Penalization Technique Research Articles

A Kirchhoff Type Equation in pmb {mathbb {R}}^{N} Involving the fractional (p,\xa0q)-Laplacian

In this paper, we deal with the following class of fractional (p, q)-Laplacian Kirchhoff type problem: 1+[u]s,pp(-Δ)psu+1+[u]s,qq(-Δ)qsu+V(εx)(|u|p-2u+|u|q-2u)=f(u)inRN,u∈Ws,p(RN)∩Ws,q(RN),u>0inRN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} \\left( 1+[u]_{s,p}^{p}\\right) (-\\Delta )_{p}^{s}u+ \\left( 1+[u]^{q}_{s, q}\\right) (-\\Delta )_{q}^{s}u + V(\\varepsilon x) (|u|^{p-2}u + |u|^{q-2}u)= f(u) &{} \\text{ in } \\mathbb {R}^{N}, \\\\ u\\in W^{s, p}(\\mathbb {R}^{N})\\cap W^{s,q}(\\mathbb {R}^{N}), \\quad u>0 \\text{ in } \\mathbb {R}^{N}, \\end{array} \\right. \\end{aligned}$$\\end{document}where varepsilon >0, sin (0, 1), 1<p<q<frac{N}{s}<2q, (-Delta )_{t}^{s}, with tin {p, q}, is the fractional t-Laplacian operator, V:mathbb {R}^{N}rightarrow mathbb {R} is a positive continuous potential such that inf _{partial Lambda }V>inf _{Lambda } V for some bounded open set Lambda subset mathbb {R}^{N}, and f:mathbb {R}rightarrow mathbb {R} is a superlinear continuous nonlinearity with subcritical growth at infinity. By combining the method of Nehari manifold, a penalization technique, and the Lusternik–Schnirelman category theory, we study the multiplicity and concentration properties of solutions for the above problem when varepsilon rightarrow 0.

An efficient penalized estimation approach for semiparametric linear transformation models with interval-censored data.

We consider efficient estimation of flexible transformation models with interval-censored data. To reduce the dimension of semiparametric models, the unknown monotone function is approximated via a monotone B-spline. A penalization technique is used to provide computationally efficient estimation of all parameters. To accomplish model fitting and inference, an easy to implement nested iterative expectation-maximization (EM) algorithm is developed for estimation, and a simple variance-covariance estimation approach is proposed which makes large-sample inference for the regression parameters possible. Theoretically, we show that the estimator of the unknown monotone increasing function achieves the optimal rate of convergence, and the estimators of the regression parameters are asymptotically normal and efficient under the appropriate selection of the order of the smoothing parameter and the knots of the spline space. The proposed penalized procedure is assessed through extensive numerical experiments and implemented in R package PenIC. The proposed methodology is further illustrated via a signal tandmobiel study.

Penalization techniques for fatigue‐based topology optimizations of structures with embedded functionally graded lattice materials

AbstractTopology optimization is widely used in industry to determine optimal load paths while satisfying stringent constraints. Recently, functionally graded lattice materials have been integrated into structural design for weight minimization, performance improvements, among others. One subject of interest is the long‐lasting designs with high fatigue failure resistance. The focus of this article is twofold, namely, to present readers with a step‐by step guide to derive sensitivities for various fatigue performances in gradient based topology optimization and to present and compare new and old methods for cumulative damage objectives. New formulations for cumulative damage assessment that modify local damage in topology optimizations by influencing local stress, yield strength or SN‐curve intercept are presented. Effects of functionally graded materials are also investigated to measure potential improvements in damage resistance. Methods of gradient descent such as moving asymptotes and sequential quadratic programming are included to compare solutions of fatigue minimization problems. The analysis shows that traditional method of stress penalization is the most effective to obtain satisfactory load paths for fatigue minimization when compared to yield and SN‐intercept penalization. Multiple benchmark tests provided results which showed slower convergence when yield strength penalization is activated which also required significantly more computation time than other methods.

