Binary optimal control by trust-region steepest descent

We present a trust-region steepest descent method for dynamic optimal control problems with binary-valued integrable control functions. Our method interprets the control function as an indicator function of a measurable set and makes set-valued adjustments derived from the sublevel sets of a topological gradient function. By combining this type of update with a trust-region framework, we are able to show by theoretical argument that our method achieves asymptotic stationarity despite possible discretization errors and truncation errors during step determination. To demonstrate the practical applicability of our method, we solve two optimal control problems constrained by ordinary and partial differential equations, respectively, and one topological optimization problem.

Blade shape optimization and internal-flow characteristics of the backward non-volute centrifugal fan

The work proposed the double parameter optimization method of the non-volute centrifugal fan’s blade profile based on the steepest descent method. Total-pressure efficiency improvement at the high-flow area was taken as an optimization objective. A method of applying the steepest descent method to modify the blade profile of backward centrifugal fan is proposed in this paper. The gradient descent direction was analyzed to design the blade profile and obtain the optimal blade profile at a high-flow rate. Besides, numerical simulations were carried out to analyze the aerodynamic performance and the internal flow characteristics of the centrifugal fan by the computational fluid dynamics method. Numerical results showed that the blade profile along the gradient descent was optimized to effectively increase the total pressure and the total pressure efficiency of the original model at the high-flow rate. The steepest descent method for local optimization could improve the fan blade design.

The exact worst-case convergence rate of the gradient method with fixed step lengths for L-smooth functions

In this paper, we study the convergence rate of the gradient (or steepest descent) method with fixed step lengths for finding a stationary point of an L-smooth function. We establish a new convergence rate, and show that the bound may be exact in some cases, in particular when all step lengths lie in the interval (0, 1/L]. In addition, we derive an optimal step length with respect to the new bound.

Comparison Between Steepest Descent Method and Conjugate Gradient Method by Using Matlab

The Steepest descent method and the Conjugate gradient method to minimize nonlinear functions have been studied in this work. Algorithms are presented and implemented in Matlab software for both methods. However, a comparison has been made between the Steepest descent method and the Conjugate gradient method. The obtained results in Matlab software has time and efficiency aspects. It is shown that the Conjugate gradient method needs fewer iterations and has more efficiency than the Steepest descent method. On the other hand, the Steepest descent method converges a function in less time than the Conjugate gradient method.

Rupture Kinematics and Coseismic Slip Model of the 2021 Mw 7.3 Maduo (China) Earthquake: Implications for the Seismic Hazard of the Kunlun Fault

The 21 May 2021 Maduo earthquake was the largest event to occur on a secondary fault in the interior of the active Bayanhar block on the north-central Tibetan plateau in the last twenty years. A detailed kinematic study of the Maduo earthquake helps us to better understand the seismogenic environments of the secondary faults within the block, and its relationship with the block-bounding faults. In this study, firstly, SAR images are used to obtain the coseismic deformation fields. Secondly, we use a strain model-based method and steepest descent method (SDM) to resolve the three-dimensional displacement components and to invert the coseismic slip distribution constrained by coseismic displacement fields, respectively. The three-dimensional displacement fields reveal a dominant left-lateral strike-slip motion, local horizontal displacement variations and widely distributed near-fault subsidence/uplift deformation. We prefer a five-segment fault slip model, with well constrained fault geometry featuring different dip angles and striking, constrained by InSAR observations. The peak coseismic slip is estimated to be ~5 m near longitude 98.9°E at a depth of ~4–7 km. Overall, the distribution of the coseismic slip on the fault is highly correlated to the measured surface displacement offsets along the entire rupture. We observe the moderate shallow slip deficit and limited afterslip deformation following the Maduo earthquake, it may indicate the effects of off-fault deformation during the earthquake and stable interseismic creep on the fault. The occurrence of the Maduo earthquake on a subsidiary fault updates the importance and the traditional estimate of the seismic hazards for the Kunlun fault.

Mechanical Compound Fault Analysis Method Based on Shift Invariant Dictionary Learning and Improved FastICA Algorithm

For mechanical compound fault, it is of great significance to employ the vibration signal of a single-channel compound fault to analyze and realize the separation of multiple fault sources, which is essentially the problem of single-channel blind source separation. Shift invariant K-means singular value decomposition (shift invariant K-SVD) dictionary learning is suitable to extract the periodic and repeated fault features of a rotating machinery fault, hence in this article a single-channel compound fault analysis method is put forward which combines shift invariant K-SVD with improved fast independent component analysis (improved FastICA) algorithm. Firstly, based on single-channel compound fault signal, the shift invariant K-SVD algorithm can be used for learning multiple latent components that can be constructed as a virtual multi-channel signal. Then the improved FastICA algorithm is utilized to realize the separation of multiple fault source signals. With regard to the FastICA algorithm, the third-order convergence Newton iteration method is adopted to improve convergence speed. Moreover, in order to address the problem that FastICA is very sensitive to initialization, a steepest descent method can be applied. The experimental analysis of the compound fault of rolling bearing verifies that the presented method is effective to separate multiple fault source signals and the improved FastICA algorithm can increase convergence rate and overcome the problem of sensitivity to initialization.