Статті в журналах: "Adjoint discret"

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

Ni, Angxiu. "Backpropagation in hyperbolic chaos via adjoint shadowing." Nonlinearity 37, no. 3 (January 30, 2024): 035009. http://dx.doi.org/10.1088/1361-6544/ad1aed.

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Abstract To generalise the backpropagation method to both discrete-time and continuous-time hyperbolic chaos, we introduce the adjoint shadowing operator S acting on covector fields. We show that S can be equivalently defined as: S is the adjoint of the linear shadowing operator S; S is given by a ‘split then propagate’ expansion formula; S ( ω ) is the only bounded inhomogeneous adjoint solution of ω. By (a), S adjointly expresses the shadowing contribution, a significant part of the linear response, where the linear response is the derivative of the long-time statistics with respect to system parameters. By (b), S also expresses the other part of the linear response, the unstable contribution. By (c), S can be efficiently computed by the nonintrusive shadowing algorithm in Ni and Talnikar (2019 J. Comput. Phys. 395 690–709), which is similar to the conventional backpropagation algorithm. For continuous-time cases, we additionally show that the linear response admits a well-defined decomposition into shadowing and unstable contributions.

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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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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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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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Wu, Hangkong, Xuanlong Da, Dingxi Wang, and Xiuquan Huang. "Multi-Row Turbomachinery Aerodynamic Design Optimization by an Efficient and Accurate Discrete Adjoint Solver." Aerospace 10, no. 2 (January 21, 2023): 106. http://dx.doi.org/10.3390/aerospace10020106.

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This paper proposes an approach that combines manual differentiation (MD) and automatic differentiation (AD) to develop an efficient and accurate multi-row discrete adjoint solver. In this approach, the structures of adjoint codes generated using an AD tool are first analyzed. Then, the AD-generated codes are manually adjusted to reduce memory and CPU time consumption. This manual adjustment is performed by replacing the automatically generated low-efficient differentiated codes with manually developed ones. To demonstrate the effectiveness of the proposed approach, the single-stage transonic compressor–NASA Stage 35 and the 1.5-stage Aachen turbine–are used. The solution information exchange at a rotor-stator/stator-rotor interface is achieved by a conservative, non-reflective, and robust discrete adjoint mixing plane method. The results show that the discrete adjoint solver developed by hybrid automatic and manual differentiation is more economical in computational cost than that developed purely by an AD tool and has higher sensitivity accuracy than the adjoint solver with the constant eddy viscosity (CEV) assumption. Moreover, the multi-row turbomachinery design optimizations can be efficiently performed by the discrete adjoint solver developed by the hybrid automatic and manual differentiation.

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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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8

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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Agarwal, Ravi P., Safi S. Rabie та Samir H. Saker. "On Discrete Weighted Lorentz Spaces and Equivalent Relations between Discrete ℓp-Classes". Fractal and Fractional 7, № 3 (14 березня 2023): 261. http://dx.doi.org/10.3390/fractalfract7030261.

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In this paper, we study some relations between different weights in the classes Bp,Bp*,Mp and Mp* that characterize the boundedness of the Hardy operator and the adjoint Hardy operator. We also prove that these classes generate the same weighted Lorentz space Λp. These results will be proven by using the properties of classes Bp,Bp*,Mp and Mp*, including the self-improving properties and also the properties of the generalized Hardy operator Hp, the adjoint operator Sq and some fundamental relations between them connecting their composition to their sum.

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Cao, Junying, Zhongqing Wang, and Ziqiang Wang. "A Uniform Accuracy High-Order Finite Difference and FEM for Optimal Problem Governed by Time-Fractional Diffusion Equation." Fractal and Fractional 6, no. 9 (August 28, 2022): 475. http://dx.doi.org/10.3390/fractalfract6090475.

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In this paper, the time fractional diffusion equations optimal control problem is solved by 3−α order with uniform accuracy scheme in time and finite element method (FEM) in space. For the state and adjoint state equation, the piecewise linear polynomials are used to make the space variables discrete, and obtain the semidiscrete scheme of the state and adjoint state. The priori error estimates for the semidiscrete scheme for state and adjoint state equation are established. Furthermore, the 3−α order uniform accuracy scheme is used to make the time variable discrete in the semidiscrete scheme and construct the full discrete scheme for the control problems based on the first optimal condition and ‘first optimize, then discretize’ approach. The fully discrete scheme’s stability and truncation error are analyzed. Finally, two numerical examples are denoted to show that the theoretical analysis are correct.

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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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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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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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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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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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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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17

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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18

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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19

Esquível, Manuel L., Nadezhda P. Krasii, and Philippe L. Didier. "On a Schrödinger Equation in the Complex Space Variable." AppliedMath 4, no. 4 (December 19, 2024): 1555–87. https://doi.org/10.3390/appliedmath4040083.

