Semilinear Models Research Articles

Weakly nonlinear versus semi-linear models of the nonlinear evolution of the centrifugal instability

We carry out a weakly nonlinear analysis of the centrifugal instability for a columnar vortex in a rotating fluid, and compare the results to those of the semi-linear model derived empirically by Yim et al. (J. Fluid Mech., vol. 897, 2020, A34). The asymptotic analysis assumes that the Reynolds number is close to the instability threshold so that the perturbation is only marginally unstable. This leads to two coupled equations that govern the evolutions of the amplitude of the perturbation and of the mean flow under the effect of the Reynolds stresses due to the perturbation. These equations differ from the Stuart–Landau amplitude equation or coupled amplitude equations involving a mean field that have been derived previously. In particular, the amplitude does not saturate to a constant as in the supercritical Stuart–Landau equation, but decays afterwards reflecting the instability disappearance when the mean flow tends toward a neutrally stable profile in the direct numerical simulations (DNS). These equations resemble those of the semi-linear model except that the perturbation in the weakly nonlinear model keeps at leading order the structure of the eigenmode of the unperturbed base flow. The predictions of the weakly nonlinear equations are compared to those of the semi-linear model and to DNS for the Rossby number $Ro=-4$ and various Reynolds numbers and wavenumbers. They are in good agreement with the DNS when the growth rate is sufficiently small. However, the agreement deteriorates and becomes only qualitative for parameters away from the marginal values, whereas the semi-linear model continues to be in better agreement with the DNS.

Probing the pressure dependence of sound speed and attenuation in bubbly media: Experimental observations, a theoretical model and numerical calculations

The problem of attenuation and sound speed of bubbly media has remained partially unsolved. Comprehensive data regarding pressure-dependent changes of the attenuation and sound speed of a bubbly medium are not available. Our theoretical understanding of the problem is limited to linear or semi-linear theoretical models, which are not accurate in the regime of large amplitude bubble oscillations. Here, by controlling the size of the lipid coated bubbles (mean diameter of ≈5.4μm), we report the first time observation and characterization of the simultaneous pressure dependence of sound speed and attenuation in bubbly water below, at and above microbubbles resonance (frequency range between 1–3 MHz). With increasing acoustic pressure (between 12.5–100 kPa), the frequency of the peak attenuation and sound speed decreases while maximum and minimum amplitudes of the sound speed increase. We propose a nonlinear model for the estimation of the pressure dependent sound speed and attenuation with good agreement with the experiments. The model calculations are validated by comparing with the linear and semi-linear models predictions. One of the major challenges of the previously developed models is the significant overestimation of the attenuation at the bubble resonance at higher void fractions (e.g. 0.005). We addressed this problem by incorporating bubble–bubble interactions and comparing the results to experiments. Influence of the bubble–bubble interactions increases with increasing pressure. Within the examined exposure parameters, we numerically show that, even for low void fractions (e.g. 5.1×10-6) with increasing pressure the sound speed may become 4 times higher than the sound speed in the non-bubbly medium.

Coupled micro-macroscopic modeling of layered composites with finite deformations

A coupled micro- and macroscopic modeling of an incompressible layered composite material under finite deformations is performed based on the asymptotic homogenization method and universal semi-linear models. The deformation diagrams of the periodicity cell are calculated. A method for searching for effective transversally isotropic properties of LCM based on the obtained diagrams is considered. Cylindrical bending of an LCM plate is simulated using the model of a transversally isotropic medium. The microstresses in the periodicity cell are calculated based on the homogenized stresses.

Computing Semilinear Sparse Models for Approximately Eventually Periodic Signals

Some elements of the theory and algorithmics corresponding to the computation of semilinear sparse models for discrete-time signals are presented. In this study, we will focus on approximately eventually periodic discrete-time signals, that is, signals that can exhibit an aperiodic behavior for an initial amount of time, and then become approximately periodic afterwards. The semilinear models considered in this study are obtained by combining sparse representation methods, linear autoregressive models and GRU neural network models. Firstly, each block model is fitted independently, using some reference data corresponding to some signal under consideration. Secondly, some mixing parameters are fitted, in order to obtain a signal model consisting of a linear combination of the previously fitted blocks, using the aforementioned reference data. Along the process, sparse representations of some of the matrices of parameters of the resulting model are computed. Some prototypical computational implementations are presented as well.

