Local saddles of relaxed averaged alternating reflections algorithms on phase retrieval

Phase retrieval can be expressed as a non-convex constrained optimization problem to identify one phase minimizer one a torus. Many iterative transform techniques have been proposed to identify the minimizer, e.g., relaxed averaged alternating reflections(RAAR) algorithms. In this paper, we present one optimization viewpoint on the RAAR algorithm. RAAR algorithm is one alternating direction method of multipliers(ADMM) with one penalty parameter. Pairing with multipliers (dual vectors), phase vectors on the primal space are lifted to higher-dimensional vectors, the RAAR algorithm is one continuation algorithm, which searches for local saddles in the primal-dual space. The dual iteration approximates one gradient ascent flow, which drives the corresponding local minimizers in a positive-definite Hessian region. Altering penalty parameters, the RAAR eliminates the stagnation of these corresponding local minimizers in the primal space and thus screens out many stationary points corresponding to non-local minimizers.

Robustness of nonlinear parameter identification in the presence of process noise using control-based continuation

In this study, we consider the experimentally obtained, periodically forced response of a nonlinear structure in the presence of process noise. Control-based continuation is used to measure both the stable and unstable periodic solutions, while different levels of noise are injected into the system. Using these data, the robustness of the control-based continuation algorithm and its ability to capture the noise-free system response are assessed by identifying the parameters of an associated Duffing-like model. We demonstrate that control-based continuation extracts system information more robustly, in the presence of a high level of noise, than open-loop parameter sweeps and so is a valuable tool for investigating nonlinear structures.

The Calculation Process of the Limit Cycle for Lorenz System

This work focuses on the bifurcation behavior before chaos phenomenon happens. Traditional numerical method is unable to solve the unstable limit cycle of nonlinear system. One algorithm is introduced to solve the unstable one, which is based one of the continuation method is called DEPAR approach. Combined with analytic and numerical method, the two stable and symmetrical equilibrium solutions exist through Fork bifurcation and the unstable and symmetrical limit cycles exist through Hopf bifurcation of Lorenz system. With the continuation algorithm, the bifurcation behavior and its phase diagram is solved. The results demonstrate the unstable periodical solution is around the equilibrium solution, besides the trajectory into the unstable area cannot escape but only converge to the equilibrium solution.

Numerical Continuation on a Graphical Processing Unit for Kinematic Synthesis

This paper presents an implementation of a homotopy path tracking algorithm for polynomial numerical continuation on a graphical processing unit (GPU). The goal of this algorithm is to track homotopy curves from known roots to the unknown roots of a target polynomial system. The path tracker solves a set of ordinary differential equations to predict the next step and uses a Newton root finder to correct the prediction so the path stays on the homotopy solution curves. In order to benefit from the computational performance of a GPU, we organize the procedure so it is executed as a single instruction set, which means the path tracker has a fixed step size and the corrector has a fixed number iterations. This trade-off between accuracy and GPU computation speed is useful in numerical kinematic synthesis where a large number of solutions must be generated to find a few effective designs. In this paper, we show that our implementation of GPU-based numerical continuation yields 85 effective designs in 63 s, while an existing numerical continuation algorithm yields 455 effective designs in 2 h running on eight threads of a workstation.

Resonance Responses and Bifurcations of Circular Mesh Antenna in Thermal Environment

In this paper, resonance responses and bifurcations of the circular mesh antenna with 1:3 internal resonance in thermal environment are studied. Considering this internal resonance, based on the equivalent circular mesh antenna model, which is a composite laminated circular cylindrical shell clamped along a generatrix and with the radial pre-stretched membranes at both ends in thermal environment, we obtain the amplitude-frequency response curves of the system in the state-parameter space by using the prediction-correction continuation algorithm in order to study the bifurcation characteristics of the equivalent circular mesh antenna model. Meanwhile, the fold bifurcation points and Antronov-Hopf bifurcation points are detected and located on these resonance response curves. We find that the nonlinear vibrations of the equivalent circular mesh antenna model have the hardening characteristics, and the temperature excitation has noticeable effects on the stabilities and bifurcations of this system.

A New Inexact Non-Interior Continuation Algorithm for Second-Order Cone Programming

Second-order cone programming has received considerable attention in the past decades because of its wide range of applications. Non-interior continuation method is one of the most popular and efficient methods for solving second-order cone programming partially due to its superior numerical performances. In this paper, a new smoothing form of the well-known Fischer-Burmeister function is given. Based on the new smoothing function, an inexact non-interior continuation algorithm is proposed. Attractively, the new algorithm can start from an arbitrary point, and it solves only one system of linear equations inexactly and performs only one line search at each iteration. Moreover, under a mild assumption, the new algorithm has a globally linear and locally Q-quadratical convergence. Finally, some preliminary numerical results are reported which show the effectiveness of the presented algorithm.