Homotopy Continuation Algorithm Research Articles

A Polyhedral Homotopy Algorithm for Real Zeros

We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients satisfying certain concavity conditions. It operates entirely over the real numbers and tracks the optimal number of solution paths. In more technical terms; we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba.

Coupling Matrix Extraction by Numerical Solving of Polynomial Systems by Homotopy Continuation

In this letter, the homotopy continuation (HC) approach for a problem of coupling matrix (CM) extraction is presented. HC method is used to solve the system of polynomial equations that are formulated by comparing polynomial coefficients of given filter scattering parameters S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">11</sub> and S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">21</sub> and those being described based on well-known relations between scattering parameters and CM. HC requires an initial ( t=0) system of equations with a known solution. During the HC algorithm run, the system and solution are transformed to the final ( t=1) system and solution that is our goal. In our approach, the initial system ( t=0) and its solution are formulated by comparing polynomial coefficients of S-parameters obtained for CM including entries of any, even random, values. To validate the method a numerical experiment has been presented, i.e., CM extraction for measured sixth-order (two regular and two parasitic cross-couplings) filter. The experiment demonstrated excellent results obtained with the proposed method.

A TFC-based homotopy continuation algorithm with application to dynamics and control problems

A method for solving zero-finding problems is developed by tracking homotopy paths, which define connecting channels between an auxiliary problem and the objective problem. Current algorithms’ success highly relies on empirical knowledge, due to manually, inherently selected homotopy paths. This work introduces a homotopy method based on the Theory of Functional Connections (TFC). The TFC-based method implicitly defines infinite homotopy paths, from which the most promising ones are selected. A two-layer continuation algorithm is devised, where the first layer tracks the homotopy path by monotonously varying the continuation parameter, while the second layer recovers possible failures resorting to a TFC representation of the homotopy function. Compared to pseudo-arclength methods, the proposed TFC-based method retains the simplicity of direct continuation while allowing a flexible path switching. Numerical simulations illustrate the effectiveness of the presented method.

A homotopy continuation inversion of geoelectrical sounding data

In nonlinear inversion of geophysical data, improper initial approximation of the model parameters usually leads to local convergence of the normal Newton iteration methods, despite enforcing constraints on the physical properties. To mitigate this problem, we present a globally convergent Homotopy continuation algorithm to solve the nonlinear least squares problem through a path-tracking strategy in model space. The proposed scheme is based on introducing a new functional to replace the quadratic Tikhonov-Phillips functional. The algorithm implementation includes a sequence of predictor-corrector steps to find the best direction of the solution. The predictor calculates an approximate solution of the corresponding new function in the Homotopy in consequence of using a new value of the continuation parameter at each step of the algorithm. The predicted approximate solution is then corrected by applying the corrector step (e.g., Gauss-Newton method). The global convergence of the Homotopy algorithm is compared with a conventional iterative method through the synthetic and real 1-D resistivity data sets. Furthermore, a bootstrap-based uncertainty analysis is provided to quantify the error in the inverted models derived from the case study. The results of blocky and smooth inversion demonstrate that the presented optimization method outperforms the standard algorithm in the sense of stability, rate of convergence, and the recovered models.

Unified Selective Harmonic Elimination for Cascaded H-Bridge Asymmetric Multilevel Inverter

The unequal dc-link voltages in the cascaded H-bridge (CHB) asymmetric multilevel inverters (AMLIs) lead to asymmetry of the equations for selective harmonic elimination (SHE), which increases the difficulty in solving the switching angles. On the other hand, the unequal dc-link voltages provide a large amount of synthesized waveforms which can be used to optimize the performance of inverters. However, as most of the traditional SHE approaches are derived according to a given output waveform, it cannot handle multiple synthesized waveforms. In this paper, the unified SHE equations are derived for all the possible synthesized waveforms under a specific switching angles distribution among the cells. By using the polynomial homotopy continuation (PHC) algorithm, all the possible solutions of the unified SHE equations can be found without the selection of initial values. As the waveforms generated by H-bridge are limited to three levels, some of the solutions are not physically realizable and should be discarded, hence, the criterion is proposed to distinguish these specious solutions. The case of a CHB converter with two cells and six switching angles is studied; in total, there exist 86 groups of candidate solutions and 14 of them are physically realizable. Experiments are also shown to validate the correctness of this unified SHE approach for CHB AMLIs.

Monolithic homotopy continuation with predictor based on higher derivatives

The predictor component of a monolithic homotopy continuation algorithm is augmented with higher derivative information for use as an efficient, robust, and scalable continuation algorithm suitable for application to large sparse systems of nonlinear algebraic equations. Convergence of the algorithm is established analytically, and efficiency studies are performed by applying the method to a practical computational aerodynamics problem.