ИССЛЕДОВАНИЕ ЗАДАЧИ ГРАНИЧНОГО УПРАВЛЕНИЯ И ФИНАЛЬНОГО НАБЛЮДЕНИЯ ДЛЯ МАТЕМАТИЧЕСКОЙ МОДЕЛИ НЕЛИНЕЙНОЙ ФИЛЬТРАЦИИ

The article is devoted to studying the problem of boundary control and final observation for a degenerate mathematical model of non-linear filtration, based on the Oskolkov equation, with the initial condition of Showalter–Sidorov. This model belongs to the class of semilinear models of the Sobolian type, in which the nonlinear operator is p-coercive and s-monotonic. The paper for the first time considers the problem of boundary control and final observation for the semilinear model of the Sobolian type and establishes the conditions of the existence of the control-state pair of the matter being studied.

Tumor perfusion evaluation using dynamic contrast-enhanced ultrasound after electrochemotherapy and IL-12 plasmid electrotransfer in murine melanoma

Electrochemotherapy with bleomycin (ECT BLM) is an effective antitumor treatment already used in clinical oncology. However, ECT alone is still considered a local antitumor therapy because it cannot induce systemic immunity. When combined with adjuvant gene electrotransfer of plasmid DNA encoding IL-12 (GET pIL-12), the combined therapy leads to a systemic effect on untreated tumors and distant metastases. Although the antitumor efficacy of both therapies alone or in combination has been demonstrated at both preclinical and clinical levels, data on the predictors of efficacy of the treatments are still lacking. Herein, we evaluated the results of dynamic contrast-enhanced ultrasound (DCE-US) as a predictive factor for ECT BLM and GET pIL-12 in murine melanoma. Melanoma B16F10 tumors grown in female C57Bl/6NCrl mice were treated with GET pIL-12 and ECT BLM. Immediately after therapy, 6 h and 1, 3, 7 and 10 days later, tumors were examined by DCE-US. Statistical analysis was performed to inspect the correlation between tumor doubling time (DT) and DCE-US measurements using semilinear regression models and Bland–Altman plots. Therapeutic groups in which DCE-US showed reduced tumor perfusion had longer tumor DTs. It was confirmed that the DCE-US parameter peak enhancement (PE), reflecting relative blood volume, had predictive value for the outcome of therapy: larger PE correlated with shorter DT. In addition, perfusion heterogeneity was also associated with outcome: tumors that had more heterogeneous perfusion had faster growth, i.e., shorter DTs. This study demonstrates that DCE-US can be used as a method to predict the efficacy of electroporation-based treatment.

Nonexistence of global solutions for generalized Tricomi equations with combined nonlinearity

In the present paper, we investigate the blow-up dynamics for local solutions to the semilinear generalized Tricomi equation with combined nonlinearity. As a result, we enlarge the blow-up region in comparison to the ones for the corresponding semilinear models with either power nonlinearity or nonlinearity of derivative type. Our approach is based on an iteration argument to establish lower bound estimates for the space average of local solutions. Finally, we obtain upper bound estimates for the lifespan of local solutions as byproduct of our iteration argument.

Global existence of solutions for weakly coupled systems of semi-linear structurally damped σ-evolution models

ABSTRACT In this paper, we study the Cauchy problems for weakly coupled systems of semi-linear structurally damped σ-evolution models with different power nonlinearities. By assuming additional regularity on the initial data, with , we use and estimates for solutions to the corresponding linear Cauchy problems to prove the global (in time) existence of small data Sobolev solutions to the weakly coupled systems of semi-linear models from suitable function spaces for any and . Moreover, not only the nonexistence of global solution is indicated by using test function method, but also a lifespan estimate is discussed when σ and are integers.

Influence of strong time-dependent oscillations in semilinear damped wave models

We study the global (in time) existence of small data solutions to some Cauchy problems for semilinear damped wave models with strong time-dependent oscillations. The goal is to understand the influence of strong oscillations in the coefficients on solutions to some semilinear models with an “effective-like” damping term, where the data are supposed to belong to different classes of regularity.

