Variants of the Uzawa method for three-order block saddle point problem

In this work, we consider numerical methods for solving a class of block three-by-three saddle point problems, which arise from finite element methods for solving time-dependent Maxwell equations and a class of quadratic programs. We present a variant of Uzawa method with two variable parameters for the saddle point problems. These two parameters can be updated easily in each iteration, similar to the evaluation of the two iteration parameters in the conjugate gradient method. We show that the new iterative method converges to the unique solution of the saddle point problems under a reasonable condition. Numerical experiments highlighting the performance of the proposed method for problems are presented.

An adaptive algorithm based on wavelets is proposed for the fast numerical solution of control problems governed by elliptic boundary value problems with Dirichlet boundary control. A quadratic cost functional representing Sobolev norms of the state and a regularization in terms of the control is to be minimized subject to linear constraints in weak form. In particular, the constraints are formulated as a saddle point problem that allows to handle the varying boundary conditions explicitly. In the framework of (biorthogonal) wavelets, a representer for the functional is derived in terms of l2-norms of wavelet expansion coefficients and the constraints are written in form of an l2 automorphism. Standard techniques from optimization are then used to deduce the resulting first order necessary conditions as a (still infinite) system in l2. Applying the machinery developed in [8,9] which has been extended to control problems in [14], an adaptive method is proposed which can be interpreted as an inexact gradient method for the control. In each iteration step, in turn the primal and the adjoint saddle point system are solved up to a prescribed accuracy by an adaptive iterative Uzawa algorithm for saddle point problems which has been proposed in [10]. Under these premises, it can be shown that the adaptive algorithm containing now three layers of iterations is asymptotically optimal. This means that the convergence rate achieved for computing the solution up to a desired target tolerance is asymptotically the same as the wavelet-best N-term approximation of the solution, and the total computational work is proportional to the number of computational unknowns.

This paper proposes a new adaptation algorithm named normalized correlation-Newton (NC-Newton) algorithm and a novel variable q-norm control method (NC-Newton-Varq-norm) for complex-domain adaptive filters. First, stochastic models are presented for two types of impulse noise intruding adaptive filters: one is present in observation noise and another at filter input. After reviewing q-norm and NC-Newton algorithm, we propose a variable q-norm control method. Analysis of the NC-Newton-Varq-norm algorithm is developed for theoretically calculating transient and steady-state convergence behavior. Through experiment with some examples, we demonstrate effectiveness of the proposed variable q-norm control method in improving filter convergence speed while preserving robustness of the NC-Newton algorithm in impulsive noise environments. Good agreement between simulated and theoretical convergence behavior validates the analysis.

With the access of large-scale new energy sources, fast frequency regulation resources of the power system are scarce. A variable parameter control method for charging pile load aggregator to participate in the fast frequency market is proposed in this paper. It is that charging pile load aggregator actively adjusts the active power according to the frequency of the local power system, and the response speed is within 0-2 seconds. In addition, the variable parameter control method proposed in this paper specifically controls the power of charging pile according to the speed and amplitude of frequency drop, which has strong pertinence and simple control scheme, and further improves the frequency stability of the power grid.

A new SOR-Like method for the saddle point problems

In this paper, the nonlinear inexact Uzawa (NIU) methods for saddle point problems are studied. A further improved convergence result of the NIU algorithm for stabilized saddle point problems is presented, which results in much smaller convergence factor. This result can also be viewed as a generalization of some previous work for classical saddle point problems. Based on this result, we generalize some adaptive versions of the NIU algorithm with variable iteration parameters for classical saddle point problems to solve stabilized saddle point problems. Convergence analyses of these algorithms are presented. It shows that they converge under similar conditions as those in classical cases, which are more practical and without estimates on the extreme eigenvalues of the involved preconditioned systems. Numerical experiments are given to demonstrate the effectiveness of these algorithms over some original NIU algorithms.

In this paper, wavelet techniques are employed for the fast numerical solution of a control problem governed by an elliptic boundary value problem with boundary control. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint, the elliptic boundary value problem, is formulated in an appropriate weak form that allows to handle varying boundary conditions explicitly: the boundary conditions are treated by Lagrange multipliers, leading to a saddle point problem. This is combined with a fictitious domain approach in order to cover also more complicated boundaries. Deviating from standard approaches, we then use (biorthogonal) wavelets to derive an equivalent infinite discretized control problem which involves only e2-norms and -operators. Classical methods from optimization yield the corresponding optimality conditions in terms of two weakly coupled (still infinite) saddle point problems for which a unique solution exists. For deriving finite-dimensional systems which are uniformly invertible, stability of the discretizations has to be ensured. This together with the e2-setting circumvents the problem of preconditioning: all operators have uniformly bounded condition numbers independent of the discretization. In order to numerically solve the resulting (finite-dimensional) linear system of the weakly coupled saddle point problems, a fully iterative method is proposed which can be viewed as an inexact gradient scheme. It consists of a gradient algorithm as an outer iteration which alternatingly picks the two saddle point problems, and an inner iteration to solve each of the saddle point problems, exemplified in terms of the Uzawa algorithm. It is proved here that this strategy converges, provided that the inner systems are solved sufficiently well. Moreover, since the system matrix is well-conditioned, it is shown that in combination with a nested iteration strategy this iteration is asymptotically optimal in the sense that it provides the solution on discretization level J with an overall amount of arithmetic operations that is proportional to the number of unknows NJ on that level. Finally, numerical results are provided.