Computing the Feasible Spaces of Optimal Power Flow Problems

The solution to an optimal power flow (OPF) problem provides a minimum cost operating point for an electric power system. The performance of OPF solution techniques strongly depends on the problem's feasible space. This paper presents an algorithm that is guaranteed to compute the entire feasible spaces of small OPF problems to within a specified discretization tolerance. Specifically, the feasible space is computed by discretizing certain of the OPF problem's inequality constraints to obtain a set of power flow equations. All solutions to the power flow equations at each discretization point are obtained using the Numerical Polynomial Homotopy Continuation algorithm. To improve computational tractability, “bound tightening” and “grid pruning” algorithms use convex relaxations to preclude consideration of many discretization points that are infeasible for the OPF problem. The proposed algorithm is used to generate the feasible spaces of two small test cases.

Matrix-free monolithic homotopy continuation with application to computational aerodynamics

A matrix-free monolithic homotopy continuation algorithm is developed which allows for approximate numerical solutions to nonlinear systems of equations without the need to solve a linear system, thereby avoiding the formation of any Jacobian or preconditioner matrices. The algorithm can converge from an arbitrary starting guess, under suitable conditions, and can give a sufficiently accurate approximation to the converged solution such that a rapid locally convergent method such as Newton’s method will converge successfully. Several forms of the algorithm are presented, as are augmentations to the algorithms which can lead to improved efficiency or stability. The method is validated and the stability and efficiency are investigated numerically based on a computational aerodynamics flow solver.

Analysis of Submarines Motion Stability based on Homotopy Continuation Theory

In this paper the motion stability of submarine was analyzed by using the homotopy continuation algorithm tool MATCONT, the Lyapunov motion stability theory and the bifurcation theory. Then the numerical simulation was used to verify the analysis results and method. The paper proposed an easy and effective method for analysis of submarines motion stability.

A monolithic homotopy continuation algorithm with application to computational fluid dynamics

A new class of homotopy continuation methods is developed suitable for globalizing quasi-Newton methods for large sparse nonlinear systems of equations. The new continuation methods, described as monolithic homotopy continuation, differ from the classical predictor–corrector algorithm in that the predictor and corrector phases are replaced with a single phase which includes both a predictor and corrector component. Conditional convergence and stability are proved analytically. Using a Laplacian-like operator to construct the homotopy, the new algorithm is shown to be more efficient than the predictor–corrector homotopy continuation algorithm as well as an implementation of the widely-used pseudo-transient continuation algorithm for some inviscid and turbulent, subsonic and transonic external aerodynamic flows over the ONERA M6 wing and the NACA 0012 airfoil using a parallel implicit Newton–Krylov finite-difference flow solver.

Crack identification of beam structures using homotopy continuation algorithm

An approach based on homotopy continuation algorithm is presented to identify the parameters of a cracked beam. Euler–Bernoulli finite beam element with a fully opened crack model is adopted to establish the dynamic equation of the structural system. In the inverse problem, the homotopy equation is derived from minimizing the error between the calculated and the simulated measured acceleration responses. The range of homotopy parameter is divided into a number of divisions. Newton iterative method is employed to estimate the solution at each of these division points. The solution at the last division point corresponds to the homotopy equation matching the objective function. Numerical simulations with a simply supported beam and two-span beam show that the proposed method is very accurate compared to an existing method for both single and multiple cracks identification. The effects of type of excitation, division of the homotopy parameter and measurement noise on the identified results are discussed. It is noted that there is no need for an accurate set of initial values with the proposed approach.

Parallelization of Eigenvalue-Based Dimensional Reductions via Homotopy Continuation

This paper investigates a homotopy-based method for embedding with hundreds of thousands of data items that yields a parallel algorithm suitable for running on a distributed system. Current eigenvalue-based embedding algorithms attempt to use a sparsification of the distance matrix to approximate a low-dimensional representation when handling large-scale data sets. The main reason of taking approximation is that it is still hindered by the eigendecomposition bottleneck for high-dimensional matrices in the embedding process. In this study, a homotopy continuation algorithm is applied for improving this embedding model by parallelizing the corresponding eigendecomposition. The eigenvalue solution is converted to the operation of ordinary differential equations with initialized values, and all isolated positive eigenvalues and corresponding eigenvectors can be obtained in parallel according to predicting eigenpaths. Experiments on the real data sets show that the homotopy-based approach is potential to be implemented for millions of data sets.

On Homotopy Continuation for Speech Restoration.

In this paper, a homotopy-based method is employed for the recovery of speech recordings from missing or corrupted samples taken in a noisy environment. The model for the acquisition device is a compressed sensing scenario using Gabor frames. To recover an approximation of the speech file, we used the basis pursuit denoising method with the homotopy continuation algorithm. We tested the proposed method with various speech recordings.

