Dimensional Nonlinear Problem Research Articles

A second-order fitted scheme combined with time two-grid technique for two-dimensional nonlinear time fractional telegraph equations involving initial singularity

In this paper, we derive the improved regularity for two-dimensional nonlinear time fractional telegraph equations by virtue of the technic of decomposition at first. Then, the famous L2-1σ formula is adopted to approximate the Caputo derivative and the central finite difference method is used for spatial discretization. The convergence accuracy of the proposed method is second order in both temporal and spatial direction. Meanwhile, for the sake of reducing the time of solving the high dimensional nonlinear problems, an efficient time two-grid algorithm is proposed. Furthermore, stability analysis of the proposed scheme is studied by the energy method. At last, numerical experiments are presented to verify the validity of the theoretical statements.

Accuracy analysis of satellite antenna panel expansion based on BP neural network

AbstractLarge deployable space mechanisms are widely used in the field of aerospace and have been paid increasingly high attention recently. The satellite antenna expansion system is the classic large deployable space mechanism. However, during the expansion of the satellite antenna deployable mechanism, the expansion accuracy is affected by the existing various uncertain factors which even result in scraping the satellite. For example, the hinge locking error has the significant influence on the deployment accuracy of the satellite antenna panel. In term of this issue, considering the advantage of the back‐propagation (BP) neural network for the high dimensional nonlinear problem, this paper mainly adopts it to analyze the impact of hinge locking error on the expansion accuracy of the antenna panel. The results show that the probability of the actual flatness deviation of the satellite antenna panel falling in the required accuracy area is 99.56%, and the probability of the actual pointing angle deviation falling in the required accuracy area is 99.84%.

Efficient discrete singular convolution differential quadrature algorithm for solitary wave solutions for higher dimensions in shallow water waves

In this research, discrete singular convolution, that depends on Regularized Shannon kernel, is used to look for the efficient solution of (4 + 1) dimensional nonlinear Fokas equation. The governing system of nonlinear five-dimensional Fokas equation is transformed into a system of nonlinear ordinary differential equations via discrete singular convolution quadrature technique. Consequently, the 4th order Runge–Kutta technique is employed to solve this system. The validity of this approach is achieved by comparing the obtained numerical solution with the exact solution, where the error is ≤ 1 × 10−5. Moreover, a parametric analysis is presented to examine the influence of the used approach on solitary wave solution. The computed solutions display strength, reliability and efficiency of the implemented techniques that can be effectively work for nonlinear equations in a like manner to linear models. Our treatment in this work works to solve higher dimensional nonlinear problems of physical and numerical features.

A class of localized soliton and fractal pattern solutions of the (2 + 1)-dimensional modified dispersive long wave model

The variable separation approach is an important tool to obtain exact localized and fractal structure solutions of higher dimensional nonlinear problems. The general separable solutions involving double random functions are obtained by means of Riccati equation and the variable separation hypothesis. By choosing the suitable arbitrary functions, we obtain a class of localized excitations and fractal structures solutions of the (2 + 1)-dimensional modified dispersive long wave (MDLW) model. Such solutions present couple lump waves, breather wave solutions, dromions, oscillating multi-lumps, oscillating multi-dromoins and ring solitons of the model. Moreover, the fractal lump and fractal dromions type excitation-localized structures of the model are presented here. To visualize the dynamics of each excitation structures are demonstrated by the 3D and density plots.

Two-variable nullcline analysis of ionic general equilibrium predicts calcium homeostasis in ventricular myocytes.

Ventricular contraction is roughly proportional to the amount of calcium released from the Sarcoplasmic Reticulum (SR) during systole. While it is rather straightforward to measure calcium levels and contractibility under different physiological conditions, the complexity of calcium handling during systole and diastole has made the prediction of its release at steady state impossible. Here we approach the problem analyzing the evolution of intracellular and extracellular calcium fluxes during a single beat which is away from homeostatic balance. Using an in-silico subcellular model of rabbit ventricular myocyte, we show that the high dimensional nonlinear problem of finding the steady state can be reduced to a two-variable general equilibrium condition where pre-systolic calcium level in the cytosol and in the SR must fulfill simultaneously two different equalities. This renders calcium homeostasis as a problem that can be studied in terms of its equilibrium structure, leading to precise predictions of steady state from single-beat measurements. We show how changes in ion channels modify the general equilibrium, as shocks would do in general equilibrium macroeconomic models. This allows us to predict when an enhanced entrance of calcium in the cell reduces its contractibility and explain why SERCA gene therapy, a change in calcium handling to treat heart failure, might fail to improve contraction even when it successfully increases SERCA expression.

