An efficient Levenberg–Marquardt method with a new LM parameter for systems of nonlinear equations

A modified Levenberg–Marquardt method for solving singular systems of nonlinear equations was proposed by Fan [J Comput Appl Math. 2003;21;625–636]. Using trust region techniques, the global and quadratic convergence of the method were proved. In this paper, to improve this method, we decide to introduce a new Levenberg–Marquardt parameter while also incorporate a new nonmonotone technique to this method. The global and quadratic convergence of the new method is proved under the local error bound condition. Numerical results show the new algorithm is efficient and promising.     

Time-discretization of nonlinear control systems with state-delay via Taylor–Lie series

The state-delay is always existent in the practical systems. Analysis of the delay phenomenon in a continuous-time domain is sophisticated. It is appropriate to obtain its corresponding discrete-time model for implementation via digital computer. In this paper, we propose a new scheme for the discretization of nonlinear systems using Taylor series expansion and the zero-order hold assumption. This scheme is applied to the sample-data representation of a nonlinear system with constant state time-delay. The mathematical expressions of the discretization scheme are presented and the effect of the time-discretization method on equilibrium properties of nonlinear control system with state time-delay is examined. The proposed scheme provides a finite-dimensional representation for nonlinear systems with state time-delay enabling existing controller design techniques to be applied to them. The performance of the proposed discretization procedure is evaluated using a nonlinear system. For this nonlinear system, various sampling rates and time-delay values are considered.     

An incremental primal–dual method for nonlinear programming with special structure

We propose a new class of incremental primal–dual techniques for solving nonlinear programming problems with special structure. Specifically, the objective functions of the problems are sums of independent nonconvex continuously differentiable terms minimized subject to a set of nonlinear constraints for each term. The technique performs successive primal–dual increments for each decomposition term of the objective function. The primal–dual increments are calculated by performing one Newton step towards the solution of the Karush–Kuhn–Tucker optimality conditions of each subproblem associated with each objective function term. We show that the resulting incremental algorithm is q-linearly convergent under mild assumptions for the original problem.     

Direct discontinuous Galerkin method for nonlinear reaction–diffusion systems in pattern formation

Nonlinear reaction–diffusion systems are often employed in mathematical modeling for pattern formation. Most of the work to date has been concerned within one-dimensional or rectangular domains. However, it is recognised that in most applications multidimensional complex geometrical domains are typically more important. In this paper we solve reaction–diffusion systems by combining direct discontinuous Galerkin (DDG) finite element methods with implicit integration factor (IIF) time integration method, on triangular meshes. This allows us solve the nonlinear algebraic systems on an element-by-element bases with significant gains in computational time. Numerical solutions of two reaction–diffusion systems, the well-studied Schnakenberg model and chloride–iodide–malonic acid (CIMA) reactive model, are presented to demonstrate effects of various domain geometries on the resulting biological patterns. Our numerical results are in good agreement with other numerical and analytical results, and with experimental results.     

An energy-preserving exponentially-fitted continuous stage Runge–Kutta method for Hamiltonian systems

Recently, the symplectic exponentially-fitted methods for Hamiltonian systems with periodic or oscillatory solutions have been attracting a lot of interest. As an alternative to them, in this paper, we propose a class of energy-preserving exponentially-fitted methods. For this aim, we show sufficient conditions for energy-preservation in terms of the coefficients of continuous stage Runge–Kutta (RK) methods, and extend the theory of exponentially-fitted RK methods in the context of continuous stage RK methods. Then by combining these two theories, we derive second and fourth order energy-preserving exponentially-fitted schemes.     

