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Journal articles on the topic 'Discrete adjoints'

Author: Grafiati

Published: 4 June 2021

Last updated: 2 February 2022

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1

Zhao, Shunliu, Matthew G. Russell, Amir Hakami, Shannon L. Capps, Matthew D. Turner, Daven K. Henze, Peter B. Percell, et al. "A multiphase CMAQ version 5.0 adjoint." Geoscientific Model Development 13, no. 7 (July 2, 2020): 2925–44. http://dx.doi.org/10.5194/gmd-13-2925-2020.

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Abstract. We present the development of a multiphase adjoint for the Community Multiscale Air Quality (CMAQ) model, a widely used chemical transport model. The adjoint model provides location- and time-specific gradients that can be used in various applications such as backward sensitivity analysis, source attribution, optimal pollution control, data assimilation, and inverse modeling. The science processes of the CMAQ model include gas-phase chemistry, aerosol dynamics and thermodynamics, cloud chemistry and dynamics, diffusion, and advection. Discrete adjoints are implemented for all the science processes, with an additional continuous adjoint for advection. The development of discrete adjoints is assisted with algorithmic differentiation (AD) tools. Particularly, the Kinetic PreProcessor (KPP) is implemented for gas-phase and aqueous chemistry, and two different automatic differentiation tools are used for other processes such as clouds, aerosols, diffusion, and advection. The continuous adjoint of advection is developed manually. For adjoint validation, the brute-force or finite-difference method (FDM) is implemented process by process with box- or column-model simulations. Due to the inherent limitations of the FDM caused by numerical round-off errors, the complex variable method (CVM) is adopted where necessary. The adjoint model often shows better agreement with the CVM than with the FDM. The adjoints of all science processes compare favorably with the FDM and CVM. In an example application of the full multiphase adjoint model, we provide the first estimates of how emissions of particulate matter (PM2.5) affect public health across the US.

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2

Bixby, Robert E., and Collette R. Coullard. "Adjoints of Binary Matroids." European Journal of Combinatorics 9, no. 2 (March 1988): 139–47. http://dx.doi.org/10.1016/s0195-6698(88)80038-6.

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3

Foniok, Jan, Jaroslav Nešetřil, and Claude Tardif. "Interleaved adjoints of directed graphs." European Journal of Combinatorics 32, no. 7 (October 2011): 1018–24. http://dx.doi.org/10.1016/j.ejc.2011.03.013.

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4

Bachem, A., and W. Kern. "Adjoints of oriented matroids." Combinatorica 6, no. 4 (December 1986): 299–308. http://dx.doi.org/10.1007/bf02579255.

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5

Sandu, Adrian, and Lin Zhang. "Discrete second order adjoints in atmospheric chemical transport modeling." Journal of Computational Physics 227, no. 12 (June 2008): 5949–83. http://dx.doi.org/10.1016/j.jcp.2008.02.011.

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6

Alexe, Mihai, and Adrian Sandu. "On the discrete adjoints of adaptive time stepping algorithms." Journal of Computational and Applied Mathematics 233, no. 4 (December 2009): 1005–20. http://dx.doi.org/10.1016/j.cam.2009.08.109.

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7

Szenicer, Alexandre, Kuangdai Leng, and Tarje Nissen-Meyer. "A complexity-driven framework for waveform tomography with discrete adjoints." Geophysical Journal International 223, no. 2 (July 20, 2020): 1247–64. http://dx.doi.org/10.1093/gji/ggaa349.

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Summary We develop a new approach for computing Fréchet sensitivity kernels in full waveform inversion by using the discrete adjoint approach in addition to the widely used continuous adjoint approach for seismic waveform inversion. This method is particularly well suited for the forward solver AxiSEM3D, a combination of the spectral-element method (SEM) and a Fourier pseudo-spectral method, which allows for a sparse azimuthal wavefield parametrization adaptive to wavefield complexity, leading to lower computational costs and better frequency scaling than conventional 3-D solvers. We implement the continuous adjoint method to serve as a benchmark, additionally allowing for simulating off-axis sources in axisymmetric or 3-D models. The kernels generated by both methods are compared to each other, and benchmarked against theoretical predictions based on linearized Born theory, providing an excellent fit to this independent reference solution. Our verification benchmarks show that the discrete adjoint method can produce exact kernels, largely identical to continuous kernels. While using the continuous adjoint method we lose the computational advantage and fall back on a full-3-D frequency scaling, using the discrete adjoint retains the speedup offered by AxiSEM3D. We also discuss the creation of a data-coverage based mesh to run the simulations on during the inversion process, which would allow to exploit the flexibility of the Fourier parametrization and thus the speedup offered by our method.

