Parallel Full Space SQP Lagrange--Newton--Krylov--Schwarz Algorithms for PDE-Constrained Optimization Problems

Prudencio, Ernesto E.; Byrd, Richard H.; Cai, Xiao‐Chuan

doi:10.1137/040602997

Cited by 52 publications

(67 citation statements)

References 42 publications

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“…Leibfritz and Sachs [17] analyze an interior point method that benefits from a reformulation of the quadratic programming subproblems as mixed linear complementarity problems. Our approach has some features in common with the algorithms of Biros and Ghattas [1,2], Haber and Ascher [11], and Prudencio, Byrd and Cai [20] as we follow a full space SQP method and perform a line search to promote convergence. Unlike these papers, however, we present conditions that guarantee the global convergence of inexact SQP steps.…”

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confidence: 99%

An Inexact SQP Method for Equality Constrained Optimization

Byrd¹,

Curtis²,

Nocedal³

2008

SIAM J. Optim.

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Abstract. We present an algorithm for large-scale equality constrained optimization. The method is based on a characterization of inexact sequential quadratic programming (SQP) steps that can ensure global convergence. Inexact SQP methods are needed for large-scale applications for which the iteration matrix cannot be explicitly formed or factored and the arising linear systems must be solved using iterative linear algebra techniques. We address how to determine when a given inexact step makes sufficient progress toward a solution of the nonlinear program, as measured by an exact penalty function. The method is globalized by a line search. An analysis of the global convergence properties of the algorithm and numerical results are presented.

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confidence: 99%

An Inexact SQP Method for Equality Constrained Optimization

Byrd¹,

Curtis²,

Nocedal³

2008

SIAM J. Optim.

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“…From the necessary condition given in eqn (48), the volume fraction of liquid at each point in space will be iterated, within the constraints, via a fixed-point Newton method shown in eqn (59). (59) A survey of the literature on adjoint-based methods for optimal control [40][41][42][43] gave insight on how to construct a solution algorithm. However, there were no methods specifically for distributed control with free initial and final data and final time for unsteady shock attenuation.…”

Section: Optimal Control Of Shock Wave Attenuation Using Liquid Watermentioning

confidence: 99%

Optimal Control of Shock Wave Attenuation using Liquid Water Droplets with Application to Ignition Overpressure in Launch Vehicles

Moshman¹,

Sritharan²,

Hobson³

2011

International Journal of Flow Control

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This paper presents the first solution of an optimal control problem concerning unsteady blast wave attenuation where the control takes the form of the initial distribution of liquid water droplets. An appropriate two-phase flow model is adopted for compressible homogeneous two-phase flows. The dynamical system includes an empirical model for water droplet vaporization, the dominant mechanism for attenuating the jump in pressure across the shock front. At the end of the simulation interval, an appropriate target state is defined such that the jump in pressure of the target state is less than that of the simulated blast wave. Given the nature of the non-linear system, the final time must also be a free variable. A novel control algorithm is presented which can satisfy all necessary conditions of the optimal system and avoid taking a variation at the shock front. The adjoint-based method is applied to NASA's problem of Ignition Overpressure blast waves generated during ignition of solid grain rocket segments on launch vehicles. Results are shown for a range of blast waves that are plausible to see in the launch environment of the shuttle. Significant parameters of effective droplet distributions are identified. INTRODUCTIONThere exists a wide range of multi-phase and multi-fluid flow regimes. Much care must be taken in selecting an appropriate model and numeric method. Presently, the flow of interest consists of a gaseous carrier phase with many suspended water droplets. These two fluids' parameters are drastically different and the vast majority of numeric methods in the literature are ill-adapted since they introduce artificial diffusion and mixing at interfaces [1][2][3]. Other methods are poorly suited to mixtures where thermodynamic equilibrium cannot be reasonably assumed except at interfaces. Since the water droplets of interest will be smaller than 1 mm, their properties can be assumed to be homogeneous between computational cells and, therefore, many interfaces will be present in each cell. It will be impractical to use interface tracking methods or solve separate sets of equations for each fluid and to couple their properties at the interface. Instead, the following calculations implement a method proposed by [4] which has the advantage of solving the same set of differential equations throughout the domain. The method uses different equations of state for each phase and allows a difference in temperature at the gas-liquid interfaces. To close the system, a volume fraction variable α is introduced and a non-conservative PDE is appended to the conservative system. The method assumes that both phases are compressible which yields a hyperbolic system. Ignition overpressure (IOP) is a phenomenon present at the start of an ignition sequence in launch vehicles using solid-grain propellants. When the grain is ignited the pressure inside the combustion chamber quickly rises several orders of magnitude. This drives hot combustion products toward the nozzle and out to the open atmosphere at supersonic spee...

