Nonlinear Operator Research Articles

Preserving mesh quality in shape optimization

AbstractIn PDE‐constrained shape optimization, a lot of computational effort is used to update the geometry at every iteration step. This iterative process can result in loss of element quality and degeneracy of the underlying mesh, which is particularly relevant for applications involving large deformations of some parts of the domain. To avoid remeshing, we model the mesh deformation using the method of mappings. We use an extension equation that maps a boundary control variable to a deformation field defined over the entire domain. By using the nonlinear extension operator proposed in our previous work, we increase the set of reachable shapes, allowing us to model large deformations. We discuss how the choice of parameters of the extension equations affects the mesh quality and we study the influence of the extension factor on the convergence properties of the iterative solvers.

Pointwise eigenvector estimates by landscape functions: Some variations on the Filoche–Mayboroda–van den Berg bound

AbstractLandscape functions are a popular tool used to provide upper bounds for eigenvectors of Schrödinger operators on domains. We review some known results obtained in the last 10 years, unify several approaches used to achieve such bounds, and extend their scope to a large class of linear and nonlinear operators. We also use landscape functions to derive lower estimates on the principal eigenvalue—much in the spirit of earlier results by Donsker–Varadhan and Bañuelos–Carrol—as well as upper bounds on heat kernels. Our methods solely rely on order properties of operators: We devote special attention to the case where the relevant operators enjoy various forms of elliptic or parabolic maximum principle. Additionally, we illustrate our findings with several examples, including ‐Laplacians on domains and graphs as well as Schrödinger operators with magnetic and electric potential, also by means of elementary numerical experiments.

Learning nonlinear integral operators via recurrent neural networks and its application in solving integro-differential equations

In this paper, we propose using LSTM-RNNs (Long Short-Term Memory-Recurrent Neural Networks) to learn and represent nonlinear integral operators that appear in nonlinear integro-differential equations (IDEs). The LSTM-RNN representation of the nonlinear integral operator allows us to turn a system of nonlinear integro-differential equations into a system of ordinary differential equations for which many efficient solvers are available. Furthermore, because the use of LSTM-RNN representation of the nonlinear integral operator in an IDE eliminates the need to perform a numerical integration in each numerical time evolution step, the overall temporal cost of the LSTM-RNN-based IDE solver can be reduced to O(nT) from O(nT2) if a nT-step trajectory is to be computed. We illustrate the efficiency and robustness of this LSTM-RNN-based numerical IDE solver with a model problem. Additionally, we highlight the generalizability of the learned integral operator by applying it to IDEs driven by different external forces. As a practical application, we show how this methodology can effectively solve the Dyson’s equation for quantum many-body systems.

A new adaptive grey seasonal model for time series forecasting tasks

PurposeThis paper intends to construct a new adaptive grey seasonal model (AGSM) to promote the application of the grey forecasting model in quarterly GDP.Design/methodology/approachFirstly, this paper constructs a new accumulation operation that embodies the new information priority by using a hyperparameter. Then, a new AGSM is constructed by using a new grey action quantity, nonlinear Bernoulli operator, discretization operation, moving average trend elimination method and the proposed new accumulation operation. Subsequently, the marine predators algorithm is used to quickly obtain the hyperparameters used to build the AGSM. Finally, comparative analysis experiments and ablation experiments based on China's quarterly GDP confirm the validity of the proposed model.FindingsAGSM can be degraded to some classical grey prediction models by replacing its own structural parameters. The proposed accumulation operation satisfies the new information priority rule. In the comparative analysis experiments, AGSM shows better prediction performance than other competitive algorithms, and the proposed accumulation operation is also better than the existing accumulation operations. Ablation experiments show that each component in the AGSM is effective in enhancing the predictive performance of the model.Originality/valueA new AGSM with new information priority accumulation operation is proposed.

Machine learning for a class of partial differential equations with multi-delays based on numerical Gaussian processes

Delay partial differential equations (PDEs) are widely utilized in many fields, such as climate prediction and epidemiology. But observation data in real world is often noisy and discrete. And in order to expand the applications of delay PDEs, we consider numerical Gaussian processes to solve these models. In this paper, numerical Gaussian processes for predicting the latent solution of a type of delay PDEs with multi-delays are investigated, and various delay PDEs are studied, including problems governed by variable-order fractional order operators and nonlinear operators, so as to adapt to the needs of practical applications. Numerical Gaussian processes are very good at fitting latent solution of PDEs, when all observation data is noisy and discontinuous. And the methodology can clearly quantify the uncertainty of the predicted solution. For complex boundaries controlled by ODEs, we consider mixed boundary conditions of delay PDEs in this paper. And we also apply Runge-Kutta methods to enhance the prediction accuracy of these problems. Finally, we design seven numerical examples to investigate the efficiency of NGPs and how the noisy data influences the solution of our studied problems.

