Nonlinear Operator Research Articles

Modified version of a simplified Landweber iterative method for nonlinear ill-posed operator equations

Modified version of a simplified Landweber iterative method for nonlinear ill-posed operator equations

Influence of the beam-beam collisions on the energetic particle distribution in large helical device

The influence of Coulomb collisions between energetic particles in a neutral beam injection (NBI)-heated plasma is numerically investigated utilizing the improved collision operator for the orbit-following type of Monte Carlo codes. This nonlinear collision operator enables the collisional effect by non-Maxwellian plasmas including energetic particles to be taken into account. We have implemented it into GNET, one of the orbit-following Monte Carlo codes solving drift kinetic equation in the 5D phase-space, and performed simulations for tangentially injected NBI cases (beam energy: 180 keV), taking account of beam–beam Coulomb collisions. It is found that the beam–beam collisions cause energetic particle orbits to transition from passing to trapped and enhance the collisional transport, which results in the deterioration of the energetic particle confinement.

A CONSTRUCTIVE INVESTIGATION BY THE METHODS OF TWO-SIDED APPROXIMATIONS OF BOUNDARY PROBLEMS FOR SECOND-ORDER SEMILINEAR ELLIPTIC EQUATIONS

The paper considers the first boundary value problem for a second-order semilinear elliptic equation. Problems of this class often arise in the modeling of processes occurring in chemistry, physics, biology, etc. A special place among the methods of analysis of the problems under consideration is occupied by the so-called constructive research methods, which allow not only to prove the existence of a problem solution, but also offer an algorithm for finding it with a given accuracy. For a constructive study of a nonlinear boundary value problem, it is proposed to use two versions of the method of two-sided approximations. Both methods are based on the transition from a differential problem to an equivalent nonlinear integral equation (using the Green’s function or using the Green-Rvachev’s quasi-function), which is analyzed by methods of the theory of nonlinear operators in semi-ordered Banach spaces. Conclusions about the existence of positive solutions of the constructed integral equations and the two-sided convergence of successive approximations to these solutions are made on the basis of the results of V.I. Opojcev on the solvability of nonlinear equations with a heterotone operator. The practical implementation of the method of two-sided approximations based on the use of the Green's function has certain limitations associated with the need to have an explicit expression for this function, which narrows the range of areas in which the method can actually be applied. The method of two-sided approximations, based on the use of the Green-Rvachev’s quasi-function, which can be constructed using the apparatus of the R-functions theory for regions of sufficiently arbitrary geometry, is free from this shortcoming. The proposed methods are illustrated by computational experiments for elliptic equations with Laplace and Helmholtz operators and heterotone power nonlinearity in a number of two- and three-dimensional domains. The results of both methods of two-sided approximations were compared with each other.

Beyond mean-field: Condensate coupled with pair excitations

We prove existence results for a system of partial differential equations describing the approximate condensate wavefunction and pair-excitation kernel of a dilute T = 0 Bose gas in the stationary setting, in the presence of a trapping potential and repulsive pairwise atomic interactions. Notably, the Hartree-type equation for the condensate in this system contains contributions from non-condensate particles, and the pair excitation kernel satisfies a nonlinear operator equation. The techniques employed include a direct variational principle and also an iterative procedure for constructing solutions.

THE GREEN-RVACHOV QUASI-FUNCTION METHOD IN THE NUMERICAL ANALYSIS OF MICROELECTROMECHANICAL SYSTEMS USING TWO-SIDED APPROXIMATIONS

The work is devoted to developing a bilateral iterative method of numerical analysis of one electrostatic microelectromechanical system based on the use of the Green – Rvachev quasi-function. Microelectromechanical systems are miniature devices that combine electronic and mechanical components of micron size. Electrostatically activated microelectromechanical systems have certain disadvantages that limit the range of their operation. One of these is the pull-in instability of the functional components of the system, which occurs if the applied voltage difference is above a particular critical value. The mathematical model of the system under consideration in the work is a semi-linear elliptic equation with the Laplace operator and the Dirichlet condition. To construct an approximate solution of the problem, it is proposed to use the methods of nonlinear analysis in semi-ordered spaces, particularly the results of V. I. Opoitsev on the solvability of nonlinear operator equations with a heterotonic operator. The boundary value problem simulating the most straightforward microelectromechanical system under the action of external pressure is reduced to the Urison integral equation using the Green – Rvachev quasi-function method, which makes it possible to expand the application of the method of two-sided approximations for problems in areas of somewhat arbitrary geometry. The paper substantiates the possibility of constructing iterative sequences with two-sided convergence to a positive solution of the problem: the computational scheme is given, the conditions for its convergence to the desired solution are obtained, and the a posteriori error estimate is derived. The method is illustrated by a computational experiment for a problem considered in a rectangular area. The computational experiment results are presented in the form of the surface and lines of the level of the approximate solution, and the two-sided nature of the convergence of the proposed method is also graphically illustrated.

Inversion of VLF-R data using a non-linear rank order smoothing constraint and its implementation to image surface rupture of the North Anatolian Fault in Başiskele, Kocaeli, NW Turkey

Inversion of electromagnetic induction data is often realized using smoothing operators for regularizing the process. These operators are applied using derivative matrices, penalizing differences between neighboring mesh cells. The resulting smoothing effect is similar to that of low-pass Gaussian filters and may cause difficulties due to the smoothed boundaries between structures. To prevent such ambiguities, non-linear rank filters are applied widely in image processing applications to filter low frequency variations while keeping boundaries. Correspondingly, we have defined a non-linear rank order smoothing operator for regularization. The effects of the defined parameters are demonstrated on synthetic models and the operator is observed to be able to provide better defined boundaries. The implementation of the operator on field data is realized using single-frequency VLF-R data collected on profiles traversing the surface rupture of the North Anatolian Fault in Kocaeli, Turkey. In the study area, ERT data are also collected on a single profile, and its smooth inversions are realized individually and jointly with VLF-R data for comparisons. Inversions show that the recovered subsurface structure using the rank order smoothing is closer to that indicated by the ERT data and previous studies. According to the recovered models, the clayey fillings in the fault rupture are acting as a barrier affecting the horizontal fluid flow in the near-surface and the resulting water accumulations around the fault rupture may cause susceptibility for earthquake induced damages.

Analysing heart rate variability in preterm infants: the effect of temporal adjustment of NN peaks and missing data.

The measurement of heart rate variability (HRV) in preterm infants provides important information on function to clinicians. Measuring the underlying electrocardiogram (ECG) in the neonatal intensive care unit is a challenge and there is a trade off between extracting accurate measurements of the HRV and the amount of ECG processed due to contamination. Knowledge on the effects of 1) quantization in the time domain and 2) missing data on the calculation of HRV features will inform clinical implementation. In this paper, we studied multiple 5 minute epochs from 148 ECG recordings on 56 extremely preterm infants. We found that temporal adjustment of NN peaks improves the estimate of the NN interval resulting in HRV features (m = 9) that are better correlated with age (median percentage increase in correlation of individual features: 0.2%, IQR: 0.0 to 5.6%; correlation with age predictor and age from 0.721 to 0.787). Improved (sub-sample) quantization of the NN intervals (via interpolation) reduced the overall value of HRV features (median percentage reduction in feature value: -1.3%, IQR: -18.8 to 0.0; m = 9), primarily through a reduction in the energy of high-frequency oscillations. HRV features were also robust to missing data, with measures such as mean NN, fractal dimension and the smoothed nonlinear energy operator (SNEO) less susceptible to missing data than features such as VLF, LF, and HF. Furthermore, age predictions derived from a combination of HRV measures were more robust to missing data than individual HRV measures.Clinical Relevance-Poor quantization in time when estimating the NN peak and the presence of missing data confound HRV measures, particularly spectral measures.

Learning Symbolic Expressions: Mixed-Integer Formulations, Cuts, and Heuristics

In this paper, we consider the problem of learning a regression function without assuming its functional form. This problem is referred to as symbolic regression. An expression tree is typically used to represent a solution function, which is determined by assigning operators and operands to the nodes. Cozad and Sahinidis propose a nonconvex mixed-integer nonlinear program (MINLP), in which binary variables are used to assign operators and nonlinear expressions are used to propagate data values through nonlinear operators, such as square, square root, and exponential. We extend this formulation by adding new cuts that improve the solution of this challenging MINLP. We also propose a heuristic that iteratively builds an expression tree by solving a restricted MINLP. We perform computational experiments and compare our approach with a mixed-integer program–based method and a neural network–based method from the literature. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by the Applied Mathematics activity within the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research [Grant DE-AC02-06CH11357]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0050 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0050 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Fuzzy contractor and nonlinear operator equation in fuzzy normed spaces

Fuzzy contractor and nonlinear operator equation in fuzzy normed spaces

Subordinations by η-convex functions for a class of nonlinear integral operators

The aim of the present paper is to investigate some subordination implications for a class of nonlinear integral operators associated with multivalent functions in the open unit disk. Some known results associated with a class of nonlinear averaging integral operators are also pointed out as special cases of the main findings which are presented here.

