Homotopy Method Research Articles

Heat-Equation-Based Smoothing Homotopy Method for Nonlinear Optimal Control Problems

This paper presents a new heat-equation-based smoothing homotopy method for solving nonlinear optimal control problems with the indirect method. The surrogates, derived from the heat equation solution, are first incorporated into the necessary conditions as replacement of the terminal state and costate variables. The homotopy process is then applied to the heat conduction time: a longer time results in a more uniform temperature distribution and greater stability against initial temperature variations, thereby making the corresponding homotopy problem much easier to solve; zero time implies no heat conduction and reverts to the original problem. Furthermore, capitalizing on the heat equation’s boundedness characteristic, an efficient approach for determining the hypersensitive parameters is proposed, thus obviating the necessity of manual tuning. Challenging numerical examples are provided to demonstrate the superior performance of the proposed method, indicating that its convergence and efficiency are significantly enhanced compared to the original smoothing homotopy methods.

Multicriteria Optimization of Stochastic Robust Control of the Tracking System

A multicriteria optimization of stochastic robust control with two degrees of freedom of a tracking system with anisotropic regulators has been developed to increase accuracy and reduce sensitivity to uncertain object parameters. Such objects are located on a moving base, on which sensors for angles, angular velocities and angular accelerations are installed. Improvements in the accuracy of control with two degrees of freedom include closed-loop feedback control and open-loop feedback control through the use of reference and perturbation effects. The multicriteria optimization of the stochastic robust control tracking system with two degrees of freedom with anisotropic controllers is reduced to the iterative solution of a system of four coupled Riccati equations, the Lyapunov equation, and the determination of the anisotropy norm of the system by an expression of a special form, which is numerically solved using the homotopy method, which includes vectorization matrices and iterations according to Newton's method. The objective vector of robust control is calculated in the form of a solution of a vector game, the vector gains of which are direct indicators of the quality that the system should achieve in different modes of its operation. The calculation of the vector gains of this game is related to the simulation of a synthesized system with anisotropic regulators for different modes of operation with different input signals and object parameter values. The solutions of this vector game are calculated on the basis of a set of Pareto-optimal solutions taking into account the binary relations of preferences on the basis of the metaheuristic algorithm of multi-swarm Archimedes optimization. Based on the results of the synthesis of stochastic robust control of a tracking system with two degrees of freedom with anisotropic controllers, it is shown that the use of synthesized controllers made it possible to increase the accuracy of system control, reduce the time of transient processes by 3–5 times, reduce the variance of errors by 2.7 times, and reduce the sensitivity of the system to the change of object parameters compared to typical regulators.

Modified homotopy approach for diffractive production in the saturation region

In this paper we continue to develop the homotopy method for solving of the nonlinear evolution equation for the diffractive production in deep inelastic scattering. We introduce part of the nonlinear corrections as a first step of this approach. This simplified nonlinear evolution equation is solved analytically taking into account the initial and boundary conditions for the process. At the second step of our approach we demonstrated that the perturbative procedure can be used for the remaining parts of the nonlinear corrections. It turns out that these corrections are small and can be estimated in the regular iterative procedure. Published by the American Physical Society 2024

Improved Solutions of OHAM Approximate Procedure for Classes of Nonlinear ODEs

The primary purpose of this study is to apply the Optimal Homotopy Asymptotic Method (OHAM) to various nonlinear initial value problems of different orders to evaluate its accuracy, convergence, and computational efficiency. OHAM is considered a highly effective technique for solving nonlinear differential equations and is commonly used in scientific and engineering disciplines. It combines the strengths of homotopy and asymptotic methods. OHAM offers a straightforward approach to controlling and adjusting the convergence of the series solution. This is achieved through the utilization of an auxiliary function that incorporates multiple convergent control parameters with one order of approximation, which are optimally determined. The OHAM approach heavily relies on the auxiliary function H(p) which allows for the flexible and efficient solving of nonlinear differential equations. By carefully constructing and optimizing the parameters of H( p), the convergence of the solution series can be effectively controlled. As a result, OHAM proves to be a versatile and effective approach for solving various mathematical and engineering problems. Several examples have been solved. Numerical comparisons, displayed in tables and shown graphically in figures, prove and confirm the capability, efficiency, and better accuracy with less computational work.