Filtration model with multiple particle capture

Grout filtration in porous soil is used in construction industry to create underground waterproof walls. When the suspension flows through the pores, various forces act on the suspended particles, blocking them on the frame of the porous medium. A one-dimensional model of deep bed filtration for a monodisperse suspension in a porous medium with several particle capture mechanisms is considered. The mathematical model includes the equation of mass balance of suspended and retained particles and the kinetic equation of deposit growth with a piecewise-smooth linear-constant filtration function and a nonlinear concentration function. The solution of the nonlinear model is obtained by the finite difference method using an explicit difference scheme with second-order approximation. To construct the asymptotics of a complex model, the solutions of simplified linear and semilinear models and their combination are used. In the zone of a smooth filtration function, the best approximation of the solution of a complex model is determined by a certain linear combination of simple solutions. In another area, solution of a simplified problem with a piecewise-smooth filtration function and a linear concentration function is closest to the solution of a nonlinear model. Calculations show that in the zone of a smooth filtration function, a combination of simple solutions defines a solution approximation with second-order of smallness. In the area where it is necessary to take into account the non-smoothness of the filtration function, the approximation of a solution has a first order of smallness.

An application of L1 estimates for oscillating integrals to parabolic like semi-linear structurally damped σ-evolution models

We study the following Cauchy problems for semi-linear structurally damped σ-evolution models:utt+(−Δ)σu+μ(−Δ)δut=f(u,ut),u(0,x)=u0(x),ut(0,x)=u1(x) with σ≥1, μ>0 and δ∈(0,σ2). Here the function f(u,ut) stands for the power nonlinearities |u|p and |ut|p with a given number p>1. We are interested in investigating L1 estimates for oscillating integrals in the presentation of the solutions to the corresponding linear models with vanishing right-hand sides by applying the theory of modified Bessel functions and Faà di Bruno's formula. By assuming additional Lm regularity on the initial data, we use (Lm∩Lq)−Lq and Lq−Lq estimates with q∈(1,∞) and m∈[1,q), to prove the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on Lq spaces.

Inference in Linear Regression Models with Many Covariates and Heteroscedasticity

ABSTRACTThe linear regression model is widely used in empirical work in economics, statistics, and many other disciplines. Researchers often include many covariates in their linear model specification in an attempt to control for confounders. We give inference methods that allow for many covariates and heteroscedasticity. Our results are obtained using high-dimensional approximations, where the number of included covariates is allowed to grow as fast as the sample size. We find that all of the usual versions of Eicker–White heteroscedasticity consistent standard error estimators for linear models are inconsistent under this asymptotics. We then propose a new heteroscedasticity consistent standard error formula that is fully automatic and robust to both (conditional) heteroscedasticity of unknown form and the inclusion of possibly many covariates. We apply our findings to three settings: parametric linear models with many covariates, linear panel models with many fixed effects, and semiparametric semi-linear models with many technical regressors. Simulation evidence consistent with our theoretical results is provided, and the proposed methods are also illustrated with an empirical application. Supplementary materials for this article are available online.

A Family of Non Linear Models in a Market with Semi Markov Regimes: Application to the Commodity and the Derivative Market

This paper introduces a family of coupled semi Markov regime switching multidimensional non linear models for general asset prices. Two particular instances of the models are explored. The first instance is one modeling commodity prices. Estimation formulas for historical parameters are developed. The second instance of the family of models introduced is one generalizing Heston model. It allows for semi Markov regime switching of Heston parameters. We develop a general semi closed formula for vanilla option prices given the risk neutral option parameters.

Fully nonlinear statistical and machine‐learning approaches for hydrological frequency estimation at ungauged sites

AbstractThe high complexity of hydrological systems has long been recognized. Despite the increasing number of statistical techniques that aim to estimate hydrological quantiles at ungauged sites, few approaches were designed to account for the possible nonlinear connections between hydrological variables and catchments characteristics. Recently, a number of nonlinear machine‐learning tools have received attention in regional frequency analysis (RFA) applications especially for estimation purposes. In this paper, the aim is to study nonlinearity‐related aspects in the RFA of hydrological variables using statistical and machine‐learning approaches. To this end, a variety of combinations of linear and nonlinear approaches are considered in the main RFA steps (delineation and estimation). Artificial neural networks (ANNs) and generalized additive models (GAMs) are combined to a nonlinear ANN‐based canonical correlation analysis (NLCCA) procedure to ensure an appropriate nonlinear modeling of the complex processes involved. A comparison is carried out between classical linear combinations (CCAs combined with linear regression (LR) model), semilinear combinations (e.g., NLCCA with LR) and fully nonlinear combinations (e.g., NLCCA with GAM). The considered models are applied to three different data sets located in North America. Results indicate that fully nonlinear models (in both RFA steps) are the most appropriate since they provide best performances and a more realistic description of the physical processes involved, even though they are relatively more complex than linear ones. On the other hand, semilinear models which consider nonlinearity either in the delineation or estimation steps showed little improvement over linear models. The linear approaches provided the lowest performances.

