Convergence analysis of the Neumann–Neumann waveform relaxation method for time-fractional RC circuits

The classical waveform relaxation (WR) methods rely on decoupling the large-scale ODEs system into small-scale subsystems and then solving these subsystems in a Jacobi or Gauss–Seidel pattern. However, in general it is hard to find a clever partition and for strongly coupled systems the classical WR methods usually converge slowly and non-uniformly. On the contrary, the WR methods of longitudinal type, such as the Robin-WR method and the Neumann–Neumann waveform relaxation (NN-WR) method, possess the advantages of simple partitioning procedure and uniform convergence rate. The Robin-WR method has been extensively studied in the past few years, while the NN-WR method is just proposed very recently and does not get much attention. It was shown in our previous work that the NN-WR method converges much faster than the Robin-WR method, provided the involved parameter, namely β, is chosen properly. In this paper, we perform a convergence analysis of the NN-WR method for time-fractional RC circuits, with special attention to the optimization of the parameter β. For time-fractional PDEs, this work corresponds to the study of the NN-WR method at the semi-discrete level. We present a detailed numerical test of this method, with respect to convergence rate, CPU time and asymptotic dependence on the problem/discretization parameters, in the case of two- and multi-subcircuits.     

DerivFit:A Program for Rate Equation Parameter Fitting Using Derivatives

A C program for fitting parameters in enzymatic rate equations is presented. TheDerivFitprogram employs the reaction scheme in the form of ordinary differential equations (ODEs). The kinetic parameters are fitted to the experimental data by minimizing the sum of squared deviations of experimental points from theoretically predicted progress curves. In the minimization process we use the Gradient, Newton, and Marquardt algorithms. The gradients are calculated explicitly by solving a set of additional ODEs that are automatically attached by the program, taking advantage of a general formulation of the basic ODEs that determine the reaction's time course. The program is applied to simple enzymatic systems including slow tight-binding inhibition.     

Computing Transitional Cycles for a Deterministic Time-to-Build Growth Model

Capital investment requires a gestation lag for being productive. This time-to-build feature can lead to an autonomous system of mixed-type functional differential equations (FDEs), causing aggregate fluctuations for a deterministic economy. We present a continuation method to solve a system of mixed FDEs by solving a sequence of boundary value problems for systems of ordinary differential equations (ODEs). Unlike other iteration techniques, this method can avoid the need to predetermine both forward-looking and backward-looking difference terms. The strategy is to form a homotopy that can deform a simpler ODE system into a target FDE system, while treating the deformation process as a sequence of ODEs. We can thereby compute oscillatory cycles for a time-to-build economy while it is in transition to the steady state.     

Rational general solutions of planar rational systems of autonomous ODEs

In this paper, we provide an algorithm to compute explicit rational solutions of a rational system of autonomous ordinary differential equations (ODEs) from its rational invariant algebraic curves. The method is based on the proper rational parametrization of these curves and the fact that by linear reparametrizations, we can find the rational solutions of the given system of ODEs. Moreover, if the system has a rational first integral, we can decide whether it has a rational general solution and compute it in the affirmative case.     

Trajectory optimization under kinematical constraints for moving target search

Various recent events in the Mediterranean sea have shown the enormous importance of maritime search-and-rescue missions. By reducing the time to find floating victims, the number of casualties can be reduced. A major improvement can be achieved by employing unmanned aerial systems for autonomous search missions. In this context, the need for efficient search trajectory planning methods arises. Existing approaches either consider K-step-lookahead optimization without accounting for kinematics of fixed-wing platforms or propose a suboptimal myopic method. A few approaches consider both aspects, however only applicable to stationary target search. The contribution of this article consists of a novel method for Markovian target search-trajectory optimization. This is a unified method for fixed-wing and rotary-wing platforms, taking kinematical constraints into account. It can be classified as K-step-lookahead planning method, which allows for anticipation to the estimated future position and motion of the target. The method consists of a mixed integer linear program that optimizes the cumulative probability of detection. We show the applicability and effectiveness in computational experiments for three types of moving targets: diffusing, conditionally deterministic, and Markovian. This approach is the first K-step-lookahead method for Markovian target search under kinematical constraints.     

生化系统稳态优化的一种新算法

针对一类生化系统的稳态优化问题, 在已有间接优化方法(IOM)的线性优化问题中引入一个反映S–系统解和原模型解一致性的等式约束, 应用Lagrangian乘子法将修正后的非线性优化问题转化为一个等价的线性优化问题, 提出了一种改进的稳态优化新算法. 该优化算法不仅可以收敛到正确的系统最优解, 而且可用现有的线性规划算法去计算. 最后将算法应用于几个生化系统的稳态优化中, 结果表明, 本文提出的优化算法是有效的.     