Exploring a Dynamic Homotopy Technique to Enhance the Convergence of Classical Power Flow Iterative Solvers in Ill-Conditioned Power System Models

This paper presents a dynamic homotopy technique that can be used to calculate a preliminary result for a power flow problem (PFP). This result can then be used as an initial estimate to efficiently solve the PFP using either the classical Newton-Raphson (NR) method or its fast decoupled version (FDXB) while still maintaining high accuracy. The preliminary stage for the dynamic homotopy problem is formulated and solved by employing integration techniques, where implicit and explicit schemes are studied. The dynamic problem assumes an initial condition that coincides with the initial estimate for a traditional iterative method such as NR. In this sense, the initial guess for the FPF is adequately set as a flat start, which is a starting for the case when this initialization is of difficult assignment for convergence. The static homotopy method requires a complete solution of a PFP per homotopy pathway point, while the dynamic homotopy is based on numerical integration methods. This approach can require only one LU factorization at each point of the pathway. Allocating these points properly helps avoid several PFP resolutions to build the pathway. The hybrid technique was evaluated for large-scale systems with poor conditioning, such as a 109,272-bus model and other test systems under stressed conditions. A scheme based on the implicit backward Euler scheme demonstrated the best performance among other numerical solvers studied. It provided reliable partial results for the dynamic homotopy problem, which proved to be suitable for achieving fast and highly accurate solutions using both the NR and FDXB solvers.

A new method for identifying elastic parameters of isotropic materials based on square specimens

This paper proposes a new impulse excitation technique using a square plate. First, the functional relationship between the modal frequency of the specimen and the geometrical dimensions and mechanical parameters was established by using the finite element method. Then, the continuous functional relationship derived by a homotopy method allowed the frequency ratios to be related to the thickness-to-length ratio and Poisson’s ratio. By measuring the frequency ratios and thickness-to-length ratio, Poisson’s ratio could be calculated using this functional relationship. When the density and Poisson’s ratio were known, Young’s modulus could be identified inversely in conjunction with the finite element analysis. Finally, a comparison test between this method and the traditional impulse excitation technique was designed and implemented, and the results showed that this method has advantages in both testing efficiency and accuracy. The study provides a new idea for system identification, which has important application value and promotion significance.

Mainardi smoothing homotopy method for solving nonlinear optimal control problems

This paper proposes a new type of smoothing homotopy method for solving general nonlinear optimal control problems via the indirect method, leveraging the Mainardi kernel as the smoothing kernel. The Mainardi kernel is firstly derived from the fundamental solution of the time-fractional diffusion-wave equation, representing a generalized form of the Gaussian kernel. By altering the fractional derivative order, the kernel can seamlessly switch between non-Gaussian and Gaussian forms. Then, parts of the two-point boundary value problems are convolved with the smoothing kernel, and the resulting surrogates are incorporated into the necessary conditions, replacing the terminal state and costate variables. The homotopy process is used for the smoothing parameter: increasing this parameter enhances the smoothing effect, simplifying the homotopy problems, while a parameter value of zero implies no smoothing, reverting to the original problems. Additionally, explicit expressions for the Mainardi kernel at specific derivative orders are derived, avoiding the inefficiencies associated with fractional derivatives. Simulation examples demonstrate that the proposed method offers significant advantages in flexibility and convergence compared to the traditional Gaussian smoothing homotopy method.

Some results on Quillen’s conjecture via equivalent-poset techniques

We extend the main theorem of Aschbacher and Smith on the Quillen conjecture from p>5 to the remaining odd primes p = 3,5 . In the process, we develop further combinatorial and homotopical methods for studying the poset of non-trivial elementary abelian p -subgroups of a finite group. The techniques lead to a number of further results on the conjecture, often reducing dependence on the CFSG; in particular, we also provide some partial results toward the case of p=2 .

