Discrete second order adjoints in atmospheric chemical transport modeling

Sandu, Adrian; Zhang, Lin

doi:10.1016/j.jcp.2008.02.011

Cited by 38 publications

(36 citation statements)

References 68 publications

(80 reference statements)

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“…Any cost function that depends on the solution along the entire trajectory can be brought to the form (2) by introducing additional "quadrature variables" [11]. In this paper we assume that the functions f and Ψ are at least twice continuously differentiable.…”

Section: Continuous Soamentioning

confidence: 99%

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Second-order adjoints for solving PDE-constrained optimization problems

Cioaca

Alexe

Sandu

2012

Optimization Methods and Software

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Inverse problems are of utmost importance in many fields of science and engineering. In the variational approach inverse problems are formulated as PDE-constrained optimization problems, where the optimal estimate of the uncertain parameters is the minimizer of a certain cost functional subject to the constraints posed by the model equations. The numerical solution of such optimization problems requires the computation of derivatives of the model output with respect to model parameters. The first order derivatives of a cost functional (defined on the model output) with respect to a large number of model parameters can be calculated efficiently through first order adjoint sensitivity analysis. Second order adjoint models give second derivative information in the form of matrix-vector products between the Hessian of the cost functional and user defined vectors. Traditionally, the construction of second order derivatives for large scale models has been considered too costly. Consequently, data assimilation applications employ optimization algorithms that use only first order derivative information, like nonlinear conjugate gradients and quasi-Newton methods.In this paper we discuss the mathematical foundations of second order adjoint sensitivity analysis and show that it provides an efficient approach to obtain Hessian-vector products. We study the benefits of using of second order information in the numerical optimization process for data assimilation applications. The numerical studies are performed in a twin experiment setting with a two-dimensional shallow water model. Different scenarios are considered with different discretization approaches, observation sets, and noise levels. Optimization algorithms that employ second order derivatives are tested against widely used methods that require only first order derivatives. Conclusions are drawn regarding the potential benefits and the limitations of using high-order information in large scale data assimilation problems.

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Section: Continuous Soamentioning

confidence: 99%

“…al. [11] have developed a rigorous approach to deriving discrete second order adjoints for Runge Kutta and Rosenbrock methods, and have applied them to chemical transport models. Alexe et.…”

Section: Introductionmentioning

confidence: 99%

Second-order adjoints for solving PDE-constrained optimization problems

Cioaca

Alexe

Sandu

2012

Optimization Methods and Software

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“…The overall simulation accuracy can be considerably improved by data assimilation [3,4]. Thus, the overall simulation error also depends on the amount of information carried by external measurements, and by the quality of the data assimilation system used [5][6][7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Chemical Mechanism Solvers in Air Quality Models

et al. 2011

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The solution of chemical kinetics is one of the most computationally intensive tasks in atmospheric chemical transport simulations. Due to the stiff nature of the system, implicit time stepping algorithms which repeatedly solve linear systems of equations are necessary. This paper reviews the issues and challenges associated with the construction of efficient chemical solvers, discusses several families of algorithms, presents strategies for increasing computational efficiency, and gives insight into implementing chemical solvers on accelerated computer architectures.

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“…We use the dual consistent upwind spatial discretization given in [37]. The particular form of the cost functional (91) implies that the discrete adjoint system has a forcing term only at the observation times t k (93), where it is necessary to add the observation mismatch [77]:…”

Section: Space-time Consistency and Accuracy Of The Discrete Adjoint mentioning

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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Discrete second order adjoints in atmospheric chemical transport modeling

Cited by 38 publications

References 68 publications

Second-order adjoints for solving PDE-constrained optimization problems

Second-order adjoints for solving PDE-constrained optimization problems

Chemical Mechanism Solvers in Air Quality Models

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

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