Lie symmetries, reduction and exact solutions of the (1+2)-dimensional nonlinear problem modeling the solid tumour growth

The well known nonlinear model for describing the solid tumour growth (Byrne et al. 2003) is under study using an approach based on Lie symmetries. It is shown that the model in the two-dimensional (in space) approximation forms a (1+2)-dimensional boundary value problem, which admits a highly nontrivial Lie symmetry. The special case involving the power-law nonlinearities is examined in details. The symmetries derived are applied for the reduction of the nonlinear boundary value problem in question to problems of lower dimensionality. Finally, the reduced problems with correctly-specified coefficients were exactly solved and the exact solutions derived were analysed, in particular, some plots were build in order to understand the time-space behaviour of these solutions and to discuss their biological interpretation.

Globally Constructed Adaptive Local Basis Set for Spectral Projectors of Second Order Differential Operators

Spectral projectors of second order differential operators play an important role in quantum physics and other scientific and engineering applications. In order to resolve local features and to obtain converged results, typically the number of degrees of freedom needed is much larger than the rank of the spectral projector. This leads to significant cost in terms of both computation and storage. In this paper, we develop a method to construct a basis set that is adaptive to the given differential operator. The basis set is systematically improvable, and the local features of the projector is built into the basis set. As a result the required number of degrees of freedom is only a small constant times the rank of the projector. The construction of the basis set uses a randomized procedure, and only requires applying the differential operator to a small number of vectors on the global domain, while each basis function itself is supported on strictly local domains and is discontinuous across the global domain. The spectral projector on the global domain is systematically approximated from such a basis set using the discontinuous Galerkin (DG) method. The global construction procedure is very flexible, and allows a local basis set to be consistently constructed even if the operator contains a nonlocal potential term. We verify the effectiveness of the globally constructed adaptive local basis set using one-, two- and three-dimensional linear problems with local potentials, as well as a one dimensional nonlinear problem with nonlocal potentials resembling the Hartree-Fock problem in quantum physics.

A spectral method for an elliptic equation with a nonlinear Neumann boundary condition

Let Ω be an open region in ℝd, d ≥ 2, that is diffeomorphic to 𝔹d. Consider solving −Δu + γu = 0 on Ω with the Neumann boundary condition $\frac {\partial u}{\partial \mathbf {n}}=b\left (\cdot ,u\right ) $ over ∂Ω. The function b is a nonlinear function of u. The problem is reformulated in a weak form, and then a spectral Galerkin method is used to create a sequence of finite dimensional nonlinear problems. An error analysis shows that under suitable assumptions, the solutions of the finite dimensional problems converge to those of the original problem. To carry out the error analysis, the original problem and the spectral method is converted to a nonlinear integral equation over H1/2 (Ω) , and the reformulation is analyzed using tools for solving nonlinear integral equations. Numerical examples are given to illustrate the method. In our error analysis, we assume the existence and local uniqueness of a solution. For the case of three dimensions and a nonlinearity b that is given by the Stefan–Boltzmann law, we will provide an existence proof in the final section.

Evaluation of different metaheuristic optimization algorithms in feature selection and parameter determination in SVM classification

In various studies, classification method and input features are two main factors that have significant effects on the results. In high dimensional non-linear problems, SVM was suggested as the superior classification. In these cases, the kernel parameters are often determined by the trial-and-error approach, which leads to reduce reliability of the results and automation level. On the other hand, because of the similarity of pixel spectral behavior, the classification accuracy will be reduced using only spectral bands in complex urban areas. To overcome this limitation, using additional features (e.g., textural and elevational features) was suggested in many studies. However, due to high variety of textural features in terms of type and direction, the presence probability of dependent features will increase by using all the features, which results in classification inefficiency. In addition to relatively high automation, in this paper, metaheuristic optimization algorithms were used to optimize simultaneously SVM in feature selection (FS) and parameter determination (PD) process as a solution due to being independent of image type and scene. There are few comprehensive evaluations in this field in various studies. For this purpose, a comprehensive research of the most efficient optimization algorithms in SVM (i.e., ACOR, GA, ICA, and PSO) was carried out in different ways and by different input features. Moreover, the results were compared to random forest (RF) classification in terms of FS process and accuracy. The optimized SVMs were implemented on two different image scenes (i.e., simple suburban and complex urban areas) in order to show the robustness of the optimized methods in terms of image type and scene. The results were evaluated by five quantitative criteria and McNemar’s test. Also, the approximate time calculations, the number of optimized features, and parameters were presented for each image scene. In comparison with using only spectral bands, the results showed that the optimized SVMs improved the overall accuracy (OA) by 12% and kappa coefficient (KC) by 15% using independent features (Z score of 320 at 95% confidence interval). Moreover, the ICA in conjunction with SVM can provide more accurate results rather than other optimization algorithms. By applying optimized features, OA and KC were improved by 4.88 and 5.9% in the simple suburban scene and 21.82 and 40.21% in the complex urban scene in comparison with using all the input features, respectively. The higher improvement was in the complex image scene because FS is more important in these scenes.