Differentiable homotopy methods for gradually reinforcing the training of fully connected neural networks

Deep fully connected neural networks (FCNNs) are the workhorses of deep learning and are broadly applicable due to their “agnostic” structure. Generally, the learning capability of FCNNs improves with the increase in the number of layers and the width of each layer, which, however, comes at an increased computational cost in training. To alleviate this difficulty, in this paper, we develop a gradually reinforced differentiable homotopy (GRDH) method to train the FCNNs. Explicitly speaking, by introducing an extra variable t ranging between zero and one, we design a series of auxiliary functions, which are continuous and monotonically increasing in t. With the above functions, we formulate an optimization problem for training an artificial FCNN, which progressively incorporates more layers and nodes into the neural network as t changes from one to zero and eventually becomes an FCNN with a target number of layers or width of nodes. We prove that the set of solutions to the artificial problem contains an everywhere differentiable path, which starts from a uniquely given point at t=1 and ends at the weights and biases of the target FCNN as t goes to zero. The proposed GRDH method is a novel method that incorporates the differentiable homotopy methods into the training of deep learning methods, and retains the satisfactory theoretical convergence property the classical homotopy methods possess. To promote the application of the GRDH method, we implement it and another efficient method called HTA to train the same FCNNs and find that the GRDH method outperforms the HTA both in the computational time and number of iterations for obtaining a solution with similar (even higher) accuracy. Numerical results further confirm the effectiveness of the GRDH method to solve classification problems.

On computation of solution for [formula omitted] dimensional fractional order general wave equation

In this research article, we handle a class of (2+1) dimensional wave equations under fractional-order derivatives using an iterative integral transform due to Laplace. The concerned derivative is taken in the Caputo’s sense. The proposed method is a purely algebraic manipulation approach to compute solutions without a priori knowledge of geometry and physical meaning related to the proposed problem. In fact, in this procedure, we combine two novel techniques, Laplace transforms (LT) and iterative procedures to form a hybrid technique for computation of the solution to the proposed problem. The method is rapidly convergent. Here, we give various examples for the validation of our proposed method. The superiority of the method over the existing numerical method is that it does not require any prior discretization or collocation of functions. Also, the method is independent of axillary parameters as needed in the homotopy methods, because such auxiliary parameters control the efficiency of the mentioned methods. The proposed procedure is simple and straightforward. In addition, some comparison between exact and approximate solutions is also given. The proposed method is compared with the homotopy perturbation method (HPM) which shows that the proposed technique is more efficient and easy to implement.

Strong nonlinear mixing evolutions within phononic frequency combs

Phononic frequency combs (PFCs) represent an emerging attractive nonlinear vibrational phenomenon characterized by equidistant spectral lines. Despite the extensive experimental studies, the complex nonlinear mixing nature of PFCs continues to present significant challenges in solving and investigating their complete dynamics, which is difficult to achieve by existing computational approaches. In this paper, the entire solution space within a representative PFC induced by a 1:2 internal resonance is elucidated by conducting continuation computations and numerical long-time integrations. The proposed continuation approach is achieved by integrating our developed semi-analytical residue-regulating homotopy method (RRHM) with a pseudo arc-length continuation technique. In this solution space, we unearth wide-range nonlinear evolutions including overlapping intervals between the periodic and quasi-periodic branches, abundant multivalued sub-intervals, cyclic-fold (CF) bifurcations, and torus-doubling (TD) routes to chaos. In addition, multiple coexistences of a chaotic attractor and a periodic orbit, a chaotic attractor and a quasi-periodic orbit, as well as a periodic orbit and three quasi-periodic orbits are identified. Furthermore, we meticulously dissect and distinguish non-smooth variations in PFC morphology, which manifest as multiple jumps in comb spacing as the excitation frequency is swept across. This study could serve as a general guide for a comprehensive exploration of PFC dynamics and can offer insights to inform and inspire related experimental studies.

Dynamic characteristics of vertically irregular structures with random fields of different probability distributions based on stochastic homotopy method

Considering the uncertainty of elastic modulus in different materials as well as the strong nonlinearity between the structural frequency and elastic modulus, it is a great challenge for computing the dynamic characteristics of vertically irregular structures. In this research, a new stochastic homotopy approach is presented to compute the basic dynamic characteristics of the vertically irregular structures, where the elastic modulus of different materials are assumed as the random fields with different probability distributions. To obtain the dynamic characteristics of the vertically irregular structures, a stochastic eigenvalue equation is established firstly. Then the stochastic eigenvalue and eigenvector are represented by the homotopy series, and the stochastic residual error in relation to the stochastic eigenvalue equation is minimized to obtain the coefficients of the homotopy series. Afterwards, the basic dynamic characteristics, including stochastic natural frequencies and modal shapes, of the vertically irregular structures are obtained. In comparison to the sampling-based stochastic surrogate model methods like the Kriging and non-intrusive arbitrary polynomial chaos methods, the presented approach can efficiently yield more stable statistical moments of the natural frequencies. Meanwhile the proposed method can generate the statistical moments of the modal shapes, which is difficult to achieve with the stochastic surrogate model methods. Finally, the effectiveness of the suggested method is testified through two examples of a reinforced concrete-steel column and a realistic reinforced concrete-steel frame structure.