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We study a separable Hilbert space of smooth curves taking values in the Segal–Bergmann space of analytic functions in the complex plane, and two of its subspaces that are the domains of unbounded non self-adjoint linear partial differential operators of the first and second order. We show how to build a Hilbert basis for this space. We study these first- and second-order partial derivation non-self-adjoint operators defined on this space, showing that these operators are defined on dense subspaces of the initial space of smooth curves; we determine their respective adjoints, compute their respective commutators, determine their eigenvalues and, under some normalisation conditions on the eigenvectors, we present examples of a discrete set of eigenvalues. Using these derivation operators, we study a Schrödinger-type equation, building particular solutions given by their representation as smooth curves on the Segal–Bergmann space, and we show the existence of general solutions using an Fourier–Hilbert base of the space of smooth curves. We point out the existence of self-adjoint operators in the space of smooth curves that are obtained by the composition of the partial derivation operators with multiplication operators, showing that these operators admit simple sequences of eigenvalues and eigenvectors. We present two applications of the Schrödinger-type equation studied. In the first one, we consider a wave associated with an object having the mass of an electron, showing that two waves, when considered as having only a free real space variable, are entangled, in the sense that the probability densities in the real variable are almost perfectly correlated. In the second application, after postulating that a usual package of information may have a mass of the order of magnitude of the neutron’s mass attributed to it—and so well into the domain of possible quantisation—we explore some consequences of the model.

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20

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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21

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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22

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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Yang, L., and J. Yang. "Gradient-based aerodynamic shape optimization using a discrete adjoint approach on a graphics processing unit." Journal of Physics: Conference Series 2784, no. 1 (June 1, 2024): 012008. http://dx.doi.org/10.1088/1742-6596/2784/1/012008.

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Abstract This paper presents a gradient-based aerodynamic shape optimization framework that utilizes a Graphics Processing Unit (GPU) for solving both flow and adjoint equations. It is built based on a GPU-accelerated flow solver that has been developed previously. Hence, the focus of this work is on how to solve the adjoint equations on the GPU and subsequently compute the gradients. The adjoint equations are right-preconditioned by a block Incomplete Lower Upper (ILU) preconditioner and solved by a restarted Generalized Minimum Residual (GMRES) method. The exact residual Jacobian matrix in the adjoint equations is computed using finite difference and a distance-2 graph coloring algorithm. With the adjoint-based gradients, the steepest descent method with momentum is employed for constrained aerodynamic shape optimization of a wing-body configuration at a transonic flow condition.

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24

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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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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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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27

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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Popa, Ioan-Lucian, Traian Ceaușu, Larisa Elena Biriș, Tongxing Li, and Akbar Zada. "GENERALIZED EXPONENTIALLY STABLE LINEAR TIME-VARYING DISCRETE BEHAVIORS." Annals of the Academy of Romanian Scientists Series on Mathematics and Its Application 12, no. 1-2 (2020): 256–73. http://dx.doi.org/10.56082/annalsarscimath.2020.1-2.256.

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This paper presents a new approach to formulating exponential behaviors like stability/instability for the linear time-varying systems and for the adjoint one. The classical concept of uniform exponential stability is generalized. Using this generalized concepts, some results extending existing uniform exponential stability conditions for linear time-varying systems are derived. As special cases for these results, some conditions are derived for the adjoint system. A characterization of the generalized concepts in terms of Lyapunov sequences is also given. Also, an example is included to further illustrate the connection with the classical concept of uniform exponential stability.

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29

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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30

Henze, D. K., and J. H. Seinfeld. "Development of the adjoint of GEOS-Chem." Atmospheric Chemistry and Physics Discussions 6, no. 5 (October 19, 2006): 10591–648. http://dx.doi.org/10.5194/acpd-6-10591-2006.

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Abstract. We present the adjoint of the global chemical transport model GEOS-Chem, focusing on the chemical and thermodynamic relationships between sulfate – ammonium – nitrate aerosols and their gas-phase precursors. The adjoint model is constructed from a combination of manually and automatically derived discrete adjoint algorithms and numerical solutions to continuous adjoint equations. Explicit inclusion of the processes that govern secondary formation of inorganic aerosol is shown to afford efficient calculation of model sensitivities such as the dependence of sulfate and nitrate aerosol concentrations on emissions of SOx, NOx, and NH3. The adjoint model is extensively validated by comparing adjoint to finite difference sensitivities, which are shown to agree within acceptable tolerances; most sets of comparisons have a nearly 1:1 correlation and R2>0.9. We explore the robustness of these results, noting how insufficient observations or nonlinearities in the advection routine can degrade the adjoint model performance. The potential for inverse modeling using the adjoint of GEOS-Chem is assessed in a data assimilation framework through a series of tests using simulated observations, demonstrating the feasibility of exploiting gas- and aerosol-phase measurements for optimizing emission inventories of aerosol precursors.

