The Adjoint Method and Its Application to Trajectory Optimization

Jurovics, Stephen; McIntyre, John E.

doi:10.2514/8.6284

Cited by 26 publications

(5 citation statements)

References 3 publications

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“…where ®« (31) Carrying through the previous analysis it is clear that the maximum principle must still apply along the constraint, but the Green's vector for the constrained section of the trajectory is defined by Equation 31, where X" does not depend on any particular qk and is defined by…”

Section: Necessary Conditions For Discrete Systemsmentioning

confidence: 99%

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Green's Functions and Optimal Systems. Necessary Conditions and Iterative Technique

Denn¹,

Aris²

1965

Ind. Eng. Chem. Fund.

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Section: Necessary Conditions For Discrete Systemsmentioning

confidence: 99%

“…Jurovics and McIntyre (31) attempted to extend the Goodman and Lance scheme to systems with unspecified final time, but apparently failed to obtain convergence (25, 30). Jazwinski (30), however, has reported successful application of such a method.…”

Section: Iterative Solutionmentioning

confidence: 99%

Green's Functions and Optimal Systems. Necessary Conditions and Iterative Technique

Denn¹,

Aris²

1965

Ind. Eng. Chem. Fund.

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“…The backpropagation technique given by Lailly (1983) makes use of the adjoint state method, which has been used extensively in optimal control theory since at least 1956 (Goodman & Lance 1956; Jurovics & McIntyre 1962; Lee & Marcus 1967). The matrix derivation of the backpropagation technique given in eqs (18) to (21) was derived by one of us (Shin 1988) several years ago, but the result has not received much attention in the literature.…”

Section: The Inverse Problem In Thespace–frequency Domainmentioning

confidence: 99%

Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion

Pratt

Shin

Hicks³

1998

Geophys. J. Int.

1,309

150

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Summary By specifying a discrete matrix formulation for the frequency–space modelling problem for linear partial differential equations (‘FDM’ methods), it is possible to derive a matrix formalism for standard iterative non‐linear inverse methods, such as the gradient (steepest descent) method, the Gauss–Newton method and the full Newton method. We obtain expressions for each of these methods directly from the discrete FDM method, and we refer to this approach as frequency‐domain inversion (FDI). The FDI methods are based on simple notions of matrix algebra, but are nevertheless very general. The FDI methods only require that the original partial differential equations can be expressed as a discrete boundary‐value problem (that is as a matrix problem). Simple algebraic manipulation of the FDI expressions allows us to compute the gradient of the misfit function using only three forward modelling steps (one to compute the residuals, one to backpropagate the residuals, and a final computation to compute a step length). This result is exactly analogous to earlier backpropagation methods derived using methods of functional analysis for continuous problems. Following from the simplicity of this result, we give FDI expressions for the approximate Hessian matrix used in the Gauss–Newton method, and the full Hessian matrix used in the full Newton method. In a new development, we show that the additional term in the exact Hessian, ignored in the Gauss–Newton method, can be efficiently computed using a backpropagation approach similar to that used to compute the gradient vector. The additional term in the Hessian predicts the degradation of linearized inversions due to the presence of first‐order multiples (such as free‐surface multiples in seismic data). Another interpretation is that this term predicts changes in the gradient vector due to second‐order non‐linear effects. In a numerical test, the Gauss–Newton and full Newton methods prove effective in helping to solve the difficult non‐linear problem of extracting a smooth background velocity model from surface seismic‐reflection data.

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“…In addition, however, more sophisticated applications are desired, such as to discern latent variables from empirical data, augment extant models through the incorporation of novel variables to enhance their congruence with empirical evidence, and facilitate the creation of parsimonious predictive models characterized by reduced dimensionality. In many cases, these issues can be addressed using the socalled adjoint method 1,8,[10][11][12][13][14][15][16][17] , which is a powerful, reliable, and efficient method for parameter estimation to hundreds of thousands of parameters, offering the multifaceted set of capabilities mentioned above.…”

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confidence: 99%

Tutorial: a beginner’s guide to building a representative model of dynamical systems using the adjoint method

Lettermann,

Jurado,

Betz

et al. 2024

Commun Phys

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Building a representative model of a complex dynamical system from empirical evidence remains a highly challenging problem. Classically, these models are described by systems of differential equations that depend on parameters that need to be optimized by comparison with data. In this tutorial, we introduce the most common multi-parameter estimation techniques, highlighting their successes and limitations. We demonstrate how to use the adjoint method, which allows efficient handling of large systems with many unknown parameters, and present prototypical examples across several fields of physics. Our primary objective is to provide a practical introduction to adjoint optimization, catering for a broad audience of scientists and engineers.

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The Adjoint Method and Its Application to Trajectory Optimization

Cited by 26 publications

References 3 publications

Green's Functions and Optimal Systems. Necessary Conditions and Iterative Technique

Green's Functions and Optimal Systems. Necessary Conditions and Iterative Technique

Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion

Tutorial: a beginner’s guide to building a representative model of dynamical systems using the adjoint method

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