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8

Faure, Christèle, and Isabelle Charpentier. "Comparing Global Strategies for Coding Adjoints." Scientific Programming 9, no. 1 (2001): 1–10. http://dx.doi.org/10.1155/2001/485915.

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From a computational point of view, sensitivity analysis, calibration of a model, or variational data assimilation may be tackled after the differentiation of the numerical code representing the model into an adjoint code. This paper presents and compares methodologies to generate discrete adjoint codes. These methods can be implemented when hand writing adjoint codes, or within Automatic Differentiation (AD) tools. AD has been successfully applied to industrial codes that were large and general enough to fully validate this new technology. We compare these methodologies in terms of execution time and memory requirement on a one dimensional thermal-hydraulic module for two-phase flow modeling. With regard to this experiment, some development axes for AD tools are extracted as well as methods for AD tool users to get efficient adjoint codes semi-automatically. The next objective is to generate automatically adjoint codes as efficient as hand written ones.

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9

Larour, Eric, Jean Utke, Anton Bovin, Mathieu Morlighem, and Gilberto Perez. "An approach to computing discrete adjoints for MPI-parallelized models applied to Ice Sheet System Model 4.11." Geoscientific Model Development 9, no. 11 (November 1, 2016): 3907–18. http://dx.doi.org/10.5194/gmd-9-3907-2016.

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Abstract. Within the framework of sea-level rise projections, there is a strong need for hindcast validation of the evolution of polar ice sheets in a way that tightly matches observational records (from radar, gravity, and altimetry observations mainly). However, the computational requirements for making hindcast reconstructions possible are severe and rely mainly on the evaluation of the adjoint state of transient ice-flow models. Here, we look at the computation of adjoints in the context of the NASA/JPL/UCI Ice Sheet System Model (ISSM), written in C++ and designed for parallel execution with MPI. We present the adaptations required in the way the software is designed and written, but also generic adaptations in the tools facilitating the adjoint computations. We concentrate on the use of operator overloading coupled with the AdjoinableMPI library to achieve the adjoint computation of the ISSM. We present a comprehensive approach to (1) carry out type changing through the ISSM, hence facilitating operator overloading, (2) bind to external solvers such as MUMPS and GSL-LU, and (3) handle MPI-based parallelism to scale the capability. We demonstrate the success of the approach by computing sensitivities of hindcast metrics such as the misfit to observed records of surface altimetry on the northeastern Greenland Ice Stream, or the misfit to observed records of surface velocities on Upernavik Glacier, central West Greenland. We also provide metrics for the scalability of the approach, and the expected performance. This approach has the potential to enable a new generation of hindcast-validated projections that make full use of the wealth of datasets currently being collected, or already collected, in Greenland and Antarctica.

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10

Gou, Tianyi, and Adrian Sandu. "Continuous versus discrete advection adjoints in chemical data assimilation with CMAQ." Atmospheric Environment 45, no. 28 (September 2011): 4868–81. http://dx.doi.org/10.1016/j.atmosenv.2011.06.015.

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11

Alfter, M., W. Kern, and A. Wanka. "On adjoints and dual matroids." Journal of Combinatorial Theory, Series B 50, no. 2 (December 1990): 208–13. http://dx.doi.org/10.1016/0095-8956(90)90075-b.

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12

Jando, Dörte. "Efficient goal-oriented global error estimators for BDF methods using discrete adjoints." Journal of Computational and Applied Mathematics 316 (May 2017): 195–212. http://dx.doi.org/10.1016/j.cam.2016.09.032.

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13

Capps, S. L., D. K. Henze, A. Hakami, A. G. Russell, and A. Nenes. "ANISORROPIA: the adjoint of the aerosol thermodynamic model ISORROPIA." Atmospheric Chemistry and Physics Discussions 11, no. 8 (August 19, 2011): 23469–511. http://dx.doi.org/10.5194/acpd-11-23469-2011.

