Method For Multiple Solutions Research Articles

Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs

The local minimax method (LMM) proposed by Li and Zhou (2001) and Li and Zhou (2002) is an efficient method to solve nonlinear elliptic partial differential equations (PDEs) with certain variational structures for multiple solutions. The steepest descent direction and the Armijo-type step-size search rules are adopted in Li and Zhou (2002) and play a significant role in the performance and convergence analysis of traditional LMMs. In this paper, a new algorithm framework of the LMMs is established based on general descent directions and two normalized (strong) Wolfe-Powell-type step-size search rules. The corresponding algorithm framework named as the normalized Wolfe-Powell-type LMM (NWP-LMM) is introduced with its feasibility and global convergence rigorously justified for general descent directions. As a special case, the global convergence of the NWP-LMM algorithm combined with the preconditioned steepest descent (PSD) directions is also verified. Consequently, it extends the framework of traditional LMMs. In addition, conjugate gradient-type (CG-type) descent directions are utilized to speed up the NWP-LMM algorithm. Finally, extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared for different algorithms in the LMM’s family to indicate the effectiveness and robustness of our algorithms. In practice, the NWP-LMM combined with the CG-type direction indeed performs much better than its known LMM companions.

An Efficient Spectral Trust-Region Deflation Method for Multiple Solutions

It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct solutions and each solution may carry a unique physical meaning. One typical approach for finding multiple solutions is to use the Newton method with different initial guesses that ideally fall into the basins of attraction confining the solutions. In this paper, we propose a fast and accurate numerical method for multiple solutions comprised of three ingredients: (i) a well-designed spectral-Galerkin discretization of the underlying PDE leading to a nonlinear algebraic system (NLAS) with multiple solutions; (ii) an effective deflation technique to eliminate a known (founded) solution from the other unknown solutions leading to deflated NLAS; and (iii) a viable nonlinear least-squares and trust-region (LSTR) method for solving the NLAS and the deflated NLAS to find the multiple solutions sequentially one by one. We demonstrate through ample examples of differential equations and comparison with relevant existing approaches that the spectral LSTR-Deflation method has the merits: (i) it is quite flexible in choosing initial values, even starting from the same initial guess for finding all multiple solutions; (ii) it guarantees high-order accuracy; and (iii) it is quite fast to locate multiple distinct solutions and explore new solutions which are not reported in literature.

DEA Cross-Efficiency Aggregation with Deviation Degree Based on Standardized Euclidean Distance

Data envelopment analysis (DEA) has been extended to cross-efficiency to provide better discrimination and ranking of decision-making units (DMUs). Current researches about cross-efficiency mainly focus on the non-uniqueness of optimal solution of linear programming and information aggregation. As a common distance metric, standardized Euclidean distance is introduced to define the discrimination power between two vectors and the deviation degree for measuring the difference between the individual preference and group ideal preference. Based on above definitions, an alternative method is presented to compare multiple optimal solutions, and further, a universal weighted cross-efficiency model considering both dynamic adjustment of weights and preference formulation is constructed for evaluation and ranking. Two numerical examples are given to illustrate the effectiveness of the comparison method for multiple optimal solutions and weights determination method of DMUs, respectively. At last, a practical application aimed at evaluating environmental treatment efficiency in western area of China is given. Comparative analysis shows that our model could be more moderate, flexible, and general than some available models and methods, which can extend the theoretical research of cross-efficiency evaluation.

Normalized Goldstein-type local minimax method for finding multiple unstable solutions of semilinear elliptic PDEs

Normalized Goldstein-type local minimax method for finding multiple unstable solutions of semilinear elliptic PDEs

Convergence analysis of a spectral-Galerkin-type search extension method for finding multiple solutions to semilinear problems

本文提出计算半线性椭圆边值问题多解的一类高效的谱 Galerkin 型搜索延拓法 (SGSEM). 该方法基于模型方程相应线性特征值问题的若干特征函数的线性组合构造多解初值, 充分利用了传统搜索延拓法构造多解初值方面的优势. 同时, 采用插值系数 Legendre-Galerkin 谱方法离散模型问题, 具有计算成本低、计算精度高的优点. 运用 Schauder 不动点定理和其他技巧, 本文严格证明了对应于每个特定真解的数值解的存在性以及限制在该真解一个充分小的邻域内的数值解的唯一性, 并证明了其谱收敛性. 数值结果验证了算法的可行性与高效性, 并展示了不同类型的多解.

Multiple Soliton Solutions of Some Nonlinear Partial Differential Equations

In this paper, we implemented an improved tanh function Method for multiple soliton solutions of new coupled Konno-Oono equation and extended (3+1)-dimensional KdV-type equation.

