On projected Newton–Krylov solvers for instationary laminar reacting gas flows

Veldhuizen, S. van; Vuik, C.; Kleijn, Chris R.

doi:10.1016/j.jcp.2009.11.004

Cited by 1 publication

(9 citation statements)

References 18 publications

(40 reference statements)

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“…Moveover, Algorithm 1 in this paper has some peculiar characteristics of its own. For example, we exploit a projected gradient direction

scriptP ({bold-italicx}^{k} - λ \nabla normalΘ ({bold-italicx}^{k})) - {bold-italicx}^{k}

to generate next iterate x k + 1 , in the case when projected Newton direction is unavailable or fails to be a direction of descent for the projected Newton–Krylov algorithms in ; in addition, we make use of a simple indicator variable FLAG NG ( = 0 or 1), to determine the switching between a projected Newton direction and a projected gradient direction. As will be shown in this section (i.e., Lemma ), the projected gradient direction

scriptP ({bold-italicx}^{k} - λ \nabla normalΘ ({bold-italicx}^{k})) - {bold-italicx}^{k}

is always descent.…”

Section: Feasible Projected Newton–krylov Methodsmentioning

confidence: 99%

“…Roughly speaking, we will use a projected gradient direction as a descent direction to calculate the next iterate when the current projected Newton direction is not descent. As will be shown in Section 2, this modification will not only ensure the global convergence of the projected Newton-Krylov methods in [1] but also inherit the computational advantages of these methods, such as the capacity of solving extreme large-scale problems mentioned earlier, the matrix-free operation, and preconditioning technique, and so on, see, for example, [7]. In other words, we extend the theory of Newton-Krylov methods in [2,8] fully to the projected case.…”

Section: Introductionmentioning

confidence: 99%

“…Note that constraints x > 0 in system (1) are special cases of box constraints. In addition to appearing in systems of discretized partial differential equations, compare with [1], system (2) also arises from optimization and equilibrium problems, compare with [3,9]. Great efforts have been made to find a solution of nonlinear equations (2) by solving an equivalently constrained minimization problem…”

Section: Introductionmentioning

confidence: 99%

“…Based on these observations, van Veldhuizen, Vuik, and Kleijn proposed a class of projected Newton–Krylov methods to find the nonnegative solutions of the advection–diffusion‐reaction equations arising from laminar reacting flow problems for chemical vapor deposition. More specifically, the authors first used the first‐order Euler backward method to discretize the advection–diffusion‐reaction equations and then employed projected Newton–Krylov methods to solve the underlying large‐scale systems of nonlinear equations in the form of

({, \begin{matrix} F (x) = 0 \\ x \geq 0, \end{matrix})

where

F : {double-struckR}^{n} \to {double-struckR}^{n}

is continuously differentiable. From the class of discretized nonlinear partial differential equations, the size of system is very large; 10 9 or more unknowns are no exception.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, as is well known in optimization, a projected gradient direction on a constraint set is usually a descent direction , see, for example . This actually motivates us to modify the projected Newton–Krylov methods in by means of introducing a series of projected gradient directions. Roughly speaking, we will use a projected gradient direction as a descent direction to calculate the next iterate when the current projected Newton direction is not descent.…”

Section: Introductionmentioning

confidence: 99%

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Globalization technique for projected Newton–Krylov methods

Chen

Vuik

2016

Numerical Meth Engineering

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SUMMARYLarge-scale systems of nonlinear equations appear in many applications. In various applications, the solution of the nonlinear equations should also be in a certain interval. A typical application is a discretized system of reaction diffusion equations. It is well known that chemical species should be positive otherwise the solution is not physical and in general blow up occurs. Recently, a projected Newton method has been developed, which can be used to solve this type of problems. A drawback is that the projected Newton method is not globally convergent. This motivates us to develop a new feasible projected Newton-Krylov algorithm for solving a constrained system of nonlinear equations. Combined with a projected gradient direction, our feasible projected Newton-Krylov algorithm circumvents the non-descent drawback of search directions which appear in the classical projected Newton methods. Global and local superlinear convergence of our approach is established under some standard assumptions. Numerical experiments are used to illustrate that the new projected Newton method is globally convergent and is a significate complementarity for Newton-Krylov algorithms known in the literature.

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“…Moveover, Algorithm 1 in this paper has some peculiar characteristics of its own. For example, we exploit a projected gradient direction

scriptP ({bold-italicx}^{k} - λ \nabla normalΘ ({bold-italicx}^{k})) - {bold-italicx}^{k}

scriptP ({bold-italicx}^{k} - λ \nabla normalΘ ({bold-italicx}^{k})) - {bold-italicx}^{k}

is always descent.…”

Section: Feasible Projected Newton–krylov Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

({, \begin{matrix} F (x) = 0 \\ x \geq 0, \end{matrix})

where

F : {double-struckR}^{n} \to {double-struckR}^{n}

is continuously differentiable. From the class of discretized nonlinear partial differential equations, the size of system is very large; 10 9 or more unknowns are no exception.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations