negative constraint (ANC) and the abundances summed-to-one constraint (ASC) automatically. Experimental results on synthetic and real hyperspectral data demonstrate that the proposed method can be used for both linear and nonlinear spectral mixture situations, and has good unmixing performances.

Overview of the Papers: Why is Linear Thinking so Dominant?

A remarkable feature of the set of papers in this issue is the consistent pattern of tending to deal with non-linear situations as if they are linear that is shown by students widely differing in age and educational systems, and studying various topics within mathematics. Methodological issues about the difficulty of interpretation of student behavior, and the need to avoid attributing more rationality than is warranted, are raised. Suggestions about educational steps that could be taken to avoid an over-dependence on linearity are made.

KADATH: A spectral solver for theoretical physics

KADATH is a library that implements spectral methods in a very modular manner. It is designed to solve a wide class of problems that arise in the context of theoretical physics. Several types of coordinates are implemented and additional geometries can be easily encoded. Partial differential equations of various types are discretized by means of spectral methods. The resulting system is solved using a Newton–Raphson iteration. Doing so, KADATH is able to deal with strongly non-linear situations. The algorithms are validated by applying the library to four different problems of contemporary physics, in the fields of gauge field theory and general relativity.

BDF-like methods for nonlinear dynamic analysis

We present several time integration algorithms of second-order accuracy that are numerically simple and effective for nonlinear elastodynamic problems. These algorithms are based on a general four-step scheme that has a resemblance to the backward differentiation formulas. We also present an extension to the composite strategy of the Bathe method. Appropriate values for the algorithmic parameters are determined based on considerations of stability and dissipativity, and less dissipative members of each algorithm have been identified. We demonstrate the convergence characteristics of the proposed algorithms with a nonlinear dynamic problem having analytic solutions, and test these algorithms with several three-dimensional nonlinear elastodynamic problems involving large deformations and rotations, employing St. Venant-Kirchhoff and compressible Neo-Hookean hyperelastic material models. These tests show that stable computations are obtained with the proposed algorithms in nonlinear situations where the trapezoidal rule encounters a well-known instability.

A variational perspective on controllability

We introduce and examine a variational approach to controllability problems based on the minimization of a suitable error functional that measures how far admissible functions are from being a solution of the underlying state law. This philosophy is based on imposing boundary, initial and desirable final conditions into the admissible class of functions, and focus on minimizing the departure of such admissible functions from being a solution of the state law. This strategy leads to considering a functional analytical framework which sometimes is a bit different from the standard one, but it permits to show that the unique continuation property is equivalent to exact controllability. Indeed, the unique continuation property can be interpreted as the fact that local minimizers of the error functional can only occur at zero error. The following two are the main ingredients in which this variational approach relies: the existence of minimizers for the error and the unique continuation property just stated. The main issue on the existence of minimizers is to overcome lack of coercivity in the classical sense. Two advantages of this approach are its natural extension to nonlinear situations, and its direct numerical implementation as a steepest descent procedure.