Archivio della ricerca dell'Università di Modena e Reggio Emiliahttps://iris.unimore.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Mon, 17 May 2021 13:03:57 GMT2021-05-17T13:03:57Z10861Inner solvers for interior point methods for large scale nonlinear programminghttp://hdl.handle.net/11380/421269Titolo: Inner solvers for interior point methods for large scale nonlinear programming
Abstract: This paper deals with the solution of nonlinear programming problems arising from elliptic control problems by an interior point scheme. At each step of the scheme, we have to solve a large scale symmetric and indefinite system; inner iterative solvers, with an adaptive stopping rule, can be used in order to avoid unnecessary inner iterations, especially when the current outer iterate is far from the solution.In this work, we analyse the method of multipliers and the preconditioned conjugate gradient method as inner solvers for interior point schemes. We discuss the convergence of the whole approach, the implementation details and report the results of numerical experimentation on a set of large scale test problems arising from the discretization of elliptic control problems. A comparison with other interior point codes is also reported.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/11380/4212692007-01-01T00:00:00ZApplication of inexact Newton method to the optimization of systems with distributed parametershttp://hdl.handle.net/11380/1060323Titolo: Application of inexact Newton method to the optimization of systems with distributed parameters
Abstract: This paper is concerned with the numerical solution of optimal control problems for distributed parameter systems by means of nonlinear programming methods. Specifically, we will consider optimal control problems for systems described by reaction diffusion convection equations subject to control and state inequality constraints. We assume that for these problems there exists a unique solution and the dynamic systems have an equilibrium state stable (in the sense of Lyapunov) and are completely controllable and observable. We transcribe these problems into large finite dimensional nonlinear programming problems by introducing suitable finite difference discretization schemes. A basic point of this transcription is a consistency condition for which the condition
of optimality of the discretized problems (Karush–Kuhn–Tucker conditions) reflect the optimality conditions of the original continuous problems (Pontryagin Maximum Principle). This relationship between the discrete and the continuous necessary optimality conditions suggests to discretize the reaction diffusion convection equation with a finite difference scheme of type FTCS (Forward Time, Centered Space) which is second order accurate in space and first order accurate in time and is numerically stable in the sense of Von Neumann under suited conditions. The numerical solutions of the above nonlinear programming problems are determined by solving the constrained system of nonlinear equations obtained by the Karush–Kuhn–Tucker (KKT) optimality conditions. For solving these KKT system we use a Modified Inexact Newton (MIN) method. The convergence properties of the MIN method are stated under standard assumptions on the KKT systems. These assumptions are the same for which, at each iteration of the MIN method, the perturbed Newton equation has a unique solution, which can be determined by solving an equality constrained quadratic programming problem with the Hestenes’ method of multipliers. Numerical studies show that the MIN method leads to satisfactory results if we take care of choosing the starting point of the MIN iterative procedure and the perturbation parameter in such a way that the damping parameter is sufficiently large for all iterations.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/11380/10603232011-01-01T00:00:00ZNumerical studies on semi-implicit and implicit methods for reaction-diffusion equationshttp://hdl.handle.net/11380/1060325Titolo: Numerical studies on semi-implicit and implicit methods for reaction-diffusion equations
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/11380/10603252014-01-01T00:00:00ZA note on the Rosenbrock formulaehttp://hdl.handle.net/11380/1060324Titolo: A note on the Rosenbrock formulae
Abstract: In this report the Rosenbrock formulae are considered. These formulae are particularly suited for the integration of stiff differential systems such as the ones arising from reaction kinetics combustion modeling.
The numerical techniques for the analysis of the A-stability and of the L-stability of a third order Rosenbrock formula are reported.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/11380/10603242013-01-01T00:00:00ZOn solving a special class of weakly nonlinear finite difference systemshttp://hdl.handle.net/11380/1060319Titolo: On solving a special class of weakly nonlinear finite difference systems
Abstract: In this paper we consider the Newton–iterative method for solving weakly nonlinear finite difference systems of the form F (u) = Au + G(u) = 0, where the Jacobian matrix G′(u) satisfies an affine invariant Lipschitz condition. We also consider a modification of the method for which we can improve the likelihood of convergence from initial approximations that may be outside the attraction ball of the Newton–iterative method. We analyse the convergence of this damped method in the framework of the line search strategy. Numerical experiments on a diffusion–convection problem show the effectiveness of the method.