Reconstruction of incidence reporting rate for SARS-CoV-2 Delta variant of COVID-19 pandemic in the US

In recent years, advanced regularization techniques have emerged as a powerful tool aimed at stable estimation of infectious disease parameters that are crucial for future projections, prevention, and control. Unlike other system parameters, i.e., incubation and recovery rates, the case reporting rate, Ψ, and the time-dependent effective reproduction number, Re(t), are directly influenced by a large number of factors making it impossible to pre-estimate these parameters in any meaningful way. In this study, we propose a novel iteratively-regularized trust-region optimization algorithm, combined with SuSvIuIvRD compartmental model, for stable reconstruction of Ψ and Re(t) from reported epidemic data on vaccination percentages, incidence cases, and daily deaths. The innovative regularization procedure exploits (and takes full advantage of) a unique structure of the Jacobian and Hessian approximation for the nonlinear observation operator. The proposed inversion method is thoroughly tested with synthetic and real SARS-CoV-2 Delta variant data for different regions in the United States of America from July 9, 2021, to November 25, 2021. Our study shows that case reporting rate during the Delta wave of COVID-19 pandemic in the US is between 12% and 37%, with most states being in the range from 15% to 25%. This confirms earlier accounts on considerable under-reporting of COVID-19 cases due to the impact of ”silent spreaders” and the limitations of testing.

Progressive Shell Qasistatics for Unstructured Meshes

Thin shell structures exhibit complex behaviors critical for modeling and design across wide-ranging applications. Capturing their mechanical response requires finely detailed, high-resolution meshes. Corresponding simulations for predicting equilibria with these meshes are expensive, whereas coarse-mesh simulations can be fast but generate unacceptable artifacts and inaccuracies. The recently proposed progressive simulation framework [Zhang et al. 2022] offers a promising avenue to address these limitations with consistent and progressively improving simulation over a hierarchy of increasingly higher-resolution models. Unfortunately, it is currently severely limited in application to meshes and shapes generated via Loop subdivision. We propose Progressive Shells Quasistatics to extend progressive simulation to the high-fidelity modeling and design of all input shell (and plate) geometries with unstructured (as well as structured) triangle meshes. To do so, we construct a fine-to-coarse hierarchy with a novel nonlinear prolongation operator custom-suited for curved-surface simulation that is rest-shape preserving, supports complex curved boundaries, and enables the reconstruction of detailed geometries from coarse-level meshes. Then, to enable convergent, high-quality solutions with robust contact handling, we propose a new, safe, and efficient shape-preserving upsampling method that ensures non-intersection and strain limits during refinement. With these core contributions, Progressive Shell Quasistatics enables, for the first time, wide generality for progressive simulation, including support for arbitrary curved-shell geometries, progressive collision objects, curved boundaries, and unstructured triangle meshes - all while ensuring that preview and final solutions remain free of intersections. We demonstrate these features across a wide range of stress-tests where progressive simulation captures the wrinkling, folding, twisting, and buckling behaviors of frictionally contacting thin shells with orders-of-magnitude speed-up in examples over direct fine-resolution simulation.

A matrix technique‐based numerical treatment of a nonlocal singular boundary value problems

The mathematical modeling of the decisive event of astrophysics, physiology, and many other areas of science and technology witness the involvement of singular boundary value problems. The nonlocal boundary conditions are more informative than local boundary conditions (initial conditions and two‐point boundary conditions) to evaluate some mathematical models. This article presents a collocation approach‐based matrix technique to approximate the solution of the fusion of a class of singular differential equations subject to nonlocal three‐point boundary conditions. The proposed strategy utilizes the truncation of the series expansion of a function belonging to in terms of Bernoulli polynomials. It transforms the singular boundary value problems into a set of nonlinear algebraic equations, which can be dealt with by any mathematical software. The Lipschitz condition on an equivalent completely continuous nonlinear operator has been used to prove the convergence analysis of the scheme. Some extremely nonlinear test examples are solved and provided in contrast with the exact solution. These numerical results are also examined against some existing numerical techniques to verify the applicability and significance of the proposed methodology. There are a few numerical examples that are application based but do not have exact solutions. In such cases, residual error norm is employed to measure the accuracy of the numerical strategies. The computational data demonstrate the superiority and validity of the proposed technique over existing numerical approaches.