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31

Henze, D. K., A. Hakami, and J. H. Seinfeld. "Development of the adjoint of GEOS-Chem." Atmospheric Chemistry and Physics 7, no. 9 (May 11, 2007): 2413–33. http://dx.doi.org/10.5194/acp-7-2413-2007.

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Abstract. We present the adjoint of the global chemical transport model GEOS-Chem, focusing on the chemical and thermodynamic relationships between sulfate – ammonium – nitrate aerosols and their gas-phase precursors. The adjoint model is constructed from a combination of manually and automatically derived discrete adjoint algorithms and numerical solutions to continuous adjoint equations. Explicit inclusion of the processes that govern secondary formation of inorganic aerosol is shown to afford efficient calculation of model sensitivities such as the dependence of sulfate and nitrate aerosol concentrations on emissions of SOx, NOx, and NH3. The accuracy of the adjoint model is extensively verified by comparing adjoint to finite difference sensitivities, which are shown to agree within acceptable tolerances. We explore the robustness of these results, noting how discontinuities in the advection routine hinder, but do not entirely preclude, the use of such comparisons for validation of the adjoint model. The potential for inverse modeling using the adjoint of GEOS-Chem is assessed in a data assimilation framework using simulated observations, demonstrating the feasibility of exploiting gas- and aerosol-phase measurements for optimizing emission inventories of aerosol precursors.

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32

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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33

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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34

Zhao, Jia, Guoliang Shi та Jun Yan. "Discreteness of spectrum for Schrödinger operators with δʹ-type conditions on infinite regular trees". Proceedings of the Royal Society of Edinburgh: Section A Mathematics 147, № 5 (14 серпня 2017): 1091–117. http://dx.doi.org/10.1017/s030821051600041x.

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This paper deals with the spectral properties of self-adjoint Schrödinger operators with δʹ-type conditions on infinite regular trees. Firstly, we discuss the semi-boundedness and self-adjointness of this kind of Schrödinger operator. Secondly, by using the form approach, we give the necessary and sufficient condition that ensures that the spectra of the self-adjoint Schrödinger operators with δʹ-type conditions are discrete.

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35

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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36

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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37

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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38

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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39

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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40

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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41

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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42

Luo, Songting, Shingyu Leung, and Jianliang Qian. "An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria." Communications in Computational Physics 10, no. 5 (November 2011): 1113–31. http://dx.doi.org/10.4208/cicp.020210.311210a.

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AbstractThe equilibrium metric for minimizing a continuous congested traffic model is the solution of a variational problem involving geodesic distances. The continuous equilibrium metric and its associated variational problem are closely related to the classical discrete Wardrop’s equilibrium. We propose an adjoint state method to numerically approximate continuous traffic congestion equilibria through the continuous formulation. The method formally derives an adjoint state equation to compute the gradient descent direction so as to minimize a nonlinear functional involving the equilibrium metric and the resulting geodesic distances. The geodesic distance needed for the state equation is computed by solving a factored eikonal equation, and the adjoint state equation is solved by a fast sweeping method. Numerical examples demonstrate that the proposed adjoint state method produces desired equilibrium metrics and outperforms the subgradient marching method for computing such equilibrium metrics.

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43

Ancourt, Kevin, Jacques Peter, and Olivier Atinault. "Adjoint and Direct Characteristic Equations for Two-Dimensional Compressible Euler Flows." Aerospace 10, no. 9 (September 12, 2023): 797. http://dx.doi.org/10.3390/aerospace10090797.

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The method of characteristics is a classical method for gaining understanding in the solution of a partial differential equation. It has recently been applied to the adjoint equations of the 2D steady-state Euler equations and the first goal of this paper is to present a linear algebra analysis that greatly simplifies the discussion of the number of independent characteristic equations satisfied along a family of characteristic curves. This method may be applied for both the direct and the adjoint problem. Our second goal is to directly derive in conservative variables the characteristic equations of 2D compressible inviscid flows. Finally, the theoretical results are assessed for a nozzle flow with a classical scheme and its dual consistent discrete adjoint.

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44

Okello, Boaz, Fredrick Nyamwala, and David Ambogo. "Spectral theory of commutative higher order difference operators with unbounded coefficients." Annals of Mathematics and Computer Science 26 (January 5, 2025): 1–27. https://doi.org/10.56947/amcs.v26.415.