Exact Penalization and Optimality Conditions for Approximate Directional Minima

In this paper, we study the concept of approximate directional efficiency for set-valued constrained and unconstrained optimization problems. In our work, we concerned with finding conditions under which the Clarke penalization technique can be applied, and we derive some optimality conditions via variational analysis tools such as limiting normal cones and its corresponding normal coderivative.

Locally penalized single-index model using B-splines and spherical coordinates

In this study, we focus on the estimation of the regression function in the single-index model based on B-splines using penalization techniques. We adopt a spherical coordinates reparameterization of an index vector to deal with an identification problem of the single-index model. To provide a spatially adaptive method, two types of penalties are applied to the estimation of the index vector and the regression function. A special penalty called the localized penalty is introduced to handle the sparsity of the index vector using the spherical coordinates, and the total variation penalty is considered to deal with the smoothing function. Using a coordinate descent algorithm with a grid search of the two tuning parameters, the entire solution paths of the index coefficients and the regression functions for tuning parameters can be obtained efficiently. The performance of the proposed estimator is studied through both numerical simulations and real data sets. An R software package pbssim is available.

A Comparison of Autometrics and Penalization Techniques under Various Error Distributions: Evidence from Monte Carlo Simulation

This work compares Autometrics with dual penalization techniques such as minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD) under asymmetric error distributions such as exponential, gamma, and Frechet with varying sample sizes as well as predictors. Comprehensive simulations, based on a wide variety of scenarios, reveal that the methods considered show improved performance for increased sample size. In the case of low multicollinearity, these methods show good performance in terms of potency, but in gauge, shrinkage methods collapse, and higher gauge leads to overspecification of the models. High levels of multicollinearity adversely affect the performance of Autometrics. In contrast, shrinkage methods are robust in presence of high multicollinearity in terms of potency, but they tend to select a massive set of irrelevant variables. Moreover, we find that expanding the data mitigates the adverse impact of high multicollinearity on Autometrics rapidly and gradually corrects the gauge of shrinkage methods. For empirical application, we take the gold prices data spanning from 1981 to 2020. While comparing the forecasting performance of all selected methods, we divide the data into two parts: data over 1981–2010 are taken as training data, and those over 2011–2020 are used as testing data. All methods are trained for the training data and then are assessed for performance through the testing data. Based on a root-mean-square error and mean absolute error, Autometrics remain the best in capturing the gold prices trend and producing better forecasts than MCP and SCAD.

Improved penalty algorithm for mixed integer PDE constrained optimization problems

Optimal control problems including partial differential equation (PDE) as well as integer constraints merge the combinatorial difficulties of integer programming and the challenges related to large-scale systems resulting from discretized PDEs. So far, the branch-and-bound framework has been the most common solution strategy for such problems. In order to provide an alternative solution approach, especially in a large-scale context, this article investigates penalization techniques. Taking inspiration from a well-known family of existing exact penalty algorithms, a novel improved penalty algorithm is derived, whose key ingredients are a basin hopping strategy and an interior point method, both of which are specialized for the problem class. A thorough numerical investigation is carried out for a standard stationary test problem. Extensions to a convection-diffusion as well as a nonlinear test problem finally demonstrate the versatility of the approach.

Computational Fluid Dynamics Using the Adaptive Wavelet-Collocation Method

Advancements to the adaptive wavelet-collocation method over the last decade have opened up a number of new possible areas for active research. Volume penalization techniques allow complex immersed boundary conditions to be used with high efficiency for both internal and external flows. Anisotropic methods make it possible to use body-fitted meshes while still taking advantage of the dynamic adaptability properties wavelet-based methods provide. The parallelization of the approach has made it possible to perform large high-resolution simulations of detonation initiation and fluid instabilities to uncover new physical insights that would otherwise be difficult to discover. Other developments include space-time adaptive methods and nonreflecting boundary conditions. This article summarizes the work performed using the adaptive wavelet-collocation method developed by Vasilyev and coworkers over the past decade.