New function method to the (n+1)-dimensional nonlinear problems

In this study, a new approach that assumes and is applied to construct the traveling wave solutions of the (N + 1)-dimensional double sine-Gordon and (N + 1)-dimensional double sinh-cosh-Gordon equations. Some new elliptic integral function solutions are respectively obtained by this method, and then these solutions are converted into the Jacobi elliptic function solutions. According these results, one can easily see that this method is very effective mathematical tool for the (N+1)-dimensional nonlinear physical problems.

Two-level Multi-surrogate Assisted Optimization method for high dimensional nonlinear problems

Curse of dimensionality is a key issue in engineering optimization. When the dimension increases, distribution of samples becomes sparse due to expanded design space. To obtain accurate and reliable results, the amount of samples often grows exponentially with the dimensions. To improve the efficiency of the surrogate with limited samples, a Two-level Multi-surrogate Assisted Optimization (TMAO) is suggested. The framework of the TMAO is to decompose a complicated problem into separable and non-separable components. In the first-level, High Dimensional Model Representation (HDMR) is utilized to determine the correlations among input variables. Then, a high dimensional problem can be decomposed into separable and non-separable components. Thus, the dimension of the original problem might be reduced significantly. Moreover, considering noises and outliers, Support Vector Regression (SVR)-HDMR is utilized to obtain more reliable surrogate. Expected Improvement (EI) criterion is suggested to generate new samples to save computational cost. In the second-level, to handle the non-separable component, a multi-surrogate assisted sampling strategy is suggested. Compared with other methods, the distinctive characteristic of the suggested sampling strategy is to use different surrogates to search potential uncertain regions. Considering the diversity of surrogates, more feature samples might be generated close to the local optimum. Even though it is still difficult to find a global solution, it could help us to find a feasible solution in practice. To verify the performance of the suggested method, several high dimensional mathematical functions are tested by the suggested method. The results demonstrate that all test functions can be successfully solved.

Variable Selection Using Adaptive Nonlinear Interaction Structures in High Dimensions

Numerous penalization based methods have been proposed for fitting a traditional linear regression model in which the number of predictors, p, is large relative to the number of observations, n. Most of these approaches assume sparsity in the underlying coefficients and perform some form of variable selection. Recently, some of this work has been extended to nonlinear additive regression models. However, in many contexts one wishes to allow for the possibility of interactions among the predictors. This poses serious statistical and computational difficulties when p is large, as the number of candidate interaction terms is of order p2. We introduce a new approach, “Variable selection using Adaptive Nonlinear Interaction Structures in High dimensions” (VANISH), that is based on a penalized least squares criterion and is designed for high dimensional nonlinear problems. Our criterion is convex and enforces the heredity constraint, in other words if an interaction term is added to the model, then the corresponding main effects are automatically included. We provide theoretical conditions under which VANISH will select the correct main effects and interactions. These conditions suggest that VANISH should outperform certain natural competitors when the true interaction structure is sufficiently sparse. Detailed simulation results are also provided, demonstrating that VANISH is computationally efficient and can be applied to nonlinear models involving thousands of terms while producing superior predictive performance over other approaches.

A trust region interior point algorithm for infinite dimensional nonlinear programming

A trust region interior point algorithm for infinite dimensional nonlinear problem, which is motivated by the application of black-box approach to the distributed parameter system optimal control problem with equality and inequality constraints on states and controls, and with bounds on the controls is formulated. By introducing a proper functional which is analogous to the Lagrange function, both equality and inequality constraints can be treated identically and the first order optimality condition is given, then based on the works of Coleman, Ulbrich and Heinkenschloss, the trust region interior point algorithm which is employed to solve the optimization problem under consideration is presented.

Calculation on Flux-MMF Relationship of Orthogonal-Core

Orthogonal-cores have various potential applications, for instance in parametric transformers and dc-ac converters. The operating characteristics of the devices can be calculated on the basis of the measured relationship of flux to MMF of the orthogonal -core. To achieve optimal design of the applied device, the relationship of flux to MMF must be determined; however, this involves solving a three dimensional nonlinear problem. In this paper, we calculate the flux-MMF relationship based on a magnetic circuit model for the orthogonal-core. The computed results agree well with experiment. The method of this study is shown to be valid for calculation of characteristics and useful for optimal design of application devices.

Basic results in the development of sensitivity and stability analysis in nonlinear programming

For large classes of mathematical programming problems, a detailed technical survey is given of key developments in sensitivity and stability analysis results, i.e. results characterizing the relationship between the optimal value function or a solution set and problem perturbations. The emphasis is on finite dimensional nonlinear problems with deterministic parametric perturbations. Precise assumptions and conclusions of key results are given in the more than 30 theorems that are stated. Some effort has been made to unify the notation and terminology and to place the results in perspective. Directions of future research and applications are indicated. Finally, an extensive bibliography is included. The paper is motivated by a desire to unify into one body of theory the many penetrating results that are now known in this crucially important area.