GAUSS-NEWTON METHOD APPLICATION IN THE PROBLEM OF PHASE FUNCTION RECONSTRUCTING FROM HILBERTOGRAMS

The problem of phase function reconstructing in Hilbert diagnostics of gaseous, condensed and reacting media is discussed in the work. The method for reconstructing the phase disturbances structure of a probing light field, based on the iterative Gauss-Newton al- gorithm, is proposed.This method does not require the second derivatives determination and greatly reduces the number of calculations.It consists in the sequential selection of a complex phase profile, which is specified by the sum of third degree Bezier curves, and the hilbertogram calculation in order to minimize the root-mean-square error between the experimental and reconstructed hilbertograms. The Jacobi matrix for the nonlinear integral operator of Hilbert visualization is calculated. The proposed algorithm was tested on test functions.The devel- opment of the method and its applications is associated with the application of the algorithm to the processing of experimental results.

Self-Organized Operational Neural Networks for The Detection of Atrial Fibrillation

Abstract Atrial fibrillation is a common cardiac arrhythmia, and its incidence increases with age. Currently, numerous deep learning methods have been proposed for AF detection. However, these methods either have complex structures or poor robustness. Given the evidence from recent studies, it is not surprising to observe the limitations in the learning performance of these approaches. This can be attributed to their strictly homogenous conguration, which solely relies on the linear neuron model. The limitations mentioned above have been addressed by operational neural networks (ONNs). These networks employ a heterogeneous network configuration, incorporating neurons equipped with diverse nonlinear operators. Therefore, in this study, to enhance the detection performance while maintaining computational efficiency, a novel model named multi-scale Self-ONNs (MSSelf-ONNs) was proposed to identify AF. The proposed model possesses a significant advantage and superiority over conventional ONNs due to their self-organization capability. Unlike conventional ONNs, MSSelf -ONNs eliminate the need for prior operator search within the operator set library to find the optimal set of operators. This unique characteristic sets MSSelf -ONNs apart and enhances their overall performance. To validate and evaluate the system, we have implemented the experiments on the well-known MIT-BIH atrial fibrillation database. The proposed model yields total accuracies and kappa coefficients of 98% and 0.95, respectively. The experiment results demonstrate that the proposed model outperform the state-of-the-art deep CNN in terms of both performance and computational complexity.

The solution set to the nonlinear operator equation XAX = BX on Hilbert spaces

We present the representation of solution sets of the operator equation XAX = BX on Hilbert spaces by taking advantage of the invariant subspace of B, including solution sets of nonzero solutions, normal solutions and self-adjoint solutions. Moreover, necessary conditions and sufficient conditions are established for existence of nonzero solutions of the operator equation XAX = BX which commute with B.

Corotational force-based beam finite element with rigid joint offsets for 3D framed structures

In numerical analysis of frame structures, modeling of the connection between structural members often requires the introduction of rigid end offsets to correctly describe the stiffness of the joints. This is typical of beam-to-column connections in civil constructions but is also common in lattice materials, where the element overlapping at the joints locally increases the stiffness of the nodes. However, when nonlinear geometric effects are included, correct simulation of such phenomena is challenging.This paper presents a geometrically nonlinear three-dimensional force-based beam finite element that efficiently accounts for the effects of rigid joint offsets. The model imposes the element equilibrium in the local reference system referring to the element deformed configuration, considering the von Kármán nonlinear terms, and exploits a corotational formulation to account for rigid large displacements and rotations of the beam. Two alternative approaches are used to introduce the rigid offsets. The first derives from an existing proposal where kinematic and static nonlinear operators are adopted to describe the behavior of the rigid portions at the element ends. The second is a novel approach, easier to implement and use, that involves modifying the integration, along the element axis, of the cross-section strains and flexibility. This can be easily defined for beam models under linear geometry assumption but requires particular attention for the analysis of frame structures under large displacements and strains. Both approaches are validated through numerical examples, including comparison with other numerical methods and experimental results available in the literature.