A numerical homogenization technique for unidirectional composites using polygonal generalized finite elements

The present study proposes a computational methodology to obtain the homogenized effective elastic properties of unidirectional fibrous composite materials by using the generalized finite-element method and penalization techniques to impose periodic boundary conditions on non-uniform polygonal unit cells. Each unit cell is described by a single polygonal finite element using Wachspress functions as base shape functions and different families of enrichment functions to account for the internal fiber influence on stresses and strains fields. The periodic boundary conditions are imposed using reflection laws between two parallel opposing faces using a Lagrange multiplier approach; this reflection law creates a distributed reaction force over the edges of the [Formula: see text]-gon from the direct application of a given deformation gradient, which simulates different macroscopic load cases on the macroscopic body the unit cell is part of. The methodology is validated through a comparison with results for similar unit cells found in the literature and its computational efficiency is compared to simple cases solved using a classic finite-element approach. This methodology showed computational advantages over the classic finite elements in both computational efficiency and total number of degrees of freedom for convergence and flexibility on the shape of the unit cell used. Finally, the methodology provides an efficient way to introduce non-circular fiber shapes and voids.

A multiplicity result for a (p,\xa0q)-Schr\xf6dinger\u2013Kirchhoff type equation

In this paper, we study a class of (p, q)-Schrödinger–Kirchhoff type equations involving a continuous positive potential satisfying del Pino–Felmer type conditions and a continuous nonlinearity with subcritical growth at infinity. By applying variational methods, penalization techniques and Lusternik–Schnirelman category theory, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values.

A space-time discretization of a nonlinear peridynamic model on a 2D lamina

Peridynamics is a nonlocal theory for dynamic fracture analysis consisting in a second order in time partial integro-differential equation. In this paper, we consider a nonlinear model of peridynamics in a two-dimensional spatial domain. We implement a spectral method for the space discretization based on the Fourier expansion of the solution while we consider the Newmark-β method for the time marching. This computational approach takes advantages from the convolutional form of the peridynamic operator and from the use of the discrete Fourier transform. We show a convergence result for the fully discrete approximation and study the stability of the method applied to the linear peridynamic model. Finally, we perform several numerical tests and comparisons to validate our results and provide simulations implementing a volume penalization technique to avoid the limitation of periodic boundary conditions due to the spectral approach.

Regression modelling of interval censored data based on the adaptive ridge procedure

A new method for the analysis of time to ankylosis complication on a dataset of replanted teeth is proposed. In this context of left-censored, interval-censored and right-censored data, a Cox model with piecewise constant baseline hazard is introduced. Estimation is carried out with the expectation maximisation (EM) algorithm by treating the true event times as unobserved variables. This estimation procedure is shown to produce a block diagonal Hessian matrix of the baseline parameters. Taking advantage of this interesting feature in the EM algorithm, a L 0 penalised likelihood method is implemented in order to automatically determine the number and locations of the cuts of the baseline hazard. This procedure allows to detect specific areas of time where patients are at greater risks for ankylosis. The method can be directly extended to the inclusion of exact observations and to a cure fraction. Theoretical results are obtained which allow to derive statistical inference of the model parameters from asymptotic likelihood theory. Through simulation studies, the penalisation technique is shown to provide a good fit of the baseline hazard and precise estimations of the resulting regression parameters.

Bounded resonant problems driven by fractional Laplacian

In this paper we study the existence of nontrivial solutions for the fractional Laplacian resonance problem with a bounded nonlinearity via Morse theory and a penalization technique.

Concentration phenomena for fractional magnetic NLS equations

We study the multiplicity and concentration of complex-valued solutions for a fractional magnetic Schrödinger equation involving a scalar continuous electric potential satisfying a local condition and a continuous nonlinearity with subcritical growth. The main results are obtained by applying a penalization technique, generalized Nehari manifold method and Ljusternik–Schnirelman theory. We also prove a Kato's inequality for the fractional magnetic Laplacian which we believe to be useful in the study of other fractional magnetic problems.