Fast generation of collision-free low-thrust trajectories for asteroid landing using deep neural networks

This study presents a fast generation method of time-optimal low-thrust trajectories based on deep neural networks (DNN) for the collision-free asteroid landings. An equivalent unconstrained differentiable problem is transformed via the introduction of smooth penalty function, so that it can be addressed with the optimal control framework. To efficiently solve the equivalent problem to generate sufficient dataset for training DNNs, a double homotopy method is developed. Specifically, the equivalent problem is connected to the time-optimal control problem without anti-collision through two consecutive simplifications, and a solving process with double backward continuation is also formulated. Besides, two DNN models are constructed to provide high-quality predictions for the minimum glide-slope constraint angle of an initial state, and the initial costates and landing time for time-optimal low-thrust trajectories with anti-collision constraints, respectively. As such, the efficiency of collision-free low-thrust landing trajectory generation is improved significantly. For the cases within the range of training dataset or within a certain extension, the low-thrust landing trajectories with a very small terminal landing error can be generated using the predicted initial costates. Meanwhile, with the high-quality initial costates, the high-precision landing trajectories can be obtained by the double homotopy method in only a few iterative steps. Finally, the promising results in a Bennu collision-free asteroid landing scenario validate the efficiency of the proposed method.

Effective potential of composite operators in the first order region of the QCD phase transition

We propose a method to determine the effective potential of QCD from the gap equation by introducing the homotopy method between the solutions of the equation of motion. Via this method, the effective potential beyond the bare vertex approximation is obtained, which then generalizes the Cornwall-Jackiw-Tomboulis effective potential for the bilocal composite operators. Moreover, the extended effective potential is set to be a function of self-energy instead of the propagator, which is the key point for the potential to be bounded from below. We then investigate the extended effective potential in the cases of phase transition of the QCD vacuum with a small current quark mass, the first order phase transition of QCD at finite temperature, and high baryon chemical potential. In the former case, the effective potential shows as an inflection point at the critical mass where the multiple solutions of the Dyson-Schwinger equation (DSE) vanishes, which is consistent with that obtained by solving the DSE directly. For the latter case, the in-medium properties, such as the latent heat and the difference of trace anomaly, of QCD is obtained. Published by the American Physical Society 2024

Multilevel Monte Carlo Methods for Stochastic Convection–Diffusion Eigenvalue Problems

We develop new multilevel Monte Carlo (MLMC) methods to estimate the expectation of the smallest eigenvalue of a stochastic convection–diffusion operator with random coefficients. The MLMC method is based on a sequence of finite element (FE) discretizations of the eigenvalue problem on a hierarchy of increasingly finer meshes. For the discretized, algebraic eigenproblems we use both the Rayleigh quotient (RQ) iteration and implicitly restarted Arnoldi (IRA), providing an analysis of the cost in each case. By studying the variance on each level and adapting classical FE error bounds to the stochastic setting, we are able to bound the total error of our MLMC estimator and provide a complexity analysis. As expected, the complexity bound for our MLMC estimator is superior to plain Monte Carlo. To improve the efficiency of the MLMC further, we exploit the hierarchy of meshes and use coarser approximations as starting values for the eigensolvers on finer ones. To improve the stability of the MLMC method for convection-dominated problems, we employ two additional strategies. First, we consider the streamline upwind Petrov–Galerkin formulation of the discrete eigenvalue problem, which allows us to start the MLMC method on coarser meshes than is possible with standard FEs. Second, we apply a homotopy method to add stability to the eigensolver for each sample. Finally, we present a multilevel quasi-Monte Carlo method that replaces Monte Carlo with a quasi-Monte Carlo (QMC) rule on each level. Due to the faster convergence of QMC, this improves the overall complexity. We provide detailed numerical results comparing our different strategies to demonstrate the practical feasibility of the MLMC method in different use cases. The results support our complexity analysis and further demonstrate the superiority over plain Monte Carlo in all cases.

