Numerical Experience with a Class of Algorithms for Nonlinear Optimization Using Inexact Function and Gradient Information

Carter, Richard G.

doi:10.1137/0914023

Cited by 40 publications

(31 citation statements)

References 11 publications

(19 reference statements)

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“…We may be able to further reduce the total number of PDE solves by employing inexact objective-function evaluations. Trust-region algorithms with inexact objective-function evaluations have been proposed in [17], [21, sect. 10.6], [72].…”

Section: Discussionmentioning

confidence: 99%

“…10.6], [72]. The approach [17], [21, sect. 10.6] requires the error estimates that allow one to reduce the error in function evaluations below a prescribed level.…”

Section: Discussionmentioning

confidence: 99%

“…10.6] requires the error estimates that allow one to reduce the error in function evaluations below a prescribed level. Therefore, asymptotic error estimates do not fit into the framework of [17], [21, sect. 10.6].…”

Section: Discussionmentioning

confidence: 99%

“…In particular, joint refinement of the objective function and its model appears to result in a very large sparse grid during the first few iterations. Therefore, it is not yet clear how to extend the conditions and corresponding theory in [17], [21, (4.4). In the context of solving single PDEs with random data, an adaptive scheme based on Taylor approximations has been proposed in [20], and an a posteriori error estimator has been proved.…”

Section: Discussionmentioning

confidence: 99%

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A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Kouri¹,

Heinkenschloss²,

Ridzal³

et al. 2013

SIAM J. Sci. Comput.

106

116

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Abstract. The numerical solution of optimization problems governed by partial differential equations (PDEs) with random coefficients is computationally challenging because of the large number of deterministic PDE solves required at each optimization iteration. This paper introduces an efficient algorithm for solving such problems based on a combination of adaptive sparse-grid collocation for the discretization of the PDE in the stochastic space and a trust-region framework for optimization and fidelity management of the stochastic discretization. The overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution of the optimization problem. It frequently uses few collocation points initially and increases the number of collocation points only as necessary, thereby keeping the number of deterministic PDE solves low while guaranteeing convergence. Currently an error indicator is used to estimate gradient errors due to adaptive stochastic collocation. The algorithm is applied to three examples, and the numerical results demonstrate a significant reduction in the total number of PDE solves required to obtain an optimal solution when compared with a Newton conjugate gradient algorithm applied to a fixed high-fidelity discretization of the optimization problem.

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Section: Discussionmentioning

confidence: 99%

“…10.6], [72]. The approach [17], [21, sect. 10.6] requires the error estimates that allow one to reduce the error in function evaluations below a prescribed level.…”

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

See 2 more Smart Citations

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Kouri¹,

Heinkenschloss²,

Ridzal³

et al. 2013

SIAM J. Sci. Comput.

106

116

View full text Add to dashboard Cite

show abstract

“…For example, in [4], the authors prove theoretically that the SQP algorithm fails to converge when the relative error of the gradient is above 50 %. Obviously, the analysis detailed in [4] cannot be readily replicated in the case of a general production optimization problem. Instead, in this work, we design a numerical experiment that induces the commonly observed gradient errors and study their impact on the optimization process.…”

Section: Introductionmentioning

confidence: 99%

Effect of Time Stepping Strategy on Adjoint-based Production Optimization

Volkov

Voskov

2014

Proceedings

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The adjoint gradient method is well recognized for its efficiency in large-scale production optimization. When implemented in a sequential quadratic programming (SQP) algorithm, adjoint gradients enable the construction of a quadratic approximation of the objective function and linear approximation of the nonlinear constraints using just one forward and one backward simulation (with multiple right-hand sides). In this work, the focus is on the performance of the adjoint gradient method with respect to the adaptive time step refinement generated in the underlying forward simulations. First, we demonstrate that the mass transfer in reservoir simulation and, as a consequence, the net-present value (NPV) function are more sensitive to the degree of the time step refinement when using production bottom-hole pressure (BHP) controls than when using production rate controls. Effects of this sensitivity on optimization process are studied using six examples of uniform time stepping with different degrees of refinements. By comparing those examples, we show that corresponding optimal solutions for target production BHPs deviate at early stages of the optimization process. It indicates an inconsistency in the evaluation of the adjoint gradients and NPV function for different time step refinements. effects of this inconsistency on the results of a constrained production optimization. Two strategies of nonlinear constraints are considered: (i) nonlinear constraints handled in the optimization process and (ii) constraints applied directly in forward simulations with a common control switch procedure. In both strategies, we observe that the progress of the optimization process is greatly influenced by the degree of the time step refinement after control update. In the case of constrained simulations, the presence of control switches combined with large time steps after control update forces adaptive refinement to vary the time step size significantly. As a result, the inconsistency of the adjoint gradients and NPV values provoke an early termination of the SQP algorithm. In the case of constrained optimization, the inconsistencies in gradient evaluations are less significant, and the performance of the optimization process is governed by a satisfaction of nonlinear constraints in SQP algorithm.

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On a Quasi-Consistent Approximations Approach to Optimization Problems with Two Numerical Precision Parameters

Pironneau

Polak

Applied Optimization

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We present a theory of quasi-consistent approximations that combines the theory of consistent approximations with the theory of algorithm implementation, presented in Polak (1997), and enables us to solve infinitedimensional optimization problems whose discretization involves two precision parameters. A typical example of such a problem is an optimal control problem with initial and final value constraints. The theory includes new algorithm models that can be used with two discretization parameters. We illustrate the applicability of these algorithm models by implementing them using an approximate steepest descent method and applying it them to a simple two point boundary value optimal control problem. Our numerical results (not only the ones in this paper) show that these new algorithms perform quite well and are fairly insensitive to the selection of user-set parameters. Also, they appear to be superior to some alternative, ad hoc schemes.

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Numerical Experience with a Class of Algorithms for Nonlinear Optimization Using Inexact Function and Gradient Information

Cited by 40 publications

References 11 publications

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

A Trust-Region Algorithm with Adaptive Stochastic Collocation for PDE Optimization under Uncertainty

Effect of Time Stepping Strategy on Adjoint-based Production Optimization

On a Quasi-Consistent Approximations Approach to Optimization Problems with Two Numerical Precision Parameters

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