A new Mertens decomposition of [formula omitted]-submartingale systems. Application to BSDEs with weak constraints at stopping times

We first introduce the concept of Yg,ξ-submartingale systems, where the nonlinear operator Yg,ξ corresponds to the first component of the solution of a reflected BSDE with generator g and lower obstacle ξ. We first show that, in the case of a left-limited right-continuous obstacle, any Yg,ξ-submartingale system can be aggregated by a process which is right-lower semicontinuous. We then prove a Mertens decomposition, by using an original approach which does not make use of the standard penalization technique. These results are in particular useful for the treatment of control/stopping game problems and, to the best of our knowledge, they are completely new in the literature. As an application, we introduce a new class of Backward Stochastic Differential Equations (in short BSDEs) with weak constraints at stopping times, which are related to the partial hedging of American options. We study the wellposedness of such equations and, using the Yg,ξ-Mertens decomposition, we show that the family of minimal time-t-values Yt, with (Y,Z) a supersolution of the BSDE with weak constraints, admits a representation in terms of a reflected backward stochastic differential equation.

On the complexity of a unified convergence analysis for iterative methods

A local and a semi-local convergence of general iterative methods for solving nonlinear operator equations in Banach spaces is developed under ω-continuity conditions. Our approach unifies existing results and provides a new way of studying iterative methods. The main idea is to find a more accurate domain containing the iterates. No extra effort is used to obtain this. Also, the results of the numerical experiments are given that confirm obtained theoretical estimates.

An Evolutionary Fake News Detection Based on Tropical Convolutional Neural Networks (TCNNs) Approach

In general, the characteristics of false news are difficult to distinguish from those of legitimate news. Even if it is wrong, people can make money by spreading false information. A long time ago, there were fake news stories, including the one about "Bat-men on the moon" in 1835. A mechanism for fact-checking statements must be put in place, particularly those that garner thousands of views and likes before being refuted and proven false by reputable sources. Many machine learning algorithms have been used to precisely categorize and identify fake news. In this experiment, an ML classifier was employed to distinguish between fake and real news. In this study, we present a Tropical Convolutional Neural Networks (TCNNs) model-based false news identification system. Convolutional neural networks (CNNs), Gradient Boost, long short-term memory (LSTMs), Random Forest, Decision Tree (DT), Ada Boost, and attention mechanisms are just a few of the cutting-edge techniques that are compared in our study. Furthermore, because tropical convolution operators are fundamentally nonlinear operators, we anticipate that TCNNs will be better at nonlinear fitting than traditional CNN. Our analysis leads us to the conclusion that the Tropical Convolutional Neural Networks (TCNNs) model with attention mechanism has the maximum accuracy of 98.93%. The findings demonstrate that TCNN can outperform regular convolutional neural network (CNN) layers in terms of expressive capability.

Compressibility effects on the secondary instabilities of the circular cylinder wake

With a growing interest in low Reynolds number compressible flows, compressibility effects on the secondary instabilities developing on the circular cylinder periodic wake are investigated. The unsteady and time-averaged two-dimensional flows are characterised for Reynolds numbers ${{Re}} \in [200;350]$ and Mach numbers up to ${{M}}_{\infty }=0.5$ , revealing different flow structures which influence the characteristics of the secondary unstable modes. The two-dimensional time-periodic solution is used as the base state for a global linear stability analysis performed by means of Floquet theory coupled with a time-stepping finite-difference approach of the nonlinear operator. The influence of compressibility on mode A and mode B secondary instabilities which are responsible for the three-dimensionalisation of the two-dimensional periodic wake is analysed. A stabilising or a destabilising effect of compressibility is observed on mode A, depending on the Reynolds number and the spanwise wavelength of the mode, while mode B is stabilised by the increase of the Mach number. Compressibility is indeed found to decrease the mode kinetic energy production due to base flow shear conversion, which drives the growth of mode B. This results in a delay of the three-dimensionalisation process of the wake due to compressibility.