Application of Runge-Kutta method for finding multiple numerical solutions to intuitionistic fuzzy differential equations

The present study is aimed to discuss multiple numerical solutions to first order ordinary differential equation which is intuitionistic fuzzy in nature, under the concept of generalised differentiability. The first order intuitionistic fuzzy differential equation which has been taken for the present study, is changed into four systems of ordinary differential equations by the (α, β)-cut representation of an intuitionistic fuzzy set. After the transformation, each system contains two pair of equations; one is for membership function and the other for non-membership function. Then, the fourth order Runge-Kutta method is applied in each pair and the competence of the method over Euler method and Modified Euler method are shown by solving a real time problem.

Eigenfunction expansion method for multiple solutions of fourth-order ordinary differential equations with cubic polynomial nonlinearity

Multiple solutions of fourth-order ordinary differential equations (ODEs) with cubic polynomial nonlinearity are presented in this paper. The ODEs are discretized by the eigenfunction expansion method. Discretization error estimates are derived. We construct an efficient polynomial homotopy to find all solutions for the system of polynomial equations on the coarse level by recursion. Two kinds of filters are suggested for removing possible spurious solutions of the discretized system of polynomial equations. Numerical experiments are included to verify the error estimates and efficiency of the proposed homotopy.

Convergence analysis of a minimax method for finding multiple solutions of hemivariational inequality in Hilbert space

In 2013, a minimax method for finding saddle points of locally Lipschitz continuous functional was designed (Yao Math. Comp. 82 2087–2136 2013). The method can be applied to numerically solve hemivariational inequality for multiple solutions. Its subsequence and sequence convergence results in functional analysis were established in the same paper. But, since these convergence results do not consider discretization, they are not convergence results in numerical analysis. In this paper, we point out what approximation problem is, when this minimax method is used to solve hemivariational inequality and the finite element method is used in discretization. Computation of the approximation problem is discussed, numerical experiment is carried out and its global convergence is verified. Finally, as element size goes to zero, convergence of solutions of the approximation problem to solutions of hemivariational inequality is proved.

Using a derivative-free optimization method for multiple solutions of inverse transport problems

Identifying unknown components of an object that emits radiation is an important problem for national and global security. Radiation signatures measured from an object of interest can be used to infer object parameter values that are not known. This problem is called an inverse transport problem. An inverse transport problem may have multiple solutions and the most widely used approach for its solution is an iterative optimization method. This paper proposes a stochastic derivative-free global optimization algorithm to find multiple solutions of inverse transport problems. The algorithm is an extension of a multilevel single linkage (MLSL) method where a mesh adaptive direct search (MADS) algorithm is incorporated into the local phase. Numerical test cases using uncollided fluxes of discrete gamma-ray lines are presented to show the performance of this new algorithm.

A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems

In this work an iterative method is proposed to predict and demonstrate the existence and multiplicity of solutions for nonlinear boundary value problems. In addition, the proposed method is capable of calculating analytical approximations for all branches of solutions. This method is a combination of reproducing kernel Hilbert space method and a shooting-like technique which takes advantage of two powerful methods for solving nonlinear boundary value problems. The formulation and implementation of this iterative method is discussed for nonlinear second order with two and three-point boundary value problems. Also, the convergence of the proposed method is proved. To demonstrate the computational efficiency, the mentioned method is implemented for some nonlinear exactly solvable differential equations including strongly nonlinear Bratu equation and nonlinear reaction–diffusion equation. It is also applied successfully to two nonlinear three-point boundary value problems with unknown exact solutions. In the last example a new branch of solutions is found which shows the power of the method to search for multiple solutions and indicates that the method may be successful in cases where purely analytic methods are not.

An orthogonal subspace minimization method for finding multiple solutions to the defocusing nonlinear Schrödinger equation with symmetry

AbstractAn orthogonal subspace minimization method is developed for finding multiple (eigen) solutions to the defocusing nonlinear Schrödinger equation with symmetry. As such solutions are unstable, gradient search algorithms are very sensitive to numerical errors, will easily break symmetry, and will lead to unwanted solutions. Instead of enforcing a symmetry by the Haar projection, the authors use the knowledge of previously found solutions to build a support for the minimization search. With this support, numerical errors can be partitioned into two components, sensitive versus insensitive to the negative gradient search. Only the sensitive part is removed by an orthogonal projection. Analysis and numerical examples are presented to illustrate the method. Numerical solutions with some interesting phenomena are captured and visualized by their solution profile and contour plots. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Eigenfunction Expansion Method for Multiple Solutions of Semilinear Elliptic Equations with Polynomial Nonlinearity

Eigenfunction expansion discretization is considered for finding multiple solutions of semilinear elliptic equations with polynomial nonlinearity. Error estimates of the discretization are derived. An efficient polynomial homotopy is constructed for computing all solutions of the resulting polynomial system on coarse level. A filter strategy on successively finer levels is designed to remove spurious solutions and to refine nonspurious solutions, simultaneously. The filtered solutions are further refined by a finite element Newton method. Numerical results are included to verify the error estimates derived, the efficiency of the homotopy, and the effectiveness of the filter strategy. A specific case of the Lazer--McKenna conjecture is numerically verified.