Forward kinematics modeling of spatial parallel linkage mechanisms based on constraint equations and the numerical solving method

SUMMARYIn order to solve general kinematics modeling problems and numerical stability problems of numerical methods for spatial parallel linkage mechanisms, a general modeling method and its numerical solving algorithm is proposed. According to the need for avoiding direct singular configurations, valid joint variable space and valid forward kinematics solutions (VKSs) are defined. Taking numerical convergence near singular points into account, the pseudo-arc length homotopy continuation algorithm is given to solve the kinematics model. Finally as an example, the joint variable space of the general Stewart platform mechanism is analyzed, which is proved to be divided into subspaces by direct singular surfaces. And then, forward kinematics solutions of 200 testing points are solved separately using the pseudo-arc length homotopy continuation algorithm, the Newton homotopy continuation algorithm and the Newton–Raphson algorithm (NRA). Comparison of the results shows that the proposed method is convergent to the same solution branch with the initial configuration on all the testing points, while the other two algorithms skip to other solution branches on some near singular testing points.

Efficient optimization-based design for the separation of heterogeneous azeotropic mixtures

Model-based design of separation processes for heterogeneous azeotropic mixtures is a challenging task. The multiplicity of the solutions of equilibrium calculations and the discontinuity due to the potential switching between homogeneous and heterogeneous mixtures on a tray severely complicate the application of deterministic optimization. This paper presents a novel computational approach to reliably determine phase stability and the correct equilibrium solutions in every iteration of the optimization. The approach builds on the decomposition of the optimization problem into a generic superstructure model and an implicit model for equilibrium and enthalpy calculations, which is integrated into the optimization problem by means of an external function. The phase states and equilibrium solutions are determined by means of a reliable homotopy continuation algorithm. An additional reformulation of equilibrium solutions and enthalpy calculations allows overcoming the discontinuity problems. Different case studies illustrate the applicability and show the potential of the proposed method.

Computing All Nash Equilibria of Multiplayer Games in Electricity Markets by Solving Polynomial Equations

The analysis of the Nash equilibrium (NE) in electricity markets with imperfect competition is subject to two major challenges: the treatment of multiplayer games and the determination of the existence of multiple market equilibria. To resolve these obstacles, a solution method, based on the payoff matrix approach and polynomial equations, is proposed in this paper to calculate all the Nash equilibria of multiplayer games in the electricity markets. First, the proposed method decomposes the game by means of a set of all pure strategies assigned with positive probabilities (support). For each possible support, the NE condition can be characterized by a set of polynomial equations with inequality constraints. Next, the homotopy continuation algorithm is employed to obtain all solutions of the polynomial system. Finally, all Nash equilibria can be found by verifying each solution of all polynomial systems. Two example systems, one involving the Cournot model and the other based on the supply function equilibrium (SFE) model, are applied to test the proposed method, respectively. The results show that the proposed method has the ability to identify all NE under certain conditions, indicating that the proposed algorithm is useful for providing insights into the theoretical analysis of electricity markets.

Plotting Robust Root Locus For Polynomial Families Of Multilinear Parameter Dependence Based On Zero Inclusion/Exclusion Tests

ABSTRACTThe Mapping Theorem by Zadeh and Desoer [17] is a sufficient condition for the zero exclusion of the image or value set of an m‐dimensional box B under a multilinear mapping f: Rm → C, where R and C denote the real line and the complex plane, respectively. In this paper, we present a sufficient condition for the zero inclusion of the value set f(B). On the basis of these two conditions and the iterative subdivision of the box B, we propose a numerical procedure for testing whether or not the value set f(B) includes the origin. The procedure is easy to implement and is more efficient than that based on constructing the value set f(B) explicitly. As an application, the proposed zero inclusion test procedure is used along with a homotopy continuation algorithm to trace out the boundary curves of the robust root loci of polynomial families with multilinear parametric uncertainties.

Quick and reliable phase stability test in VLLE flash calculations by homotopy continuation

Calculation of phase equilibria is a crucial part in modeling and simulation of many chemical engineering processes. This is in particular true for systems that exhibit two liquid phases, since methods based on local solvers (e.g. Newton type) may converge to trivial solutions. In this contribution, a new method to test for phase stability and to calculate the globally stable phases for systems that exhibit two liquid phases is presented. The method is based on a preprocessing step, in which all heterogeneous regions are determined. The actual phase stability test and flash calculation is performed using a homotopy continuation algorithm. The method is applied to the LLE flash, the bubble and dew point problem, and to a general (VLLE) p, T-flash. The algorithm is shown to be very reliable while its computational demand is small compared with other methods. Hence, the method seems to qualify for applications in simulation and optimization where flash calculations are performed many times.

Some real and unreal enumerative geometry for flag manifolds.

We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that two new classes of enumerative geometric problems on flag manifolds may have all their solutions be real and modify this method to show that another class may have no real solutions, which is a new phenomenon. This method originated in a numerical homotopy continuation algorithm adapted to the special Schubert calculus on Grassmannians and in principle gives optimal numerical homotopy algorithms for finding explicit solutions to these other enumerative problems.

The Homotopy Continuation Algorithm for the Real Nonsymmetric Eigenproblem: Further Development and Implementation

New development and implementation of a homotopy continuation algorithm for the nonsymmetric eigenvalue problem are presented, along with thorough investigation on the advantages and limitations of our code. Numerical results on substantial varieties of matrices arising in applications indicate excellent efficiency and stability of our algorithm and a drastic improvement over previous implementations.