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Abstract. We present the development of ANISORROPIA, the discrete adjoint of the ISORROPIA thermodynamic equilibrium model that treats the Na+-SO42−-HSO4−-NH4+-NO3−-Cl−-H2O aerosol system, and we demonstrate its sensitivity analysis capabilities. ANISORROPIA calculates sensitivities of an inorganic species in aerosol or gas phase with respect to the total concentrations of each species present with only a two-fold increase in computational time over the forward model execution. Due to the highly nonlinear and discontinuous solution surface of ISORROPIA, evaluation of the adjoint required a new, complex-variable version of the the model, which determines first-order sensitivities with machine precision and avoids cancellation errors arising from finite difference calculations. The adjoint is verified over an atmospherically relevant range of concentrations, temperature, and relative humidity. We apply ANISORROPIA to recent field campaign results from Atlanta, GA, USA, and Mexico City, Mexico, to characterize the inorganic aerosol sensitivities of these distinct urban air masses. The variability in the relationship between PM2.5 mass and precursor concentrations shown has important implications for air quality and climate. ANISORROPIA enables efficient elucidation of aerosol concentration dependence on aerosol precursor emissions in the context of atmospheric chemical transport model adjoints.

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14

Liu, Zheng, and Adrian Sandu. "On the properties of discrete adjoints of numerical methods for the advection equation." International Journal for Numerical Methods in Fluids 56, no. 7 (2008): 769–803. http://dx.doi.org/10.1002/fld.1547.

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15

Foniok, Jan, and Claude Tardif. "Digraph functors which admit both left and right adjoints." Discrete Mathematics 338, no. 4 (April 2015): 527–35. http://dx.doi.org/10.1016/j.disc.2014.10.018.

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16

Rauser, Florian, Jan Riehme, Klaus Leppkes, Peter Korn, and Uwe Naumann. "On the use of discrete adjoints in goal error estimation for shallow water equations." Procedia Computer Science 1, no. 1 (May 2010): 107–15. http://dx.doi.org/10.1016/j.procs.2010.04.013.

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17

Rajan, Dayanand S. "The adjoints to the derivative functor on species." Journal of Combinatorial Theory, Series A 62, no. 1 (January 1993): 93–106. http://dx.doi.org/10.1016/0097-3165(93)90073-h.

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18

Im, Bokhee, and Jonathan D. H. Smith. "Generic adjoints in comtrans algebras of bilinear spaces." Linear Algebra and its Applications 428, no. 4 (February 2008): 953–61. http://dx.doi.org/10.1016/j.laa.2007.08.028.

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19

Hannemann, Ralf, Wolfgang Marquardt, Uwe Naumann, and Boris Gendler. "Discrete first- and second-order adjoints and automatic differentiation for the sensitivity analysis of dynamic models." Procedia Computer Science 1, no. 1 (May 2010): 297–305. http://dx.doi.org/10.1016/j.procs.2010.04.033.

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20

Hannemann, Ralf, and Wolfgang Marquardt. "Continuous and Discrete Composite Adjoints for the Hessian of the Lagrangian in Shooting Algorithms for Dynamic Optimization." SIAM Journal on Scientific Computing 31, no. 6 (January 2010): 4675–95. http://dx.doi.org/10.1137/080714518.

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21

Wang, Qiming, and Zhaojie Zhou. "SUPG-Stabilized Virtual Element Method for Optimal Control Problem Governed by a Convection Dominated Diffusion Equation." Entropy 23, no. 6 (June 5, 2021): 723. http://dx.doi.org/10.3390/e23060723.

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In this paper, the streamline upwind/Petrov Galerkin (SUPG) stabilized virtual element method (VEM) for optimal control problem governed by a convection dominated diffusion equation is investigated. The virtual element discrete scheme is constructed based on the first-optimize-then-discretize strategy and SUPG stabilized virtual element approximation of the state equation and adjoint state equation. An a priori error estimate is derived for both the state, adjoint state, and the control. Numerical experiments are carried out to illustrate the theoretical findings.

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22

Fikl, Alexandru, Vincent Le Chenadec, and Taraneh Sayadi. "Control and Optimization of Interfacial Flows Using Adjoint-Based Techniques." Fluids 5, no. 3 (September 10, 2020): 156. http://dx.doi.org/10.3390/fluids5030156.

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The applicability of adjoint-based gradient computation is investigated in the context of interfacial flows. Emphasis is set on the approximation of the transport of a characteristic function in a potential flow by means of an algebraic volume-of-fluid method. A class of optimisation problems with tracking-type functionals is proposed. Continuous (differentiate-then-discretize) and discrete (discretize-then-differentiate) adjoint-based gradient computations are formulated and compared in a one-dimensional configuration, the latter being ultimately used to perform optimisation in two dimensions. The gradient is used in truncated Newton and steepest descent optimisers, and the algorithms are shown to recover optimal solutions. These validations raise a number of open questions, which are finally discussed with directions for future work.