A multiple exp-function method for nonlinear differential equations and its application

A multiple exp-function method for exact multiple wave solutions of nonlinear partial differential equations is proposed. The method is oriented towards the ease of use and capability of computer algebra systems and provides a direct and systematic solution procedure that generalizes Hirota's perturbation scheme. With the help of Maple, applying the approach to the (3+1)-dimensional potential-Yu–Toda–Sasa–Fukuyama equation yields exact explicit one-wave, two-wave and three-wave solutions, which include one-soliton, two-soliton and three-soliton type solutions. Two cases with specific values of the involved parameters are plotted for each of the two-wave and three-wave solutions.

A local min-max-orthogonal method for finding multiple solutions to noncooperative elliptic systems

A local min-max-orthogonal method together with its mathematical justification is developed in this paper to solve noncooperative elliptic systems for multiple solutions in an order. First it is discovered that a noncooperative system has the nature of a zero-sum game. A new local characterization for multiple unstable solutions is then established, under which a stable method for multiple solutions is developed. Numerical experiments for two types of noncooperative systems are carried out to illustrate the new characterization and method. Several important properties for the method are explored or verified. Multiple numerical solutions are found and presented with their profiles and contour plots. As a new bifurcation phenomenon, multiple asymmetric positive solutions to the second type of noncooperative systems are discovered numerically but are still open for mathematical verification.

Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems

In this paper, Homotopy Analysis Method (HAM) is applied to numerically approximate the eigenvalues of the fractional Sturm-Liouville problems. The eigenvalues are not unique. These multiple solutions, i.e., eigenvalues, can be calculated by starting the HAM algorithm with one and the same initial guess and linear operator \(\mathcal{L}\). It can be seen in this paper that the auxiliary parameter \(\hbar,\) which controls the convergence of the HAM approximate series solutions, has another important application. This important application is predicting and calculating multiple solutions.

Natural ventilation potential model considering solution multiplicity, window opening percentage, air velocity and humidity in China

The paper brings about a two-opening naturally ventilated building potential model considering solution multiplicity, window opening percentage, air velocity and humidity in China. Based on previous research by Yang et al. and Luo et al., this paper developed a revised model to estimate the natural ventilation potential. There are four main improvements: Firstly, counteract of wind and thermal pressure was included and method for multiple solutions was introduced accordingly. Secondly, the model considered the natural ventilation hours for windows opening percentage. Thirdly, wind velocity and humidity were considered lastly. Fourthly, cooling and heating hours were also estimated. In case study, the paper analyzes natural ventilation potential in cities in four typical climate regions in China. Results show that there are more natural ventilation hours than previous models estimated, especially in cold climate regions. Any parameter in this model can be changed according to practical situation, such as climate data, building orientation, window size, opening position, etc. All estimated information can provide reference for architects in early stages of building design. Meanwhile, this model can be used for energy efficiency estimation and indoor environment research.

A perturbation method for multiple sign-changing solutions

We develop a perturbation method for non-even functionals which produces prescribed number of sign-changing solutions. The abstract theory is applied to the perturbed subcritical elliptic equation and the perturbed Brezis–Nirenberg critical exponent problem.

The homotopy analysis method for multiple solutions of nonlinear boundary value problems

This paper investigates two basic steps of the homotopy analysis method (HAM) when applied to nonlinear boundary value problems of the chemical reaction kinetics, namely (1) the prediction and (2) the effective calculation of multiple solutions. To be specific, the approach is applied to the dual solutions of an exactly solvable reaction-diffusion model for porous catalysts with apparent reaction order n=-1. It is shown that (i) the auxiliary parameter ℏ which controls the convergence of the HAM solutions in general plays a basic role also in the prediction of dual solutions, and (ii) the dual solutions can be calculated by starting the HAM-algorithm with one and the same initial guess. It is conjectured that the features (1) and (2) hold generally in use of HAM to identify and to determine the multiple solutions of nonlinear boundary value problems.

Analysis of search-extension method for finding multiple solutions of nonlinear problem

For numerical computations of multiple solutions of the nonlinear elliptic problem Δu + f(u = 0 in Ω, u = 0 on Γ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shall complete its theoretical analysis. It is assumed that the nonlinearity is non-convex and its solution is isolated, under some conditions the corresponding linearized problem has a unique solution. By use of the compactness of the solution family and the contradiction argument, in general conditions, the high order regularity of the solution u ∈ H 1+α, α > 0 is proved. Assume that some initial value searched by suitably many eigenbases is already fallen into the neighborhood of the isolated solution, then the optimal error estimates of its nonlinear finite element approximation are shown by the duality argument and continuation method.