Fuzzy neighborhood Choquet integrals with overlap functions and their applications in attribute reduction

In fuzzy rough sets, fuzzy neighborhood operators and fuzzy rough approximate operators are mainly used by traditional methods to deal with the problem of attribute reduction (i.e., feature selection). However, they are parametric linear, while the corresponding weights are additive measures. For complex data, parametric linear operators and additive measures do not reflect correlations among data (or attributes). Therefore, it is important to develop non-additive measures and nonlinear operators for attribute reduction, which can effectively find those attributes that are not needed to reach a decision from the viewpoint of correlations. Along these lines, in this work, OI-fuzzy β-neighborhood measures (i.e., non-additive measures) and generalized Choquet integrals (i.e., nonlinear integrals) are presented by using overlap functions to deal with the problem of attribute reduction under the fuzzy β-covering approximation space. First, four pairs of OI-fuzzy β-neighborhood operators are constructed based on the implementation of overlap functions and their residual implications in a fuzzy β-covering approximation space. Furthermore, four pairs of OI-fuzzy β-neighborhood measures, as non-additive measures, are introduced to instead of rough approximate operators in data mining. Then, a novel method with respect to OI-fuzzy β-neighborhood measures is presented to deal with the attribute reduction in a fuzzy β-covering information decision table. Third, four pairs of generalized Choquet integrals based on the employment of OI-fuzzy β-neighborhood measures are constructed. On top of that, another method with respect to these generalized Choquet integrals is proposed to deal with the attribute reduction in a fuzzy β-covering information decision table. Finally, the presented methods are used to deal with the problem of classification. Several public data sets are applied to illustrate the feasibility and effectiveness of the above-mentioned proposed methods.

A new kind of double phase elliptic inclusions with logarithmic perturbation terms I: Existence and extremality results

This paper is devoted to introduce a new double phase elliptic inclusion problem (DPEI) involving nonlinear and nonhomogeneous partial differential operator which has unbalanced growth and logarithmic perturbation terms, and two multivalued functions which are defined in the domain and its boundary. The main goal of this paper is to establish the existence and extremality results to the elliptic inclusion problem under consideration. More exactly, we give the definitions of weak solutions, subsolutions and supersolutions to (DPEI). Then, under the coercive setting, an existence theorem of weak solutions to (DPEI) is obtained by employing a surjectivity theorem for pseudomonotone operators. Moreover, in the noncoercive framework, we apply the method of sub-supersolution combined with the nonsmooth calculus analysis and truncation techniques to prove that (DPEI) has at least a weak solution within an ordered interval of sub-supersolution. Finally, when the constraint set K satisfies a lattice condition, the existence of smallest and greatest elements of solution set to (DPEI) is established.

Influence of symmetric first-order divided differences on Secant-like methods

n this paper, by using symmetric first-order divided differences, we introduce a new family of Secant-like iterative methods with quadratic convergence. Afterthought, we analyze its semilocal and local behavior when the nonlinear operator F is not differentiable by imposing appropriate bounding conditions in each case. Theoretical results have also been tested by solving a problem which shows the applicability of our work.

Efficient approximation of cardiac mechanics through reduced-order modeling with deep learning-based operator approximation.

Reducing the computational time required by high-fidelity, full-order models (FOMs) for the solution of problems in cardiac mechanics is crucial to allow the translation of patient-specific simulations into clinical practice. Indeed, while FOMs, such as those based on the finite element method, provide valuable information on the cardiac mechanical function, accurate numerical results can be obtained at the price of very fine spatio-temporal discretizations. As a matter of fact, simulating even just a few heartbeats can require up to hours of wall time on high-performance computing architectures. In addition, cardiac models usually depend on a set of input parameters that are calibrated in order to explore multiple virtual scenarios. To compute reliable solutions at a greatly reduced computational cost, we rely on a reduced basis method empowered with a new deep learning-based operator approximation, which we refer to as Deep-HyROMnet technique. Our strategy combines a projection-based POD-Galerkin method with deep neural networks for the approximation of (reduced) nonlinear operators, overcoming the typical computational bottleneck associated with standard hyper-reduction techniques employed in reduced-order models (ROMs) for nonlinear parametrized systems. This method can provide extremely accurate approximations to parametrized cardiac mechanics problems, such as in the case of the complete cardiac cycle in a patient-specific left ventricle geometry. In this respect, a 3D model for tissue mechanics is coupled with a 0D model for external blood circulation; active force generation is provided through an adjustable parameter-dependent surrogate model as input to the tissue 3D model. The proposed strategy is shown to outperform classical projection-based ROMs, in terms of orders of magnitude of computational speed-up, and to return accurate pressure-volume loops in both physiological and pathological cases. Finally, an application to a forward uncertainty quantification analysis, unaffordable if relying on a FOM, is considered, involving output quantities of interest such as, for example, the ejection fraction or the maximal rate of change in pressure in the left ventricle.