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23

Hekmat, Mohamad Hamed, and Masoud Mirzaei. "Development of Discrete Adjoint Approach Based on the Lattice Boltzmann Method." Advances in Mechanical Engineering 6 (January 1, 2014): 230854. http://dx.doi.org/10.1155/2014/230854.

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The purpose of this research is to present a general procedure with low implementation cost to develop the discrete adjoint approach for solving optimization problems based on the LB method. Initially, the macroscopic and microscopic discrete adjoint equations and the cost function gradient vector are derived mathematically, in detail, using the discrete LB equation. Meanwhile, for an elementary case, the analytical evaluation of the macroscopic and microscopic adjoint variables and the cost function gradients are presented. The investigation of the derivation procedure shows that the simplicity of the Boltzmann equation, as an alternative for the Navier-Stokes (NS) equations, can facilitate the process of extracting the discrete adjoint equation. Therefore, the implementation of the discrete adjoint equation based on the LB method needs fewer attempts than that of the NS equations. Finally, this approach is validated for the sample test case, and the results gained from the macroscopic and microscopic discrete adjoint equations are compared in an inverse optimization problem. The results show that the convergence rate of the optimization algorithm using both equations is identical and the evaluated gradients have a very good agreement with each other.

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24

Towara, Markus, Michel Schanen, and Uwe Naumann. "MPI-Parallel Discrete Adjoint OpenFOAM." Procedia Computer Science 51 (2015): 19–28. http://dx.doi.org/10.1016/j.procs.2015.05.181.

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25

BERTOLAZZI, ENRICO. "DISCRETE CONSERVATION AND DISCRETE MAXIMUM PRINCIPLE FOR ELLIPTIC PDEs." Mathematical Models and Methods in Applied Sciences 08, no. 04 (June 1998): 685–711. http://dx.doi.org/10.1142/s0218202598000317.

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A class of finite volume numerical schemes for the solution of self-adjoint elliptic equations is described. The main feature of the schemes is that numerical solutions share both discrete conservation and discrete strong maximum principle without restriction on the differential operator or on the volume elements.

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26

Giles, Michael B., Mihai C. Duta, Jens-Dominik Muller, and Niles A. Pierce. "Algorithm Developments for Discrete Adjoint Methods." AIAA Journal 41, no. 2 (February 2003): 198–205. http://dx.doi.org/10.2514/2.1961.

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27

Towara, M., and U. Naumann. "A Discrete Adjoint Model for OpenFOAM." Procedia Computer Science 18 (2013): 429–38. http://dx.doi.org/10.1016/j.procs.2013.05.206.

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28

Xu, Shenren, David Radford, Marcus Meyer, and Jens-Dominik Müller. "Stabilisation of discrete steady adjoint solvers." Journal of Computational Physics 299 (October 2015): 175–95. http://dx.doi.org/10.1016/j.jcp.2015.06.036.

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29

Niwa, Yosuke, Hirofumi Tomita, Masaki Satoh, Ryoichi Imasu, Yousuke Sawa, Kazuhiro Tsuboi, Hidekazu Matsueda, et al. "A 4D-Var inversion system based on the icosahedral grid model (NICAM-TM 4D-Var v1.0) – Part 1: Offline forward and adjoint transport models." Geoscientific Model Development 10, no. 3 (March 17, 2017): 1157–74. http://dx.doi.org/10.5194/gmd-10-1157-2017.

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Abstract. A four-dimensional variational (4D-Var) method is a popular algorithm for inverting atmospheric greenhouse gas (GHG) measurements. In order to meet the computationally intense 4D-Var iterative calculation, offline forward and adjoint transport models are developed based on the Nonhydrostatic ICosahedral Atmospheric Model (NICAM). By introducing flexibility into the temporal resolution of the input meteorological data, the forward model developed in this study is not only computationally efficient, it is also found to nearly match the transport performance of the online model. In a transport simulation of atmospheric carbon dioxide (CO2), the data-thinning error (error resulting from reduction in the time resolution of the meteorological data used to drive the offline transport model) is minimized by employing high temporal resolution data of the vertical diffusion coefficient; with a low 6-hourly temporal resolution, significant concentration biases near the surface are introduced. The new adjoint model can be run in discrete or continuous adjoint mode for the advection process. The discrete adjoint is characterized by perfect adjoint relationship with the forward model that switches off the flux limiter, while the continuous adjoint is characterized by an imperfect but reasonable adjoint relationship with its corresponding forward model. In the latter case, both the forward and adjoint models use the flux limiter to ensure the monotonicity of tracer concentrations and sensitivities. Trajectory analysis for high CO2 concentration events are performed to test adjoint sensitivities. We also demonstrate the potential usefulness of our adjoint model for diagnosing tracer transport. Both the offline forward and adjoint models have computational efficiency about 10 times higher than the online model. A description of our new 4D-Var system that includes an optimization method, along with its application in an atmospheric CO2 inversion and the effects of using either the discrete or continuous adjoint method, is presented in an accompanying paper Niwa et al.(2016).