Sat, 01 Jan 2005 00:00:00 GMThttp://hdl.handle.net/11380/10603192005-01-01T00:00:00ZA study of direct and Krylov iterative sparse solver techniques to approach linear scaling of the integration of chemical kinetics with detailed combustion mechanismshttp://hdl.handle.net/11380/1003723Titolo: A study of direct and Krylov iterative sparse solver techniques to approach linear scaling of the integration of chemical kinetics with detailed combustion mechanisms
Abstract: The integration of the stiff ODE systems associated with chemical kinetics is the most computationally demanding task in most practical combustion simulations. The introduction of detailed reaction mechanisms in multi-dimensional simulations is limited by unfavorable scaling of the stiff ODE solution methods with the mechanism’s size. In this paper, we compare the efficiency and the appropriateness of direct and Krylov subspace sparse iterative solvers to speed-up the integration of combustion chemistry ODEs, with focus on their incorporation into multi-dimensional CFD codes through operator splitting. A suitable preconditioner formulation was addressed by using a general-purpose incomplete LU factorization method for the chemistry Jacobians, and optimizing its parameters using ignition delay simulations for practical fuels. All the calculations were run using a same efficient framework: SpeedCHEM, a recently developed library for gas-mixture kinetics that incorporates a sparse analytical approach for the ODE system functions. The solution was integrated through direct and Krylov subspace iteration implementations with different backward differentiation formula integrators for stiff ODE systems: LSODE, VODE,
DASSL. Both ignition delay calculations, involving reaction mechanisms that ranged from 29 to 7171 species, and multi-dimensional internal combustion engine simulations with the KIVA code were used as test cases. All solvers showed similar robustness, and no integration failures were observed when using ILUT-preconditioned Krylov enabled integrators. We found that both solver approaches, coupled with efficient function evaluation numerics, were capable of scaling computational time requirements approximately
linearly with the number of species. This allows up to three orders of magnitude speed-ups in comparison with the traditional dense solution approach. The direct solvers outperformed Krylov subspace solvers at mechanism sizes smaller than about 1000 species, while the Krylov approach allowed more than 40% speed-up over the direct solver when using the largest reaction mechanism with 7171 species.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/11380/10037232014-01-01T00:00:00ZAnalysis of the convergence of an inexact Newton method for solving Karush-Kuhn-Tucker systemshttp://hdl.handle.net/11380/421271Titolo: Analysis of the convergence of an inexact Newton method for solving Karush-Kuhn-Tucker systems
Abstract: In this paper we analyze an interior point method for solving perturbed Karush-Kuhn-Tucker systems in the framework of inexact Newton methods. This gives the possibility to revise the method to introduce an adaptive technique for changing the perturbation parameter and an inner linear solver for determining an approximate solution of the perturbed Newton equation. It makes the method more robust and highly effective for large-scale optimization problems, as those that occur in data fitting applications and in the discretization of optimal control problems governed by partial differential equations.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/11380/4212712004-01-01T00:00:00ZInner solvers for interior point methods for large scale nonlinear programminghttp://hdl.handle.net/11380/1060322Titolo: Inner solvers for interior point methods for large scale nonlinear programming
Abstract: This paper deals with the solution of nonlinear programming problems arising from elliptic control problems by an interior point scheme. At each step of the scheme, we have to solve a large scale symmetric and indefinite system; inner iterative solvers, with adaptive stopping rule, can be used in order to avoid unnecessary inner iterations, especially when the current outer iterate is far from the solution. In this work, we analyze the method of multipliers and the preconditioned conjugate gradient method as inner solvers for interior point schemes. We discuss on the convergence of the whole approach, on the implementation details and we report results of a numerical experimentation on a set of large scale test problems arising from the discretization of elliptic control problems. A comparison with other interior point codes is also reported.
Sat, 01 Jan 2005 00:00:00 GMThttp://hdl.handle.net/11380/10603222005-01-01T00:00:00ZOn solving a special class of weakly nonlinear finite-difference systemshttp://hdl.handle.net/11380/452954Titolo: On solving a special class of weakly nonlinear finite-difference systems
Abstract: In this paper, we consider the Newton-iterative method for solving weakly nonlinear finite-difference systems of the form F(u)=Au+G(u)=0, where the jacobian matrix G'(u) satisfies an affine invariant Lipschitz condition. We also consider a modification of the method for which we can improve the likelihood of convergence from initial approximations that may be outside the attraction ball of the Newton-iterative method. We analyse the convergence of this damped method in the framework of the line search strategy. Numerical experiments on a diffusion-convection problem show the effectiveness of the method.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/11380/4529542009-01-01T00:00:00ZA two-stage iterative method for solving weakly nonlinear systemshttp://hdl.handle.net/11380/452950Titolo: A two-stage iterative method for solving weakly nonlinear systems
Abstract: In this paper we consider a two-stage iterative method for solving weakly nonlinear systems generated by the discretization of semilinear elliptic boundary value problems. This method is well suited for implementation on parallel computers. Theorems about the convergence and the monotone convergence of the method are proved. An application of the method for solving real practical problems related to the study of reaction-diffusion processes and of interacting populations is described.
Tue, 01 Jan 2002 00:00:00 GMThttp://hdl.handle.net/11380/4529502002-01-01T00:00:00Z