A boundary penalization technique to remove outliers from isogeometric analysis on tensor-product meshes

We introduce a boundary penalization technique to improve the spectral approximation of isogeometric analysis (IGA). The method removes the outliers appearing in the high-frequency region of the approximate spectrum when using the Cp−1,p-th (p≥3) order isogeometric elements. We focus on the classical Laplacian (Dirichlet) eigenvalue problem in 1D to illustrate the idea and then use the tensor-product structure to generate the stiffness and mass matrices for multidimensional problems. To remove the outliers, we penalize the higher-order derivatives from both the solution and test spaces at the domain boundary. Intuitively, we construct a better approximation by weakly imposing features of the exact solution. Effectively, we add terms to the variational formulation at the boundaries with minimal extra-computational cost. We then generalize the idea to remove the outliers for the isogeometric analysis of the Neumann eigenvalue problem (for p≥2). The boundary penalization does not change the test and solution spaces. In the limiting case, when the penalty goes to infinity, we perform the dispersion analysis of C2 cubic elements for the Dirichlet eigenvalue problem and C1 quadratic elements for the Neumann eigenvalue problem. We obtain the analytical eigenpairs for the resulting matrix eigenvalue problems. Numerical experiments show optimal convergence rates for the eigenvalues and eigenfunctions of the discrete operator.

Compressible flow simulation with moving geometries using the Brinkman penalization in high-order Discontinuous Galerkin

In this work we investigate the Brinkman volume penalization technique in the context of a high-order Discontinous Galerkin method to model moving wall boundaries for compressible fluid flow simulations. High-order approximations are especially of interest as they require few degrees of freedom to represent smooth solutions accurately. This reduced memory consumption is attractive on modern computing systems where the memory bandwidth is a limiting factor. Due to their low dissipation and dispersion they are also of particular interest for aeroacoustic problems. However, a major problem for the high-order discretization is the appropriate representation of wall geometries. In this work we look at the Brinkman penalization technique, which addresses this problem and allows the representation of geometries without modifying the computational mesh. The geometry is modelled as an artificial porous medium and embedded in the equations. As the mesh is independent of the geometry with this method, it is not only well suited for high-order discretizations but also for problems where the obstacles are moving. We look into the deployment of this strategy by briefly discussing the Brinkman penalization technique and its application in our solver and investigate its behavior in fundamental one-dimensional setups, such as shock reflection at a moving wall and the formation of a shock in front of a piston. This is followed by the application to setups with two and three dimensions, illustrating the method in the presence of curved surfaces.

The inertial sea wave energy converter (ISWEC) technology: Device-physics, multiphase modeling and simulations

In this paper we investigate the dynamics of the inertial sea wave energy converter (ISWEC) device using fully-resolved computational fluid dynamics (CFD) simulations. Originally prototyped by the Polytechnic University of Turin, the device consists of a floating, boat-shaped hull that is slack-moored to the sea bed. Internally, a gyroscopic power take-off (PTO) unit converts the wave-induced pitch motion of the hull into electrical energy. The CFD model is based on the incompressible Navier–Stokes equations and utilizes the fictitious domain Brinkman penalization (FD/BP) technique to couple the device physics and water wave dynamics. A numerical wave tank is used to generate both regular waves based on fifth-order Stokes theory and irregular waves based on the JONSWAP spectrum to emulate realistic sea operating conditions. A Froude scaling analysis is performed to enable two- and three-dimensional simulations for a scaled-down (1:20) ISWEC model. It is demonstrated that the scaled-down 2D model is sufficient to accurately simulate the hull’s pitching motion and to predict the power generation capability of the converter. A systematic parameter study of the ISWEC is conducted, and its optimal performance in terms of power generation is determined based on the hull and gyroscope control parameters. It is demonstrated that the device achieves peak performance when the gyroscope specifications are chosen based on the reactive control theory. It is shown that a proportional control of the PTO control torque is required to generate continuous gyroscopic precession effects, without which the device generates no power. In an inertial reference frame, it is demonstrated that the yaw and pitch torques acting on the hull are of the same order of magnitude, informing future design investigations of the ISWEC technology. Further, an energy transfer pathway from the water waves to the hull, the hull to the gyroscope, and the gyroscope to the PTO unit is analytically described and numerically verified. Additional parametric analysis demonstrates that a hull length to wavelength ratio between one-half and one-third yields high conversion efficiency (ratio of power absorbed by the PTO unit to wave power per unit crest width). Finally, device protection during inclement weather conditions is emulated by gradually reducing the gyroscope flywheel speed to zero, and the resulting dynamics are investigated.