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30

ZHENG, C. J., H. B. CHEN, T. MATSUMOTO, and T. TAKAHASHI. "3D ACOUSTIC SHAPE SENSITIVITY ANALYSIS USING FAST MULTIPOLE BOUNDARY ELEMENT METHOD." International Journal of Computational Methods 09, no. 01 (March 2012): 1240004. http://dx.doi.org/10.1142/s021987621240004x.

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This paper presents a fast multipole boundary element approach to the 3D acoustic shape sensitivity analysis based on the discrete adjoint variable method. The Burton–Miller formula is used to conquer the fictitious eigenfrequency problem of the original boundary integral equation method in solving exterior acoustic problems. Constant elements are used to discretize the boundary, so that the hypersingular integrals contained in the formulae can be evaluated explicitly and directly. Numerical examples are presented to demonstrate the accuracy and efficiency of the present approach.

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31

Mader, Charles A., Joaquim R. R. A. Martins, Juan J. Alonso, and Edwin van der Weide. "ADjoint: An Approach for the Rapid Development of Discrete Adjoint Solvers." AIAA Journal 46, no. 4 (April 2008): 863–73. http://dx.doi.org/10.2514/1.29123.

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32

ATAKISHIYEV, N. M., and A. U. KLIMYK. "DISCRETE COORDINATE REALIZATIONS OF THE q-OSCILLATOR WHEN q>1." Modern Physics Letters A 21, no. 29 (September 21, 2006): 2205–16. http://dx.doi.org/10.1142/s0217732306021578.

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We elaborate on the Macfarlane–Biedenharn q-oscillator when q>1. In this case the position operator Q = a†+a and the momentum operator P = i(a†-a) are symmetric, but not self-adjoint. For this reason, one cannot specify spectra of Q and P. Since these operators have one-parameter families of self-adjoint extensions with different spectra, the common definition of such q-oscillator is not complete. We derive an action of Q and P (as well as of the related Hamiltonian) upon functions given on the corresponding coordinate spaces, on which Q and P are self-adjoint operators. To each self-adjoint extension of Q there corresponds an appropriate coordinate space (a spectrum of this self-adjoint extension). Thus, for every fixed q>1 one obtains a one-parameter family of non-equivalent q-oscillators in their coordinate spaces.

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33

Hekmat, Mohamad Hamed, and Masoud Mirzaei. "Continuous and Discrete Adjoint Approach Based on Lattice Boltzmann Method in Aerodynamic Optimization Part I: Mathematical Derivation of Adjoint Lattice Boltzmann Equations." Advances in Applied Mathematics and Mechanics 6, no. 5 (October 2014): 570–89. http://dx.doi.org/10.4208/aamm.2013.m226.

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AbstractThe significance of flow optimization utilizing the lattice Boltzmann (LB) method becomes obvious regarding its advantages as a novel flow field solution method compared to the other conventional computational fluid dynamics techniques. These unique characteristics of the LB method form the main idea of its application to optimization problems. In this research, for the first time, both continuous and discrete adjoint equations were extracted based on the LB method using a general procedure with low implementation cost. The proposed approach could be performed similarly for any optimization problem with the corresponding cost function and design variables vector. Moreover, this approach was not limited to flow fields and could be employed for steady as well as unsteady flows. Initially, the continuous and discrete adjoint LB equations and the cost function gradient vector were derived mathematically in detail using the continuous and discrete LB equations in space and time, respectively. Meanwhile, new adjoint concepts in lattice space were introduced. Finally, the analytical evaluation of the adjoint distribution functions and the cost function gradients was carried out.

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34

Kulkarni, Mandar D., David M. Cross, and Robert A. Canfield. "Discrete Adjoint Formulation for Continuum Sensitivity Analysis." AIAA Journal 54, no. 2 (February 2016): 758–66. http://dx.doi.org/10.2514/1.j053827.