Robust Peak Detection for Holter ECGs by Self-Organized Operational Neural Networks.

Although numerous R-peak detectors have been proposed in the literature, their robustness and performance levels may significantly deteriorate in low-quality and noisy signals acquired from mobile electrocardiogram (ECG) sensors, such as Holter monitors. Recently, this issue has been addressed by deep 1-D convolutional neural networks (CNNs) that have achieved state-of-the-art performance levels in Holter monitors; however, they pose a high complexity level that requires special parallelized hardware setup for real-time processing. On the other hand, their performance deteriorates when a compact network configuration is used instead. This is an expected outcome as recent studies have demonstrated that the learning performance of CNNs is limited due to their strictly homogenous configuration with the sole linear neuron model. This has been addressed by operational neural networks (ONNs) with their heterogenous network configuration encapsulating neurons with various nonlinear operators. In this study, to further boost the peak detection performance along with an elegant computational efficiency, we propose 1-D Self-Organized ONNs (Self-ONNs) with generative neurons. The most crucial advantage of 1-D Self-ONNs over the ONNs is their self-organization capability that voids the need to search for the best operator set per neuron since each generative neuron has the ability to create the optimal operator during training. The experimental results over the China Physiological Signal Challenge-2020 (CPSC) dataset with more than one million ECG beats show that the proposed 1-D Self-ONNs can significantly surpass the state-of-the-art deep CNN with less computational complexity. Results demonstrate that the proposed solution achieves a 99.10% F1-score, 99.79% sensitivity, and 98.42% positive predictivity in the CPSC dataset, which is the best R-peak detection performance ever achieved.

Метод Гаусса-Ньютона в задаче оптимизации расчёта осесимметричной фазовой функции по данным гильберт-диагностики

A method for reconstructing phase disturbances of a probing light field using the iterative Gauss-Newton algorithm is discussed as part of the Hilbert diagnostics development of gaseous, condensed and reacting media. In this case, the need to determine second derivatives is eliminated, which simplifies the calculations. The method consists of selecting a phase profile, which is specified by a Bezier curve, and hilbertogram calculating. The coincidence of the reference and reconstructed hilbertograms serves as a criterion for the results reliability. The Jacobian matrix for the nonlinear integral operator of Hilbert visualization is obtained. The algorithm is analyzed using a test function. The method development is associated with the algorithm application to the processing of experimental results, including the reconstruction of complex structures in which the phase function is described by several Bezier polynomials.

MMV-Net: A Multiple Measurement Vector Network for Multifrequency Electrical Impedance Tomography.

Multifrequency electrical impedance tomography (mfEIT) is an emerging biomedical imaging modality to reveal frequency-dependent conductivity distributions in biomedical applications. Conventional model-based image reconstruction methods suffer from low spatial resolution, unconstrained frequency correlation, and high computational cost. Deep learning has been extensively applied in solving the EIT inverse problem in biomedical and industrial process imaging. However, most existing learning-based approaches deal with the single-frequency setup, which is inefficient and ineffective when extended to the multifrequency setup. This article presents a multiple measurement vector (MMV) model-based learning algorithm named MMV-Net to solve the mfEIT image reconstruction problem. MMV-Net considers the correlations between mfEIT images and unfolds the update steps of the Alternating Direction Method of Multipliers for the MMV problem (MMV-ADMM). The nonlinear shrinkage operator associated with the weighted l2,1 regularization term of MMV-ADMM is generalized in MMV-Net with a cascade of a Spatial Self-Attention module and a Convolutional Long Short-Term Memory (ConvLSTM) module to better capture intrafrequency and interfrequency dependencies. The proposed MMV-Net was validated on our Edinburgh mfEIT Dataset and a series of comprehensive experiments. The results show superior image quality, convergence performance, noise robustness, and computational efficiency against the conventional MMV-ADMM and the state-of-the-art deep learning methods.

Fixed Point Results for Generalized Non-linear Operators with Convergence Analysis

The purpose of this research article is to introduce a new iteration scheme and to prove J-iterationconvergence and stability results for it. We also claim the newly introduced iterative scheme has better efficiency than some of the existing iterations in the literature. Our claim is supported by numerical example.