The independent set perturbation adjoint method: A new method of differentiating mesh‐based fluids models

Fang, F.; Pain, Christopher C.; Navon, I. M.; Gorman, Gerard J.; Piggott, Matthew D.; Allison, Peter A.

doi:10.1002/fld.2297

Cited by 9 publications

(6 citation statements)

References 46 publications

(35 reference statements)

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“…In fact the computational costs are such that the procedure will remove the benefits from solving on the reduced POD space. To overcome this, this article uses Downloaded by [University of Nebraska, Lincoln] at 19:33 11 April 2015 the approach developed by Fang et al (2009Fang et al ( , 2011 which constructs the full system of equations from the sum of a set of reference system of equations…”

Section: The Pod Discretisation Of the Electromagnetic Equationsmentioning

confidence: 99%

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Reduced order borehole induction modelling

Ardjmandpour

Pain

Fang

et al. 2014

International Journal of Computational Fluid Dynamics

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This article presents a new reduced order model based on proper orthogonal decomposition (POD) for solving the electromagnetic equation for borehole modelling applications. The method aims to accurately and efficiently predict the electromagnetic fields generated by an array induction tool -an instrument that transmits and receives electrical signals along different positions within a borehole. The motivation for this approach is in the generation of an efficient 'forward model' (which provides solutions to the electromagnetic equation) for the purpose of improving the efficiency of inversion calculations (which typically require a large number of forward solutions) that are used to determine surrounding material properties. This article develops a reduced order model for this purpose as it can be significantly more efficient to compute than standard models, for example, those based on finite elements. It is shown here how the POD basis functions are generated from the snapshot solutions of a high resolution model, and how the discretised equations can be generated efficiently. The novelty is that this is the first time such a POD model reduction approach has been developed for this application, it is also unique in its use of separate POD basis functions for the real and complex solution fields. A numerical example for predicting the electromagnetic field is used to demonstrate the accuracy of the POD method for use as a forward model. It is shown that the method retains accuracy whilst reducing the costs of the computation by several orders of magnitude in comparison to an established method.

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Section: The Pod Discretisation Of the Electromagnetic Equationsmentioning

confidence: 99%

“…Once these solutions are collected the POD basis functions and their associated discretised equations are generated through the standard singular value decomposition (SVD) approach. In this article the technique developed in Fang et al (2009Fang et al ( , 2011 is applied for the efficient generation of the POD model equations.…”

Section: Introductionmentioning

confidence: 99%

Reduced order borehole induction modelling

Ardjmandpour

Pain

Fang

et al. 2014

International Journal of Computational Fluid Dynamics

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“…However, adjoint models can be difficult to formulate for complex multi-physics problems and for new generation models that use adaptive mesh resolution. The Independent Set Perturbation adjoint method [12] attempts to address these issues by forming an adjoint which can be incorporated into the computer code in a modular way, thereby aiding the maintainability of the code. The method can be applied to complex models on unstructured meshes, however, despite its modularity, introducing the code for the adjoint still requires substantial changes to be made to the original code.…”

Section: Introductionmentioning

confidence: 99%

Well Optimisation With Goal-Based Sensitivity Maps Using Time Windows And Ensemble Perturbations

Heaney¹,

Salinas²,

Pain³

et al. 2018

Proceedings

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We present an approach for forming sensitivity maps (or sensitivites) using ensembles. The method is an alternative to using an adjoint, which can be very challenging to formulate and also computationally expensive to solve. The main novelties of the presented approach are: 1) the use of goals, by weighting the perturbation to help resolve the most important sensitivities, 2) the use of time windows, which enable the perturbations to be optimised independently for each window and 3) re-orthogonalisation of the solution through time, which helps optimise each perturbation when calculating sensitivity maps. The novel application of these methods, to the generation of sensitivity maps through ensembles, greatly reduces the number of ensemble members required to form the sensitivity maps as demonstrated in this paper.As the presented method relies solely on ensemble members obtained from the forward model, it can therefore be applied directly to forward models of arbitrary complexity arising from, for example, multi-physics coupling, legacy codes or model chains.We demonstrate the efficiency of the approach by applying the method to advection problems and also a non-linear heterogeneous multi-phase porous media problem, showing, in all cases, that the number of ensemble members required to obtain accurate sensitivity maps is relatively low, in the order of 10s.

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“…Fang et al have developed a fully adaptive 3D finite-element ocean model equipped with adjoint sensitivity analysis [12,14,15,22,23,24]. The adjoint equations are discretized on spatially-variable grids, with the forward variables interpolated onto the adjoint mesh wherever needed.…”

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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The independent set perturbation adjoint method: A new method of differentiating mesh‐based fluids models

Cited by 9 publications

References 46 publications

Reduced order borehole induction modelling

Reduced order borehole induction modelling

Well Optimisation With Goal-Based Sensitivity Maps Using Time Windows And Ensemble Perturbations

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

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