Dynamical optimization using reduced order models: A method to guarantee performance

Many methods employed for the modeling, analysis, and control of dynamical systems are based on underlying optimization schemes, e.g., parameter estimation and model predictive control. For the popular single and multiple shooting optimization approaches, in each optimization step one or more simulations of the commonly high-dimensional dynamical systems are required. This numerical simulation is frequently the biggest bottleneck concerning the computational effort.In this work, systems described by parameter dependent linear ordinary differential equations (ODEs) are considered. We propose a novel approach employing model order reduction, improved a posteriori bounds for the reduction error, and nonlinear optimization via vertex enumeration. By combining these methods an upper bound for the objective function value of the full order model can be computed efficiently by simulating only the reduced order model. Therefore, the reduced order model can be utilized to minimize an upper bound of the true objective function, ensuring a guaranteed objective function value while reducing the computational effort.The approach is illustrated by studying the parameter estimation problem for a model of an isothermal continuous tube reactor. For this system we derive an asymptotically stable reduction error estimator and analyze the speed-up of the optimization.     

The socio-ecological-economic model of a region in parallel computing

A general procedure of approximate optimal control synthesis for the socio-ecological-economic model of a region is developed. Program system DSEEmodel 1.0 is created, which involves a cluster computing device to implement parallel algorithms of scenario calculations, optimization and improvement of an approximate optimal control for the socio-ecological-economic model of a region. The program system serves for conducting multi-scenario calculations to design a sustainable development strategy for a region. In general, this is a new approach to the problem of situational control of a region, which employs supercomputers to implement the full-scale socio-ecological-economic model.     

Developments in Cartesian Genetic Programming: self-modifying CGP

Self-modifying Cartesian Genetic Programming (SMCGP) is a general purpose, graph-based, developmental form of Genetic Programming founded on Cartesian Genetic Programming. In addition to the usual computational functions, it includes functions that can modify the program encoded in the genotype. This means that programs can be iterated to produce an infinite sequence of programs (phenotypes) from a single evolved genotype. It also allows programs to acquire more inputs and produce more outputs during this iteration. We discuss how SMCGP can be used and the results obtained in several different problem domains, including digital circuits, generation of patterns and sequences, and mathematical problems. We find that SMCGP can efficiently solve all the problems studied. In addition, we prove mathematically that evolved programs can provide general solutions to a number of problems: n-input even-parity, n-input adder, and sequence approximation to π.     

Optimal setpoint control of complex heat exchanger systems

This paper presents an approach for calculating the best way of distributing the streams following through a certain class of complex heat exchanger systems in order to achieve maximum heat recovery within the system. A computer code has been developed by which the described method is demonstrated, off-line, for two real cases. This program can be readily integrated into an over-all, on-line computer control system for any complex process consisting of an exchanger system of this class. Using an accurate and detailed heat exchanger model, the exit temperatures of each exchanger are calculated by a simple mathematical procedure based on Gilmour's design method. This procedure has been included in a general model for the complete scheme of the system. The scheme is made up of a series of heat exchanger groups with parallel paths in each group. The optimal distribution of the streams within a group is found by the direct search method of Hooke and Jeeves, modified to include constraints; while the overall optimization of the system is achieved via dynamic programming.     

Dynamic switching based fuzzy control strategy for a class of distributed parameter system

In this work, a dynamic switching based fuzzy controller combined with spectral method is proposed to control a class of nonlinear distributed parameter systems (DPSs). Spectral method can transform infinite-dimensional DPS into finite ordinary differential equations (ODEs). A dynamic switching based fuzzy controller is constructed to track reference values for the multi-inputs multi-outputs (MIMO) ODEs. Only a traditional fuzzy logic system (FLS) and a rule base are used in the controller, and membership functions (MFs) for different ODEs are adjusted by scaling factors. Analytical models of the dynamic switching based fuzzy controller are deduced to design the scaling factors and analyze stability of the control system. In order to obtain a good control performance, particle swarm optimization (PSO) is adopted to design the scaling factors. Moreover, stability of fuzzy control system is analyzed by using the analytical models, definition of the stability and Lyapunov stability theory. Finally, a nonlinear rod catalytic reaction process is used as an illustrated example for demonstration. The simulation results show that performance of proposed dynamic switching based fuzzy control strategy is better than a multi-variable fuzzy logic controller.     

MPC for continuous piecewise-affine systems

A large class of hybrid systems can be described by a max–min-plus-scaling (MMPS) model (i.e., using the operations maximization, minimization, addition and scalar multiplication). First, we show that continuous piecewise-affine systems are equivalent to MMPS systems. Next, we consider model predictive control (MPC) for these systems. In general, this leads to nonlinear, nonconvex optimization problems. We present a new MPC method for MMPS systems that is based on canonical forms for MMPS functions. In case the MPC constraints are linear constraints in the inputs only, this results in a sequence of linear optimization problems such that the MPC control can often be computed in a much more efficient way than by just applying nonlinear optimization as was done in previous work.     

DUSTCD: A radiative transport code for spherically symmetric dust clouds

A new computer system with an entirely new processor design is described and demonstrated on a very small trial lattice. The new computer simulates systems of differential equations of the order of 10⁴ times faster than present day computers and we describe how the machine can be applied to lattice models in theoretical physics. A brief discussion is also given of the various mathematical approaches for studying a lattice model. We used the computer on the X - Y model. In an actual QCD program an improved computer of such a kind is designed to be 10² times faster than ordinary machines.     