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“…As far as we know no one has studied shape optimization problems using Lagrange-Newton-Krylov-Schwarz (LNKSz) method, which has the potential to solve very large problems on machines with a large number of processors. The previous work in [41,42] doesn't consider the change of the computational domain which makes the study much more difficult and interesting.…”

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confidence: 99%

“…In [12,13], a parallel full space method was introduced for boundary control problems, where a Newton-Krylov method is used together with Schur complement type preconditioners which split the Jacobian system into three sub-systems that are solved one after another. In [41,42], an overlapping Schwarz based Lagrange-Newton-Krylov approach was investigated for some boundary control problems. In these papers, the one-shot approach was shown to be very successful.…”

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confidence: 99%

Parallel One-Shot Lagrange--Newton--Krylov--Schwarz Algorithms for Shape Optimization of Steady Incompressible Flows

Chen¹,

Cai²

2012

SIAM J. Sci. Comput.

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Abstract. We propose and study a new parallel one-shot Lagrange-Newton-Krylov-Schwarz (LNKSz) algorithm for shape optimization problems constrained by steady incompressible NavierStokes equations discretized by finite element methods on unstructured moving meshes. Most existing algorithms for shape optimization problems solve iteratively the three components of the optimality system: the state equations for the constraints, the adjoint equations for the Lagrange multipliers, and the design equations for the shape parameters. Such approaches are relatively easy to implement, but generally not easy to converge as they are basically nonlinear Gauss-Seidel algorithms with three large blocks. In this paper, we introduce a fully coupled, or the so-called one-shot, approach which solves the three components simultaneously. First, we introduce a moving mesh finite element method for the shape optimization problems in which the mesh equations are implicitly coupled with the optimization problems. Second, we introduce a LNKSz framework based on an overlapping domain decomposition method for solving the fully coupled problem. Such an approach doesn't involve any sequential steps that are necessary for the Gauss-Seidel type reduced space methods. The main challenges in full space approaches are that the corresponding nonlinear system is much harder to solve because it is two to three times larger and its indefinite Jacobian problems are also much more ill-conditioned. Effective preconditioning becomes the most important component of the method. Numerically, we show that LNKSz deals with these challenges quite well. As an application, we consider the shape optimization of an artery bypass problem in 2D. Numerical experiments show that our algorithms perform well on supercomputers with hundreds of processors.Key words. Shape optimization, one-level additive Schwarz preconditioner, parallel computing, one-shot method, finite element, inexact Newton.AMS subject classifications. 49K20, 49Q10, 65F08, 65F10, 65N30, 65N55, 65Y05, 76D05, 90C06, 90C30, 93C201. Introduction. Shape optimization, or optimal shape design, aims to optimize an objective function by changing the shape of the computational domain. In the past few years, shape optimization problems have received considerable attentions. On the theoretical side there are several publications dealing with the existence of solution and the sensitivity analysis to the problems; see e.g., [19,20,21,27,39,46] and references therein. On the practical side, optimal shape design has played an important role in many industrial applications, for example, aerodynamic shape design [25,33,34,35,44], artery bypass design [2,3,6,7,40,43], microfluidic biochip design [4,24,38] and so on. Most of these optimization problems have constraints imposed by partial differential equations (PDE) or other physical and geometrical conditions. They are considerably more difficult and expensive to solve than the corresponding simulation problems, and often require large scale parallel computers for their ...

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Parallel Full Space SQP Lagrange--Newton--Krylov--Schwarz Algorithms for PDE-Constrained Optimization Problems

Cited by 52 publications

References 42 publications

An Inexact SQP Method for Equality Constrained Optimization

An Inexact SQP Method for Equality Constrained Optimization

Optimal Control of Shock Wave Attenuation using Liquid Water Droplets with Application to Ignition Overpressure in Launch Vehicles

Parallel One-Shot Lagrange--Newton--Krylov--Schwarz Algorithms for Shape Optimization of Steady Incompressible Flows

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