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35

Damm, Kyle A., Rowan J. Gollan, Peter A. Jacobs, Michael K. Smart, Seonguk Lee, Eunsa Kim, and Chongam Kim. "Discrete Adjoint Optimization of a Hypersonic Inlet." AIAA Journal 58, no. 6 (June 2020): 2621–34. http://dx.doi.org/10.2514/1.j058913.

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36

Elham, Ali, and Michel J. L. van Tooren. "Discrete adjoint aerodynamic shape optimization using symbolic analysis with OpenFEMflow." Structural and Multidisciplinary Optimization 63, no. 5 (January 27, 2021): 2531–51. http://dx.doi.org/10.1007/s00158-020-02799-7.

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AbstractThe combination of gradient-based optimization with the adjoint method for sensitivity analysis is a very powerful and popular approach for aerodynamic shape optimization. However, differentiating CFD codes is a time consuming and sometimes a challenging task. Although there are a few open-source adjoint CFD codes available, due to the complexity of the code, they might not be very suitable to be used for educational purposes. An adjoint CFD code is developed to support students for learning adjoint aerodynamic shape optimization as well as developing differentiated CFD codes. To achieve this goal, we used symbolic analysis to develop a discrete adjoint CFD code. The least-squares finite element method is used to solve the compressible Euler equations around airfoils in the transonic regime. The symbolic analysis method is used for exact integration to generate the element stiffness and force matrices. The symbolic analysis is also used to compute the exact derivatives of the residuals with respect to both design variables (e.g., the airfoil geometry) and the state variables (e.g., the flow velocity). Besides, the symbolic analysis allows us to compute the exact Jacobian of the governing equations in a computationally efficient way, which is used for Newton iteration. The code includes a build-in gradient-based optimization algorithm and is released as open-source to be available freely for educational purposes.

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37

Ren, Guojing, and Huaqing Sun. "J-Self-Adjoint Extensions for a Class of Discrete Linear Hamiltonian Systems." Abstract and Applied Analysis 2013 (2013): 1–19. http://dx.doi.org/10.1155/2013/904976.

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This paper is concerned with formallyJ-self-adjoint discrete linear Hamiltonian systems on finite or infinite intervals. The minimal and maximal subspaces are characterized, and the defect indices of the minimal subspaces are discussed. All theJ-self-adjoint subspace extensions of the minimal subspace are completely characterized in terms of the square summable solutions and boundary conditions. As a consequence, characterizations of all theJ-self-adjoint subspace extensions are given in the limit point and limit circle cases.

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38

Blower, Gordon, and Andrew McCafferty. "Discrete Tracy–Widom operators." Proceedings of the Edinburgh Mathematical Society 52, no. 3 (September 23, 2009): 545–59. http://dx.doi.org/10.1017/s001309150700140x.

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AbstractIntegrable operators arise in random matrix theory, where they describe the asymptotic eigenvalue distribution of large self-adjoint random matrices from the generalized unitary ensembles. We consider discrete Tracy–Widom operators and give sufficient conditions for a discrete integrable operator to be the square of a Hankel matrix. Examples include the discrete Bessel kernel and kernels arising from the almost Mathieu equation and the Fourier transform of Mathieu's equation.

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39

Sandilya, Ruchi, and Sarvesh Kumar. "Convergence Analysis of Discontinuous Finite Volume Methods for Elliptic Optimal Control Problems." International Journal of Computational Methods 13, no. 02 (March 2016): 1640012. http://dx.doi.org/10.1142/s0219876216400120.

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In this paper, we discuss the convergence analysis of discontinuous finite volume methods applied to distribute the optimal control problems governed by a class of second-order linear elliptic equations. In order to approximate the control, two different methodologies are adopted: one is the method of variational discretization and second is piecewise constant discretization technique. For variational discretization method, optimal order of convergence in the [Formula: see text]-norm for state, adjoint state and control variables is derived. Moreover, optimal order of convergence in discrete [Formula: see text]-norm is also derived for state and adjoint state variables. Whereas, for piecewise constant approximation of control, first order convergence is derived for control, state and adjoint state variables in the [Formula: see text]-norm. In addition to that, optimal order of convergence in discrete [Formula: see text]-norm is derived for state and adjoint state variables. Also, some numerical experiments are conducted to support the derived theoretical convergence rate.