A stable Gaussian radial basis function method for solving nonlinear unsteady convection–diffusion–reaction equations

We investigate a novel method for the numerical solution of two-dimensional time-dependent convection–diffusion–reaction equations with nonhomogeneous boundary conditions. We first approximate the equation in space by a stable Gaussian radial basis function (RBF) method and obtain a matrix system of ODEs. The advantage of our method is that, by avoiding Kronecker products, this system can be solved using one of the standard methods for ODEs. For the linear case, we show that the matrix system of ODEs becomes a Sylvester-type equation, and for the nonlinear case we solve it using predictor–corrector schemes such as Adams–Bashforth and implicit–explicit (IMEX) methods. This work is based on the idea proposed in our previous paper (2016), in which we enhanced the expansion approach based on Hermite polynomials for evaluating Gaussian radial basis function interpolants. In the present paper the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. The accuracy, robustness and computational efficiency of the method are presented by numerically solving several problems.     

A new filled function applied to global optimization

The filled function method (FFM) is an approach to find the global minimizer of multi-modal functions. The numerical applicability of conventional filled functions is limited as they are defined on either exponential or logarithmic terms. This paper proposes a new filled function that does not have such disadvantages. An algorithm is presented according to the theoretical analysis. A computer program is designed, implemented, and tested. Numerical experiments on typical testing functions show that the new approach is superior to the conventional one. The result of optimization design for an electrical machine is also reported.Scope and purposeIn the context of mathematical programming, global optimization is concerned with the theory and algorithms on minima of multi-modal functions. In general, global optimization approaches can be classified into two categories: probabilistic and deterministic. The former can usually be applied to general multi-modal functions, whereas the latter typically concentrates on some particular classes of functions. The filled function method is one of a few deterministic approaches which intend to find the global minimum for general multi-modal functions. However, the numerical performance of conventional filled functions is undesirable as they are defined on either exponential or logarithmic terms or multiple parameters. This paper proposes a new filled function that does not have the above disadvantages. The present work consists of theoretical analysis, algorithm design, computer implementation, mathematical validation, and engineering application.     

Contact shape optimization: a bilevel programming approach

We consider the problem of shape optimization of nonlinear elastic solids in contact. The equilibrium of the solid is defined by a constrained minimization problem, where the body energy functional is the objective and the constraints impose the nonpenetration condition. Then the optimization problem can be formulated in terms of a bilevel mathematical program. We describe new optimality conditions for bilevel programming and construct an algorithm to solve these conditions based on Herskovits’ feasible direction interior point method. With this approach we simultaneously carry out shape optimization and nonlinear contact analysis. That is, the present method is a “one shot” technique. We describe some numerical examples solved in a very efficient way. Received July 27, 1999     

Quadratic approximate dynamic programming for input‐affine systems

We consider the use of quadratic approximate value functions for stochastic control problems with input‐affine dynamics and convex stage cost and constraints. Evaluating the approximate dynamic programming policy in such cases requires the solution of an explicit convex optimization problem, such as a quadratic program, which can be carried out efficiently. We describe a simple and general method for approximate value iteration that also relies on our ability to solve convex optimization problems, in this case, typically a semidefinite program. Although we have no theoretical guarantee on the performance attained using our method, we observe that very good performance can be obtained in practice.Copyright © 2012 John Wiley & Sons, Ltd.     

The Method of Multiple Scales: Asymptotic Solutions and Normal Forms for Nonlinear Oscillatory Problems

The method of multiple scales is implemented in Maple V Release 2 to generate a uniform asymptotic solutionO(ϵ^r) for a weakly nonlinear oscillator.In recent work, it has been shown that the method of multiple scales also transforms the differential equations into normal form, so the given algorithm can be used to simplify the equations describing the dynamics of a system near a fixed point.These results are equivalent to those obtained with the traditional method of normal forms which uses a near-identity coordinate transformation to get the system into the “simplest” form.A few Duffing type oscillators are analysed to illustrate the power of the procedure. The algorithm can be modified to take care of systems of ODEs, PDEs and other nonlinear cases.     

Specification and Verification of Dynamic Properties in Distributed Computations

The ability to specify and verify dynamic properties of computations is essential for ascertaining the correctness of distributed applications. In this paper, we consider properties that can be encoded as general Boolean predicates over global system states. We introduce two global predicate classes called simple sequences and interval-constrained sequences for specifying desirable states in some causality-preserving order along with intervening undesired states. Our formalism is simpler than more traditional proposals and permits concise and intuitive expression of many interesting system properties. Algorithms are given for verifying formulas belonging to these predicate classes in an on-line and observer-independent manner during distributed computations. We illustrate the utility of our results by applying them to examples drawn from program testing, debugging, and dynamic reconfiguration in distributed systems.