Analysis of a fire extinguishing model with a p-Laplacian operator and with non-linear advection and reaction

We provide a new model to describe the fire extinguisher dynamics in an aircraft engine nacelle. We connect the mathematical properties introduced by a non-linear operator with the physical mechanisms of a fire extinguishing process. Some discussions about diffusion and non-linear parabolic operators are carried out to justify the governing equation. In addition, the model introduces phenomena like forced convection and fast reaction to account for the discharge of pressurized fire suppressant agent that is released from the fire bottles. The driving equation is firstly discussed analytically to obtain existence and uniqueness results concerning weak solutions. Afterward, we explore profiles of solutions and a characterization of the propagating extinguisher front is introduced. In order to provide a validity note on the analytical conceptions used, a flight test has been executed. Based on this, we obtain particular values for the parameters involved in the model. The proposed assessments permit to obtain different time levels, for which the suppressant agent extinguishes a fire. Such times depend strongly on the involved physical processes connected with diffusion, forced convection and reaction.

Abnormal discharge detection using adaptive neuro-fuzzy inference method with probability density-based feature and modified subtractive clustering

Abnormal discharge (AD) is a discharge mode in electroencephalogram (EEG) with sharp outlines. Lack of related open-access datasets and insufficient annotation hamper the development of AD detection, while cognitive neuroscience, clinical research and pilots’ neural screening demand stable and reliable AD detection methods. An adaptive neuro-fuzzy inference method for AD detection is proposed in this work. First, a probability-density-based (PD) method is proposed to extract lognormal amplitude features from EEG envelopes. Second, the subtractive clustering method (SCM) is modified so that clustering radii can adapt to cluster shapes for each dimension instead of using identical radii for each cluster. The outputs of modified SCM (mSCM), coordinates of cluster centers and adjusting rates for radius components are used to automatically initialize Gaussian membership functions of adaptive-network-based fuzzy inference system (ANFIS). Finally, we conducted multiple experiments to validate mSCM-based ANFIS, comparing it with traditional machine learning classifiers (support vector machines, multi-layer perceptrons, decision trees, and random forests) and the state-of-the-art deep learning-based time-series classification methods InceptionTime and Minirocket, using synthetic data, small-size datasets and a private EEG dataset. Results show that combining PD features with features proposed in previous studies, such as smoothed nonlinear energy operator features and discrete wavelet transform features, achieved higher accuracy (95.13 ± 0.86%) and recall (89.17 ± 3.06%) than other feature combinations. mSCM created more suitable cluster boundaries than SCM on small-size datasets, and its clustering results demonstrated potential in helping interpretation of classification rules built in the ANFIS network. Results of comparison experiments showed that mSCM-based ANFIS produced competitive results in accuracy and recall for AD detection, with lower computational cost, compared to the top 2 results obtained by InceptionTime and Minrocket.

Open mapping theorem for nonlinear operators

Many authors gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. In this paper, our aim is to bring up this calculus to the attention of researchers and demonstrate its usefulness. We first discussed multiplicative normed spaces by giving some topological properties of the relevant multiplicative normed spaces. Also, we give a generalization of the open mapping theorem and the uniform boundedness principle for nonlinear operators with a number of important examples.

A general differential quasi variational–hemivariational inequality: Well-posedness and application

In the paper we analyze a general differential quasi variational-hemivariational inequality in Banach spaces which couples the Cauchy problem for an ordinary differential equation with a quasi variational-hemivariational inequality considered with a unilateral constraint and history-dependent operators. The latter appear in all of the data: the governing nonlinear operator, the generalized directional derivative of a locally Lipschitz potential, a convex potential, and the ordinary differential equation. The results on the well-posedness and regularity of solutions are proved by exploiting a fixed point approach. The theory is illustrated by an application to a quasistatic generalized Signorini unilateral frictional contact problem for viscoplastic materials with a nonsmooth multivalued contact condition.

A modified convergence analysis for steepest descent scheme for solving nonlinear operator equation

A modified convergence analysis for steepest descent scheme for solving nonlinear operator equation

The regularity theory for the double obstacle problem for fully nonlinear operator

In this paper, we prove the existence and uniqueness of W2,p (n<p<∞) solutions of a double obstacle problem. Moreover, we show the optimal regularity of the solution and the local C1 regularity of the free boundary under a thickness assumption at the free boundary point on the intersection of two free boundaries. In the study of the regularity of the free boundary, we deal with a general problem, the no-sign reduced double obstacle problem with an upper obstacle ψ, F(D2u,x)=fχΩ(u)∩{u<ψ}+F(D2ψ,x)χΩ(u)∩{u=ψ},u≤ψinB1, where Ω(u)=B1∖{u=0}∩{∇u=0}.