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40

Darwish, A. A. "A boundary value problem with a discontinuous coefficient and containing a spectral parameter in the boundary condition." International Journal of Mathematics and Mathematical Sciences 18, no. 1 (1995): 133–40. http://dx.doi.org/10.1155/s0161171295000172.

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A singular non-self-adjoint boundary value problem is considered. This problem has a discontinuous coefficient with a spectral parameter in the boundary condition. Some solutions of the eigenvalue equation are given. The discrete spectrum is studied and the resolvent is obtained. Formulation of the adjoint problem is deduced and hence the continuous spectrum of the considered problem is given. Furthermore, the spectrum of the adjoint problem is investigated.

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41

Encinas, A. M., and M. J. Jiménez. "Bounded solutions of self-adjoint second order linear difference equations with periodic coeffients." Open Mathematics 16, no. 1 (February 23, 2018): 75–82. http://dx.doi.org/10.1515/math-2018-0007.

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AbstractIn this work we obtain easy characterizations for the boundedness of the solutions of the discrete, self–adjoint, second order and linear unidimensional equations with periodic coefficients, including the analysis of the so-called discrete Mathieu equations as particular cases.

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42

Evans, K. Franklin. "SHDOMPPDA: A Radiative Transfer Model for Cloudy Sky Data Assimilation." Journal of the Atmospheric Sciences 64, no. 11 (November 1, 2007): 3854–64. http://dx.doi.org/10.1175/2006jas2047.1.

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Abstract The spherical harmonics discrete ordinate method for plane-parallel data assimilation (SHDOMPPDA) model is an unpolarized plane-parallel radiative transfer forward model, with corresponding tangent linear and adjoint models, suitable for use in assimilating cloudy sky visible and infrared radiances. It is derived from the spherical harmonics discrete ordinate method plane-parallel (SHDOMPP, also described in this article) version of the spherical harmonics discrete ordinate method (SHDOM) model for three-dimensional atmospheric radiative transfer. The inputs to the SHDOMPPDA forward model are profiles of pressure, temperature, water vapor, and mass mixing ratio and number concentration for a number of hydrometeor species. Hydrometeor optical properties, including detailed phase functions, are determined from lookup tables as a function of mass mean radius. The SHDOMPP and SHDOMPPDA algorithms and construction of the tangent-linear and adjoint models are described. The SHDOMPPDA forward model is validated against the Discrete Ordinate Radiative Transfer Model (DISORT) by comparing upwelling radiances in multiple directions from 100 cloud model columns at visible and midinfrared wavelengths. For this test in optically thick clouds the computational time for SHDOMPPDA is comparable to DISORT for visible reflection, and roughly 5 times faster for thermal emission. The tangent linear and adjoint models are validated by comparison to finite differencing of the forward model.

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ALVAREZ-VÁZQUEZ, L. J., A. MARTÍNEZ, R. MUÑOZ-SOLA, C. RODRÍGUEZ, and M. E. VÁZQUEZ-MÉNDEZ. "THE WATER CONVEYANCE PROBLEM: OPTIMAL PURIFICATION OF POLLUTED WATERS." Mathematical Models and Methods in Applied Sciences 15, no. 09 (September 2005): 1393–416. http://dx.doi.org/10.1142/s0218202505000790.

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In this work we deal with the optimal purification of polluted areas of shallow waters by means of the injection of clear water in order to promote seawater exchange. This problem can be formulated as a control constrained optimal control problem where the control is the velocity of the injected water, the state equations are the shallow water equations together with that modelling the contaminant concentration, and the cost function measures the total amount of injected water and the fulfilment of the water quality standards. We analyze the solutions of the optimal control problem and give an optimality condition in order to characterize them. We also discretize the problem by means of a characteristics-mixed finite element method, focusing our attention on both the discrete and the discretized adjoint systems, and propose an algorithm for the numerical resolution of the discrete optimization problem. Finally, we present numerical results for some computational experiments.

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Biava, Massimo, Mark Woodgate, and George N. Barakos. "Fully Implicit Discrete-Adjoint Methods for Rotorcraft Applications." AIAA Journal 54, no. 2 (February 2016): 735–49. http://dx.doi.org/10.2514/1.j054006.

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Ren, Guojing, and Yuming Shi. "Self-adjoint extensions for discrete linear Hamiltonian systems." Linear Algebra and its Applications 454 (August 2014): 1–48. http://dx.doi.org/10.1016/j.laa.2014.04.016.