Deriving Tests of the Semi–linear Regression Model Using the Density Function of a Maximal Invariant

In the context of a general regression model in which some regression coefficients are of interest and others are purely nuisance parameters, we define the density function of a maximal invariant statistic with the aim of testing for the inclusion of regressors (either linear or non-linear) in linear or semi-linear models. This allows the construction of the locally best invariant test, which in two important cases is equivalent to the one-sided t test for a regression coefficient in an artificial linear regression model.We consider a specific semi-linear model to apply the constructed test.

Strong solutions of semilinear matched microstruture models

The subject of this article is a matched microstructure model for Newtonian fluid flows in fractured porous media. This is a homogenized model which takes the form of two coupled parabolic differential equations with boundary conditions in a given (two-scale) domain in Euclidean space. The main objective is to establish the local well-posedness in the strong sense of the flow. Two main settings are investigated: semilinear systems with linear boundary conditions and semilinear systems with nonlinear boundary conditions. With the help of analytic semigroups, we establish local well-posedness and investigate the long-time behavior of the solutions in the first case: we establish global existence and show that solutions converge to zero at an exponential rate.

Goodness-of-fit tests in semi-linear models

Specification tests for the error distribution are proposed in semi-linear models, including the partial linear model and additive models. The tests utilize an integrated distance involving the empirical characteristic function of properly estimated residuals. These residuals are obtained from an initial estimation step involving a combination of penalized least squares and smoothing techniques. A bootstrap version of the tests is utilized in order to study the small sample behavior of the procedures in comparison with more classical approaches. As an example, the tests are applied on some real data sets.

Evaluation of effect of number of electrodes in ECT sensors on image quality

It is commonly assumed that the use of more electrodes in an electrical capacitance tomography (ECT) sensor would result in higher resolution images. In this paper, the issues with the number of electrodes will be discussed. To investigate the effect of the number of electrodes on the quality of reconstructed images, ECT sensors with different number of electrodes-4, 8, 12, 16, 20, 24, and 32-are investigated. The capacitance between different electrode pairs is calculated for some typical permittivity distributions using a finite element method. The obtained capacitance data are then used to reconstruct images using the projected Landweber iteration algorithm based on both linear and semi-linear ECT models. The sensitivity distributions for the ECT sensors with different number of electrodes are analyzed. The main conclusion is that limited new information can be obtained and little improvement in the quality of images be achieved if the number of electrodes is more than 12.

Parameter estimation in semi-linear models using a maximal invariant likelihood function

In this paper, we consider the problem of estimation of semi-linear regression models. Using invariance arguments, Bhowmik and King [2007. Maximal invariant likelihood based testing of semi-linear models. Statist. Papers 48, 357–383] derived the probability density function of the maximal invariant statistic for the non-linear component of these models. Using this density function as a likelihood function allows us to estimate these models in a two-step process. First the non-linear component parameters are estimated by maximising the maximal invariant likelihood function. Then the non-linear component, with the parameter values replaced by estimates, is treated as a regressor and ordinary least squares is used to estimate the remaining parameters. We report the results of a simulation study conducted to compare the accuracy of this approach with full maximum likelihood and maximum profile-marginal likelihood estimation. We find maximising the maximal invariant likelihood function typically results in less biased and lower variance estimates than those from full maximum likelihood.

Maximal invariant likelihood based testing of semi-linear models

In this paper, we use a maximal invariant likelihood (MIL) to construct two likelihood ratio (LR) tests in the context of a semi-linear regression model. The first involves testing for the inclusion of a non-linear regressor and the second involves testing a linear regressor against the alternative of a non-linear regressor. We report the results of a Monte Carlo experiment that compares the size and power properties of the traditional LR tests with those of our proposed MIL based LR tests. Our simulation results show that in both cases, the MIL based tests have more accurate asymptotic critical values and better behaved (i.e., better centred) power curves than their classical counterparts.