Computing perfect stationary equilibria in stochastic games

The notion of stationary equilibrium is one of the most crucial solution concepts in stochastic games. However, a stochastic game can have multiple stationary equilibria, some of which may be unstable or counterintuitive. As a refinement of stationary equilibrium, we extend the concept of perfect equilibrium in strategic games to stochastic games and formulate the notion of perfect stationary equilibrium (PeSE). To further promote its applications, we develop a differentiable homotopy method to compute such an equilibrium. We incorporate vanishing logarithmic barrier terms into the payoff functions, thereby constituting a logarithmic-barrier stochastic game. As a result of this barrier game, we attain a continuously differentiable homotopy system. To reduce the number of variables in the homotopy system, we eliminate the Bellman equations through a replacement of variables and derive an equivalent system. We use the equivalent system to establish the existence of a smooth path, which starts from an arbitrary total mixed strategy profile and ends at a PeSE. Extensive numerical experiments, including relevant applications like dynamic oligopoly models and dynamic legislative voting, further affirm the effectiveness and efficiency of the method.

Double saddle‐point preconditioning for Krylov methods in the inexact sequential homotopy method

AbstractWe derive an extension of the sequential homotopy method that allows for the application of inexact solvers for the linear (double) saddle‐point systems arising in the local semismooth Newton method for the homotopy subproblems. For the class of problems that exhibit (after suitable partitioning of the variables) a zero in the off‐diagonal blocks of the Hessian of the Lagrangian, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. For discretized optimal control problems with PDE constraints, this structure is often present with the canonical partitioning of the variables in states and controls. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method allows for the parallel solution of large 3D problems.

Some Remarks on Smooth Mappings of Hilbert and Banach Spaces and Their Local Convexity Property

We analyze smooth nonlinear mappings for Hilbert and Banach spaces that carry small balls to convex sets, provided that the radii of the balls are small enough. We focus on the study of new and mildly sufficient conditions for the nonlinear mapping of Hilbert and Banach spaces to be locally convex, and address a suitably reformulated local convexity problem analyzed within the Leray–Schauder homotopy method approach for Hilbert spaces, and within the Lipschitz smoothness condition for both Hilbert and Banach spaces. Some of the results presented in this work prove to be interesting and novel, even for finite-dimensional problems. Open problems related to the local convexity property for nonlinear mappings of Banach spaces are also formulated.

Parallel finite layer method for land subsidence and its homotopy parameter inversion

Land subsidence is caused by the interaction between groundwater flow and soil deformation. At present, in the field of land subsidence calculation, there is an increasing tendency to use Biot's consolidation theory to precisely simulate the coupling processes of flow and deformation. However, there are some practical problems with large-scale numerical computation, such as large computational loads, low efficiency, and difficulty in obtaining parameters. The Finite Layer Method employs orthogonality of analytical functions to achieve decoupling among various series terms. And the method can be enhanced through parallel computing to optimize the efficiency of solving complex and extensive problems due to its decoupling property. Based on this, combined with the nonlinear homotopy method, a new approach for the reversal of land subsidence is proposed. The validity of the parallel program is confirmed through comparison with established solutions, and the impact of various factors on the efficiency of parallel computing is analyzed. The results indicate that the parallel approach enhances computational efficiency, while the homotopy inversion technique exhibits broad convergence characteristics.

Double diffusive on powell eyring fluid flow by mixed convection from an exponential stretching surface with variable viscosity/thermal conductivity

This study delves into heat and mass transference in fluids with variable thermo-physical features, focusing on the Powell-Eyring fluid, a non-Newtonian substance with unique characteristics. Our aim is to understand the behaviour of this shear-thinning fluid when interacting with an exponentially stretchable surface, considering factors like mixed convection and thermal radiation. Our research endeavours to uncover novel findings in this intricate domain, with the primary objective of comprehending how this shear-thinning fluid behaves when it encounters an exponentially stretchable surface. To address the complexities, we employ the BVP4C technique in MATLAB, transforming governing equations into nonlinear partial differential equations (PDEs) based on mass, linear momentum, and energy conservation principles. The Homotopy method aids in navigating these equations, and numerical solutions are calculated using the powerful BVP4C technique for ordinary differential equations (odes). The research stands out for its comprehensive exploration of the interplay among diverse factors, including radiation, variable viscosity, mixed convection, and activation energy constraints. Findings are presented through tables and graphs, offering valuable insights into the intricate physical phenomena within this multifaceted field.