In this paper, we further investigate the local Hermitian and skew-Hermitian splitting (LHSS) iteration method and the modified LHSS (MLHSS) iteration method for solving generalized nonsymmetric saddle point problems with nonzero (2,2) blocks. When A is non-symmetric positive definite, the convergence conditions are obtained, which generalize some results of Jiang and Cao [M.-Q. Jiang and Y. Cao, On local Hermitian and Skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 2009(231): 973-982] for the generalized saddle point problems to generalized nonsymmetric saddle point problems with nonzero (2,2) blocks. Numerical experiments show the effectiveness of the iterative methods.

A variable pressure differential fuzzy control method is proposed based on the online identification method for key parameters and the fuzzy subset inference fuzzy control method of the chilled water system network model. Firstly, a phase plane fuzzy identification method is proposed for the most unfavorable thermal loop. The study focuses on analyzing the trend of room temperature deviation and deviation change in different quadrants in the phase plane. Furthermore, we establish a chilled water pipe network model that recalculates flow variation in both the main pipe and each branch pipe section to eliminate the most unfavorable thermal loop. Finally, the test platform for the fan coil variable flow air conditioning water system was designed and constructed to meet the requirements of energy-saving regulation. Additionally, the network monitoring system for the test platform was completed. The calibration and debugging results demonstrate that the monitoring error is within ±5.0%, ensuring precise control of room temperature at the end of the branch within ±0.5 °C. Results demonstrate that our novel method exhibits superior stability in room temperature control compared to traditional linear variable pressure differential set point controls while achieving energy saving ranging from 4.7% to 6.5%.

In this paper we consider additive Schwarz-type iteration methods for saddle point problems as smoothers in a multigrid method. Each iteration step of the additive Schwarz method requires the solutions of several small local saddle point problems. This method can be viewed as an additive version of a (multiplicative) Vanka-type iteration, well-known as a smoother for multigrid methods in computational fluid dynamics. It is shown that, under suitable conditions, the iteration can be interpreted as a symmetric inexact Uzawa method. In the case of symmetric saddle point problems the smoothing property, an important part in a multigrid convergence proof, is analyzed for symmetric inexact Uzawa methods including the special case of the additive Schwarz-type iterations. As an example the theory is applied to the Crouzeix-Raviart mixed finite element for the Stokes equations and some numerical experiments are presented.

Saddle point problems appear in many scientific and engineering applications. Hence, it is important to solve them fast with high accuracy. We have proposed the method for saddle point problems using block structure in order to solve them fast. In our proposed method, first, a linear system with multiple right-hand sides is solved instead of solving the saddle point problems directly. After that, a small linear system with a dense matrix is solved. In our previous work, it has been observed that our method is faster than the conventional approach, but the accuracy of the obtained approximate solutions is worse than the conventional one. In this paper, we propose to improve the accuracy of the approximate solutions of the saddle point problems by improving the accuracy of the solution of the small linear system using the mixed precision iterative refinement technique. Numerical experiments illustrate that the proposed approach improves the accuracy of the approximate solutions to the same extent as the conventional approach.

On-orbit assembly has become a crucial aspect of space operations, where the manipulator frequently and directly interacts with objects in a complex assembly process. The traditional manipulator control has limitations in adapting to diverse assembly tasks and is vulnerable to vibration, leading to assembly failure. To address this issue, we propose a human-like variable admittance control method based on the variable damping characteristics of the human arm. By collecting the velocity and contact force of human arm operations in assembly, we analyze the damping change of human arm and establish the active compliance model based on S-type damping variation rule in assembly. Furthermore, 3 passive contact models are proposed between the end of the human arm and the environment: one-sided bevel contact, both sides bevel contact, and pin-hole contact. On the basis of these active and passive models, a typical space assembly task for a robot is designed, and a human-like variable admittance controller is established and simulated. Finally, we build a ground verification platform and complete different assembly tasks, thereby successfully verifying the safety, robustness, and adaptability of the human-like variable admittance control method.

Orifice erosion effect and variable gain control method for hydraulic servo spool valve

This paper presents a variable admittance control method to achieve intuitive human–robot interactions that consider human intentions. Human intention is classified into two categories—direct and indirect. With respect to direct intention, the concept of standard force is introduced to adjust the interacting force. The proposed variable admittance control method improves intuitiveness when velocity is used as an estimate of direct intention. In the estimation of indirect intention, a force guidance method is suggested to make a robot follow and guide a human. The proposed control methodology is adapted to a six-DOF manipulator based on a one-dimensional analysis. The experiments are conducted with a manipulator (Universal Robots, UR10) and a force/torque sensor (Robotus, RFT60-HA) to evaluate the performance. The experiments validate that variable admittance control enhances the execution time, accuracy, and comfort of the operator.

In this paper, we study a saddle point problem and propose an inexact double primal-dual algorithm for solving the saddle point problem. The sequence generated by the proposed algorithm is proved to globally converge to the saddle point of the saddle point problem in the ergodic sense, and the convergence rate of the algorithm is also obtained. For the model of total variation image processing, the performance of the algorithm is demonstrated. Simulation results show that our method is more efficient in image denoising compared with existing methods.