Viability for nonlinear multi-valued reaction–diffusion systems

In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system

{ lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      10. Model transformations for state–space self–tuning control of multivariable stochastic systems      Leang S. Sheih  Yuan L. Bao  Norman P. coleman 《随机分析与应用》2013,31(1):91-101       11. Large deviations for invariant measures of general stochastic reaction–diffusion systems      Sandra Cerrai  Michael Röckner 《Comptes Rendus Mathematique》2003,337(9):597-602 In this paper we prove a large deviations principle for the invariant measures of a class of reaction–diffusion systems in bounded domains of $\text{R}{}^{\mathrm{d}}\text{,}\text{d?1}$, perturbed by a noise of multiplicative type. We consider reaction terms which are not Lipschitz-continuous and diffusion coefficients in front of the noise which are not bounded and may be degenerate. To cite this article: S. Cerrai, M. Röckner, C. R. Acad. Sci. Paris, Ser. I 337 (2003).      12. Robust H∞ control for nonlinear uncertain stochastic T–S fuzzy systems with time delays      T. Senthilkumar  P. Balasubramaniam 《Applied Mathematics Letters》2011,24(12):1986-1994       13. On the efficiency of two variants of Kurchatov’s method for solving nonlinear systems      José Antonio Ezquerro  Angela Grau  Miquel Grau-Sánchez  Miguel Ángel Hernández 《Numerical Algorithms》2013,64(4):685-698 We consider Kurchatov’smethod and construct two variants of this method for solving systems of nonlinear equations and deduce their local R-orders of convergence in a direct symbolic computation. We also propose a generalization to several variables of the efficiency used in the scalar case and analyse the efficiencies of the three methods when they are used to solve systems of nonlinear equations.      14. An analytic–numerical method for the construction of the reference law of operation for a class of mechanical controlled systems      A. D. Mizhidon  K. A. Mizhidon 《Computational Mathematics and Mathematical Physics》2017,57(4):626-633 An analytic–numerical method for the construction of a reference law of operation for a class of dynamic systems describing vibrations in controlled mechanical systems is proposed. By the reference law of operation of a system, we mean a law of the system motion that satisfies all the requirements for the quality and design features of the system under permanent external disturbances. As disturbances, we consider polyharmonic functions with known amplitudes and frequencies of the harmonics but unknown initial phases. For constructing the reference law of motion, an auxiliary optimal control problem is solved in which the cost function depends on a weighting coefficient. The choice of the weighting coefficient ensures the design of the reference law. Theoretical foundations of the proposed method are given.      15. Martingale weak solutions of the stochastic Landau–Lifshitz–Bloch equation      Song Jiang  Qiangchang Ju  Huaqiao Wang 《Journal of Differential Equations》2019,266(5):2542-2574 The present paper studies the stochastic Landau–Lifshitz–Bloch equation which is recommended as the only valid model at temperature around the Curie temperature and is especially important for the simulation of heat-assisted magnetic recording. We study the stochastic Landau–Lifshitz–Bloch equation in the case that the temperature is raised higher than the Curie temperature. The global existence of martingale weak solutions is proved by using a new argument and regularity properties of the weak solutions are discussed.      16. An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs      Alina Chertock  Michael Herty  Alexander Kurganov 《Computational Optimization and Applications》2014,59(3):689-724 We present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. Our approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method, which utilizes advantages of both the Eulerian finite-volume central-upwind scheme (for solving the balance law) and the Lagrangian discrete characteristics method (for solving the adjoint transport equation). Experimental convergence rates as well as numerical results for optimization problems with both linear and nonlinear constraints and a duct design problem are presented.      17. An efficient nonlinear iteration scheme for nonlinear parabolic–hyperbolic system      《Journal of Computational and Applied Mathematics》2012,236(2):253-264 A nonlinear iteration method named the Picard–Newton iteration is studied for a two-dimensional nonlinear coupled parabolic–hyperbolic system. It serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy. The nonlinear iteration scheme is constructed with a linearization–discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations. It can be viewed as an improved Picard iteration, and can accelerate convergence over the standard Picard iteration. Moreover, the discretization with second-order accuracy in both spatial and temporal variants is introduced to get the Picard–Newton iteration scheme. By using the energy estimate and inductive hypothesis reasoning, the difficulties arising from the nonlinearity and the coupling of different equation types are overcome. It follows that the rigorous theoretical analysis on the approximation of the solution of the Picard–Newton iteration scheme to the solution of the original continuous problem is obtained, which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem. Moreover, such approximation is independent of the iteration number. Numerical experiments verify the theoretical result, and show that the Picard–Newton iteration scheme with second-order spatial and temporal accuracy is more accurate and efficient than that of first-order temporal accuracy.      18. Construction of weak solutions of a certain stochastic Navier–Stokes equation      《Stochastics An International Journal of Probability and Stochastic Processes》2013,85(4):573-593 We prove the existence of weak solutions of stochastic Navier–Stokes equation on a two-dimensional torus, which appears in a certain variational problem. Our equation does not satisfy the coercivity condition. We construct its weak solutions due to an approximation by a sequence of solutions of equations with enlarged viscosity terms and then by showing an a priori estimate for them.      19. Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems      Miloslav Feistauer  Václav Kučera  Karel Najzar  Jaroslava Prokopová 《Numerische Mathematik》2011,117(2):251-288 The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L 2(L 2)”- and “DG”-norm formed by the “L 2(H 1)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q.      20. Existence of martingale and stationary suitable weak solutions for a stochastic Navier–Stokes system      《Stochastics An International Journal of Probability and Stochastic Processes》2013,85(3):327-337 The existence of suitable weak solutions of 3D Navier–Stokes equations, driven by a random body force, is proved. These solutions satisfy a local balance of energy. Existence of statistically stationary solutions is also proved.

