Automatic differentiation of explicit Runge-Kutta methods for optimal control

Walther, Andrea

doi:10.1007/s10589-006-0397-3

Cited by 58 publications

(43 citation statements)

References 16 publications

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“…In light of these observations, we do not present any detailed results using the continuous adjoint formulation since we do not see any advantages to this approach for our problem. We note that these finding are consistent with those reported in [19] and [42] for general Runge-Kutta time stepping methods, in [27], where the discrete and continuous adjoint approaches were applied to automatic aerodynamic optimization, and in [29] for general variational inverse problems governed by partial differential equations. A similar gradient discrepancy between discretization/optimization versus optimization/discretization can also occur with respect to the spatial discretization -see the discussion for a shape optimization problem in [18].…”

Section: Continuous Versus Discrete Adjoint Formulationsupporting

confidence: 90%

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

et al. 2014

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An adjoint formulation for the gradient-based optimization of oil-gas compositional reservoir simulation problems is presented. The method is implemented within an automatic differentiation-based compositional flow simulator (Stanford's AD-GPRS). The development of adjoint procedures for general compositional problems is much more challenging than for oil-water problems due to the increased complexity of the code and the underlying physics. The treatment of nonlinear constraints, an example of which is a maximum gas rate specification in injection or production wells, when the control variables are well bottom-hole pressures, poses a particular challenge. Two approaches for handling these constraints are presented -a formal treatment within the optimizer, and a simpler heuristic treatment in the forward model. The relationship between discrete and continuous adjoint formulations is also elucidated. duced) relative to reference solutions range from 4.2% to 11.6%. The heuristic treatment of nonlinear constraints is shown to offer a cost-effective means for obtaining feasible solutions, which are in some cases better than those obtained using the formal constraint handling procedure.

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Section: Continuous Versus Discrete Adjoint Formulationsupporting

confidence: 90%

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

et al. 2014

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“…First, it applies AD over an adaptive step size RungKutta (RK) scheme as in (7): (7) where denotes a slope estimation and is the integration step which depends on design parameters for an adaptive step size scheme. Recent studies [4] were carried out for differentiating such schemes. The difference in [4] is that the response time is prescribed in advance at a fixed value.…”

Section: Automatic Differentiationmentioning

confidence: 99%

“…Recent studies [4] were carried out for differentiating such schemes. The difference in [4] is that the response time is prescribed in advance at a fixed value. Our approach intends to make use of it as a constrained design parameter carried out further in optimization, so, its corresponding derivatives are to be valued as explained before.…”

Section: Automatic Differentiationmentioning

confidence: 99%

Automatic Differentiation Applied for Optimization of Dynamical Systems

2010

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Abstract-Simulation is ubiquitous in many scientific areas. Applied for dynamic systems usually by employing differential equations, it gives the time evolution of system states. In order to solve such problems, numerical integration algorithms are often required. Automatic Differentiation (AD) is introduced as a powerful technique to compute derivatives of functions given in the form of computer programs in a high level programming language such as FORTRAN, C or C++. Such technique fits perfectly in combination with gradient based optimization algorithms, provided that the derivatives are valued with no truncation or cancellation error. This paper intends to use Automatic Differentiation employed for numerical integration schemes of dynamical systems simulating electromechanical actuators. Then, the resulting derivatives are used for sizing such devices by means of gradient based constrained optimization.

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“…The concept of adjoint consistency, defined, e.g., in [37,38] for elliptic problems, plays an important role in the analysis of the dual (adjoint) problem solution, in the convergence of the primal approximation, as well as in the accuracy of the target functional under consideration. Building on previous duality results for time [39,40,41] and space discretizations [37,42], we develop an unified framework for investigating dual consistency of discretizations for a general type of time-dependent PDEs.…”

Section: Introductionmentioning

confidence: 99%

Space–time adaptive solution of inverse problems with the discrete adjoint method

Alexe

Sandu

2014

Journal of Computational Physics

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Abstract. Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method. This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the intergrid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided for the discontinuous Galerkin (DG) method. The adjoint model development is considerably simplified by decoupling the adaptive mesh refinement mechanism from the forward model solver, and by selectively applying automatic differentiation on individual algorithms.In forward models discontinuous Galerkin discretizations can efficiently handle high orders of accuracy, h/p-refinement, and parallel computation. The analysis reveals that this approach, paired with Runge Kutta time stepping, is well suited for the adaptive solutions of inverse problems. The usefulness of discrete discontinuous Galerkin adjoints is illustrated on a two-dimensional adaptive data assimilation problem. Space-time adaptive solution of inverse problems with the discrete adjoint method2

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Automatic differentiation of explicit Runge-Kutta methods for optimal control

Cited by 58 publications

References 16 publications

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

Automatic Differentiation Applied for Optimization of Dynamical Systems

Space–time adaptive solution of inverse problems with the discrete adjoint method

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