Algorithm 799: revolve

Griewank, Andreas; Walther, Andrea

doi:10.1145/347837.347846

Cited by 385 publications

(215 citation statements)

References 6 publications

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“…Instead of interpreting the elementary instructions appearing in low level source code such as C or FORTRAN, the elementary instructions are reinterpreted at the highest possible level of mathematical abstraction; in particular, in the finite element case, this means interpreting the program as a sequence of variational problems. This allows for the exploitation of the mathematical structure of the model, which has several key advantages: the adjoint models are derived with the minimum of user intervention, are measured to closely approximate optimal theoretical efficiency 2 , parallelise naturally, and make use of the optimal checkpointing strategy of Griewank and Walther [43]. However for time dependent problems the variational structure of a finite element model may be obscured.…”

Section: Adjoint Modelsmentioning

confidence: 99%

“…The checkpointing scheme currently checkpoints all TimeFunction dependencies (which, as noted above, may be sub-optimal), and requires a means of restoring time dependent parameters to be provided by the user. Since the forward model can be advanced with ease more advanced checkpointing algorithms, such as the method described in Griewank and Walther [43], could in principle be used, although such algorithms are not currently implemented. Figure 3 includes the derivation of a discrete adjoint model associated with the advection-diffusion model (13), with the calculation of a total derivative of a functional.…”

Section: Deriving a Discrete Adjoint Modelmentioning

confidence: 99%

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Rapid development and adjoining of transient finite element models

Maddison

Farrell

2014

Computer Methods in Applied Mechanics and Engineering

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Recent advances in high level finite element systems have allowed for the symbolic representation of discretisations and their efficient automated implementation as model source code. This allows for the extremely compact implementation of complex non-linear models in a handful of lines of high level code. In this work we extend the high level finite element FEniCS system to introduce an abstract representation of the temporal discretisation: this enables the similarly rapid development of transient finite element models. Efficiency is achieved via aggressive optimisations that exploit the temporal structure, such as automated pre-assembly and caching of forms, and the robust re-use of matrix factorisations and preconditioner data. The resulting models are as fast or faster than hand-optimised finite element codes. The high level representation of the system remains extremely compact and easily manipulated. This structure is exploited to derive the associated discrete adjoint model automatically, with the adjoint model inheriting the performance advantages of the forward model. Combined, this provides a system for the rapid development of efficient transient models, together with their discrete adjoints.

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Section: Adjoint Modelsmentioning

confidence: 99%

Section: Deriving a Discrete Adjoint Modelmentioning

confidence: 99%

Rapid development and adjoining of transient finite element models

Maddison

Farrell

2014

Computer Methods in Applied Mechanics and Engineering

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“…Of interest to the computational uid dynamics community is the issue of uid ow control, where the numerical model used for solving the governing equations are solved by FV methods, using various slope limiters and, or the PPM scheme. It should be noted that in order to compute the adjoint variables in the backward direction, we require forward states to be available (as evident from the above piece of adjoint code) in memory or recompute them, see research on checkpointing [76,77] for discussion on the trade-o between storing in memory and recomputation.…”

Section: Data Assimilation Experimentsmentioning

confidence: 99%

A comparative study of the performance of high resolution advection schemes in the context of data assimilation

Akella

Navon

2006

Int. J. Numer. Meth. Fluids

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SUMMARYHigh resolution advection schemes have been developed and studied to model propagation of ows involving sharp fronts and shocks. So far the impact of these schemes in the framework of inverse problem solution has been studied only in the context of linear models. A detailed study of the impact of various slope limiters and the piecewise parabolic method (PPM) on data assimilation is the subject of this work, using the nonlinear viscous Burgers equation in 1-D. Also provided are results obtained in 2-D using a global shallow water equations model. The results obtained in this work may point out to suitability of these advection schemes for data assimilation in more complex higher dimensional models.

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“…The discrete adjoint method [16] is very attractive in practice since the discrete adjoint model can be generated with minimal user intervention via automatic differentiation [17]. Given the complexity of current numerical solvers, automatic generation of the adjoint code is a significant advantage over both finite-difference, and hand-coded (analytical) derivatives.…”

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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Algorithm 799: revolve

Cited by 385 publications

References 6 publications

Rapid development and adjoining of transient finite element models

Rapid development and adjoining of transient finite element models

A comparative study of the performance of high resolution advection schemes in the context of data assimilation

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

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