Large deviations for invariant measures of general stochastic reaction–diffusion systems

In this paper we prove a large deviations principle for the invariant measures of a class of reaction–diffusion systems in bounded domains of $\text{R}{}^{\mathrm{d}}\text{,}\text{d?1}$, perturbed by a noise of multiplicative type. We consider reaction terms which are not Lipschitz-continuous and diffusion coefficients in front of the noise which are not bounded and may be degenerate. To cite this article: S. Cerrai, M. Röckner, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On the efficiency of two variants of Kurchatov’s method for solving nonlinear systems

We consider Kurchatov’smethod and construct two variants of this method for solving systems of nonlinear equations and deduce their local R-orders of convergence in a direct symbolic computation. We also propose a generalization to several variables of the efficiency used in the scalar case and analyse the efficiencies of the three methods when they are used to solve systems of nonlinear equations.     

An analytic–numerical method for the construction of the reference law of operation for a class of mechanical controlled systems

An analytic–numerical method for the construction of a reference law of operation for a class of dynamic systems describing vibrations in controlled mechanical systems is proposed. By the reference law of operation of a system, we mean a law of the system motion that satisfies all the requirements for the quality and design features of the system under permanent external disturbances. As disturbances, we consider polyharmonic functions with known amplitudes and frequencies of the harmonics but unknown initial phases. For constructing the reference law of motion, an auxiliary optimal control problem is solved in which the cost function depends on a weighting coefficient. The choice of the weighting coefficient ensures the design of the reference law. Theoretical foundations of the proposed method are given.     

Martingale weak solutions of the stochastic Landau–Lifshitz–Bloch equation

The present paper studies the stochastic Landau–Lifshitz–Bloch equation which is recommended as the only valid model at temperature around the Curie temperature and is especially important for the simulation of heat-assisted magnetic recording. We study the stochastic Landau–Lifshitz–Bloch equation in the case that the temperature is raised higher than the Curie temperature. The global existence of martingale weak solutions is proved by using a new argument and regularity properties of the weak solutions are discussed.     

An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs

We present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. Our approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method, which utilizes advantages of both the Eulerian finite-volume central-upwind scheme (for solving the balance law) and the Lagrangian discrete characteristics method (for solving the adjoint transport equation). Experimental convergence rates as well as numerical results for optimization problems with both linear and nonlinear constraints and a duct design problem are presented.     

An efficient nonlinear iteration scheme for nonlinear parabolic–hyperbolic system

A nonlinear iteration method named the Picard–Newton iteration is studied for a two-dimensional nonlinear coupled parabolic–hyperbolic system. It serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy. The nonlinear iteration scheme is constructed with a linearization–discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations. It can be viewed as an improved Picard iteration, and can accelerate convergence over the standard Picard iteration. Moreover, the discretization with second-order accuracy in both spatial and temporal variants is introduced to get the Picard–Newton iteration scheme. By using the energy estimate and inductive hypothesis reasoning, the difficulties arising from the nonlinearity and the coupling of different equation types are overcome. It follows that the rigorous theoretical analysis on the approximation of the solution of the Picard–Newton iteration scheme to the solution of the original continuous problem is obtained, which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem. Moreover, such approximation is independent of the iteration number. Numerical experiments verify the theoretical result, and show that the Picard–Newton iteration scheme with second-order spatial and temporal accuracy is more accurate and efficient than that of first-order temporal accuracy.     

Construction of weak solutions of a certain stochastic Navier–Stokes equation

We prove the existence of weak solutions of stochastic Navier–Stokes equation on a two-dimensional torus, which appears in a certain variational problem. Our equation does not satisfy the coercivity condition. We construct its weak solutions due to an approximation by a sequence of solutions of equations with enlarged viscosity terms and then by showing an a priori estimate for them.     

Analysis of space–time discontinuous Galerkin method for nonlinear convection–diffusion problems

The paper presents the theory of the discontinuous Galerkin finite element method for the space–time discretization of a nonstationary convection–diffusion initial-boundary value problem with nonlinear convection and linear diffusion. The problem is not singularly perturbed with dominating convection. The discontinuous Galerkin method is applied separately in space and time using, in general, different space grids on different time levels and different polynomial degrees p and q in space and time dicretization. In the space discretization the nonsymmetric, symmetric and incomplete interior and boundary penalty (NIPG, SIPG, IIPG) approximation of diffusion terms is used. The paper is concerned with the proof of error estimates in “L ²(L ²)”- and “DG”-norm formed by the “L ²(H ¹)”-seminorm and penalty terms. A special technique based on the use of the Gauss–Radau interpolation and numerical integration has been used for the derivation of an abstract error estimate. In the “DG”-norm the error estimates are optimal with respect to the size of the space grid. They are optimal with respect to the time step, if the Dirichlet boundary condition has behaviour in time as a polynomial of degree ≤ q.     

Existence of martingale and stationary suitable weak solutions for a stochastic Navier–Stokes system

The existence of suitable weak solutions of 3D Navier–Stokes equations, driven by a random body force, is proved. These solutions satisfy a local balance of energy. Existence of statistically stationary solutions is also proved.