Simulation of gas-dynamic, pressure surges and adiabatic compression phenomena in geometrically complex respirator oxygen valves

Gas-dynamic pressure surges and adiabatic compression phenomena are generally hard to predict numerically. In this contribution, we study the effect of the pressure reserve capacity on the compressible gas-dynamics pressure surge and adiabatic compression in a fitted respirator oxygen valve geometry.A three-dimensional remeshed smoothed particle hydrodynamics method for the simulation of isotropic turbulence is used, the method is coupled with Brinkman penalisation technique for flow simulation inside the complex valve geometry. Simulations are carried out for three different pressure reserve quantities, to replicate the opening of the valve, two time-based pressure inlet boundary condition functions were simulated along with an impulsively started scenario. A geometrical sensitivity analysis is provided, where the simulation is performed on a modified valve design which exhibits a damping effect on the gas dynamics and flow characteristics, which has a favourable effect on the valve functionality and safety.It is found that the capacity of the pressure reserve has a considerable effect on the simulated flow fields (velocity, temperature), as the temperature could rise 6.0× the reference temperature, and up to 2.7× the reference velocity. The numerical results are compared with a previous study carried out by Rotarex S.A., demonstrating that the remeshed particle-mesh method coupled with Brinkman penalisation provides a good quality simulation and the results are in agreement with the reference solution.

Marginal false discovery rate for a penalized transformation survival model

Survival analysis that involves moderate/high dimensional covariates has become common. Most of the existing analyses have been focused on estimation and variable selection, using penalization and other regularization techniques. To draw more definitive conclusions, a handful of studies have also conducted inference. The recently developed mFDR (marginal false discovery rate) technique provides an alternative inference perspective and can be advantageous in multiple aspects. The existing inference studies for regularized estimation of survival data with moderate/high dimensional covariates assume the Cox and other specific models, which may not be sufficiently flexible. To tackle this problem, the analysis scope is expanded to the transformation model, which is robust and has been shown to be desirable for practical data analysis. Statistical validity is rigorously established. Two data analyses are conducted. Overall, an alternative inference approach has been developed for survival analysis with moderate/high dimensional data.

Predictive Capability of QSAR Models Based on the CompTox Zebrafish Embryo Assays: An Imbalanced Classification Problem.

The CompTox Chemistry Dashboard (ToxCast) contains one of the largest public databases on Zebrafish (Danio rerio) developmental toxicity. The data consists of 19 toxicological endpoints on unique 1018 compounds measured in relatively low concentration ranges. The endpoints are related to developmental effects occurring in dechorionated zebrafish embryos for 120 hours post fertilization and monitored via gross malformations and mortality. We report the predictive capability of 209 quantitative structure–activity relationship (QSAR) models developed by machine learning methods using penalization techniques and diverse model quality metrics to cope with the imbalanced endpoints. All these QSAR models were generated to test how the imbalanced classification (toxic or non-toxic) endpoints could be predicted regardless which of three algorithms is used: logistic regression, multi-layer perceptron, or random forests. Additionally, QSAR toxicity models are developed starting from sets of classical molecular descriptors, structural fingerprints and their combinations. Only 8 out of 209 models passed the 0.20 Matthew’s correlation coefficient value defined a priori as a threshold for acceptable model quality on the test sets. The best models were obtained for endpoints mortality (MORT), ActivityScore and JAW (deformation). The low predictability of the QSAR model developed from the zebrafish embryotoxicity data in the database is mainly due to a higher sensitivity of 19 measurements of endpoints carried out on dechorionated embryos at low concentrations.