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Milatovic, Ognjen, and Françoise Truc. "Self-Adjoint Extensions of Discrete Magnetic Schrödinger Operators." Annales Henri Poincaré 15, no. 5 (June 1, 2013): 917–36. http://dx.doi.org/10.1007/s00023-013-0261-9.

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Jones, Dominic, Jens-Dominik Müller, and Faidon Christakopoulos. "Preparation and assembly of discrete adjoint CFD codes." Computers & Fluids 46, no. 1 (July 2011): 282–86. http://dx.doi.org/10.1016/j.compfluid.2011.01.042.

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Fleischli, Benno, Luca Mangani, Armando Del Rio, and Ernesto Casartelli. "A discrete adjoint method for pressure-based algorithms." Computers & Fluids 227 (September 2021): 105037. http://dx.doi.org/10.1016/j.compfluid.2021.105037.

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Belikov, D. A., S. Maksyutov, A. Yaremchuk, A. Ganshin, T. Kaminski, S. Blessing, M. Sasakawa, and A. Starchenko. "Adjoint of the Global Eulerian–Lagrangian Coupled Atmospheric transport model (A-GELCA v1.0): development and validation." Geoscientific Model Development Discussions 8, no. 7 (July 28, 2015): 5983–6019. http://dx.doi.org/10.5194/gmdd-8-5983-2015.

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Abstract. We present the development of the Adjoint of the Global Eulerian–Lagrangian Coupled Atmospheric (A-GELCA) model that consists of the National Institute for Environmental Studies (NIES) model as an Eulerian three-dimensional transport model (TM), and FLEXPART (FLEXible PARTicle dispersion model) as the Lagrangian plume diffusion model (LPDM). The tangent and adjoint components of the Eulerian model were constructed directly from the original NIES TM code using an automatic differentiation tool known as TAF (Transformation of Algorithms in Fortran; http://www.FastOpt.com), with additional manual pre- and post-processing aimed at improving the performance of the computing, including MPI (Message Passing Interface). As results, the adjoint of Eulerian model is discrete. Construction of the adjoint of the Lagrangian component did not require any code modification, as LPDMs are able to track a significant number of particles back in time and thereby calculate the sensitivity of observations to the neighboring emissions areas. Eulerian and Lagrangian adjoint components were coupled at the time boundary in the global domain.The results are verified using a series of test experiments. The forward simulation shown the coupled model is effective in reproducing the seasonal cycle and short-term variability of CO2 even in the case of multiple limiting factors, such as high uncertainty of fluxes and the low resolution of the Eulerian model. The adjoint model demonstrates the high accuracy compared to direct forward sensitivity calculations and fast performance. The developed adjoint of the coupled model combines the flux conservation and stability of an Eulerian discrete adjoint formulation with the flexibility, accuracy, and high resolution of a Lagrangian backward trajectory formulation.

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Vitale, Salvatore, Tim A. Albring, Matteo Pini, Nicolas R. Gauger, and Piero Colonna. "Fully turbulent discrete adjoint solver for non-ideal compressible flow applications." Journal of the Global Power and Propulsion Society 1 (November 22, 2017): Z1FVOI. http://dx.doi.org/10.22261/jgpps.z1fvoi.

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Abstract Non-Ideal Compressible Fluid-Dynamics (NICFD) has recently been established as a sector of fluid mechanics dealing with the flows of dense vapors, supercritical fluids, and two-phase fluids, whose properties significantly depart from those of the ideal gas. The flow through an Organic Rankine Cycle (ORC) turbine is an exemplary application, as stators often operate in the supersonic and transonic regime, and are affected by NICFD effects. Other applications are turbomachinery using supercritical CO2 as working fluid or other fluids typical of the oil and gas industry, and components of air conditioning and refrigeration systems. Due to the comparably lower level of experience in the design of this fluid machinery, and the lack of experimental information on NICFD flows, the design of the main components of these processes (i.e., turbomachinery and nozzles) may benefit from adjoint-based automated fluid-dynamic shape optimization. Hence, this work is related to the development and testing of a fully-turbulent adjoint method capable of treating NICFD flows. The method was implemented within the SU2 open-source software infrastructure. The adjoint solver was obtained by linearizing the discretized flow equations and the fluid thermodynamic models by means of advanced Automatic Differentiation (AD) techniques. The new adjoint solver was tested on exemplary turbomachinery cases. Results demonstrate the method effectiveness in improving simulated fluid-dynamic performance, and underline the importance of accurately modeling non-ideal thermodynamic and viscous effects when optimizing internal flows influenced by NICFD phenomena.

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