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We have established the necessary and sufficient conditions for any two even higher order symmetric difference maps to generate commuting minimal difference operators. We have done this through construction of appropriate comparison algebras of the self-adjoint operator extensions of the minimal operators generated and application of asymptotic summation. The results show that if the first difference on the coefficients tends to zero whenever the coefficients are allowed to be unbounded and that the difference maps considered have the same order, then they generate minimal operators that commute and the corresponding self-adjoint operators commute too. We have further shown that the self-adjoint operator extensions of the respective minimal operators can be expressed as the composite of the independent self-adjoint operator extensions if the generated minimal difference operators have closed ranges. Finally, we have shown that the spectra of these self-adjoint operator extensions are the whole of the real line if the coefficients are unbounded. These results therefore, extend the existing results in the continuous case to discrete setting.

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45

Jun, Deng, Gao Zhenghong, Huang Jiangtao, Zhao Ke, and Xia Lu. "Design method of aircraft boundary characteristics based on upwind adjoint equation." Journal of Applied Artificial Intelligence 1, no. 3 (October 18, 2024): 192–206. http://dx.doi.org/10.59782/aai.v1i3.324.

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The boundary characteristics of an aircraft determine its safety and flight performance, and have always been the difficulty and focus of aircraft design. This paper aims to improve the accuracy, efficiency and robustness of the adjoint equation solution for complex flow problems by introducing the adjoint equation of the upwind format and different flux limiter processing forms, thereby expanding the application scope of the adjoint optimization method in the design of the boundary characteristics of aircraft. Firstly, the basic principle of discrete adjoint gradient solution is introduced. On this basis, the inviscid term of the adjoint equation and the variational form of its boundary conditions are derived. According to the processing method of the flux limiter, the adjoint equation with first-order accuracy, second-order accuracy and mixed accuracy is formed. Then, the boundary processing measures of the adjoint equation are studied. Through the ONERA M6 wing gradient accuracy and robustness verification example, the solution performance of the adjoint equation of the upwind format and the center format is compared, and the influence of the limiter and boundary processing measures on the convergence and gradient accuracy of the adjoint equation is analyzed. Through the cruise aerodynamic optimization design and boundary characteristics optimization design examples of the CRM wing-body assembly, the effectiveness of the solver for the cruise performance design and boundary characteristics design of the aircraft is verified. The calculation and design results show that the solution method of the upwind adjoint equation established in this paper is robust, has high gradient accuracy, and can be applied to solving the difficult problems of aircraft boundary characteristic design.

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46

Ma, Chuang, Jiangtao Huang, Daochun Li, Jun Deng, Gang Liu, Lin Zhou, and Cheng Chen. "Discrete Adjoint Optimization Method for Low-Boom Aircraft Design Using Equivalent Area Distribution." Aerospace 11, no. 7 (July 3, 2024): 545. http://dx.doi.org/10.3390/aerospace11070545.

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This paper introduces a low-boom aircraft optimization design method guided by equivalent area distribution, which effectively improves the intuitiveness and refinement of inverse design. A gradient optimization method based on discrete adjoint equations is proposed to achieve the fast solution of the gradient information of target equivalent area distribution relative to design variables and to drive the aerodynamic shape update to the optimal solution. An optimization experiment is carried out based on a self-developed supersonic civil aircraft configuration with engines. The results show that the equivalent area distribution adjoint equation can accurately solve the gradient information. After optimization, the sonic boom level of the aircraft was reduced by 13.2 PLdB, and the drag coefficient was reduced by 60.75 counts. Moreover, the equivalent area distribution adjoint optimization method has outstanding advantages, such as high sensitivity and fast convergence speed, and can take both the low sonic boom and the low drag force of the aircraft into account, providing a powerful tool for the comprehensive optimization design of supersonic civil aircraft by considering sonic boom and aerodynamic force.

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47

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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48

Kadchenko, S. I. "NUMERICAL METHOD FOR SOLVING INVERSE SPECTRAL PROBLEMS GENERATED BY PERTURBED SELF-ADJOINT OPERATORS." Vestnik of Samara University. Natural Science Series 19, no. 9.1 (June 5, 2017): 5–11. http://dx.doi.org/10.18287/2541-7525-2013-19-9.1-5-11.

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A new numerical method for solving inverse spectral problems generated by perturbed self-adjoint operators from their spectral characteristics is developed. The method was tested on the problems for perturbed Sturm - Liouville operator. The results of numerous calculations have shown its computational efficiency. The simple algebraic formulas for finding the eigenvalues of discrete operators was found. At that the calculation of eigenvalues of perturbed self-adjoint operator can start from any number, no matter known the eigenvalues from previous numbers or not.

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49

BIRIŞ, LARISA ELENA, TRAIAN CEAUŞU, IOAN-LUCIAN POPA, and "NICOLAE MARIAN" SEIMEANU. ""Lyapunov Conditions for One-Sided Discrete-Time Random Dynamical Systems"." Carpathian Journal of Mathematics 38, no. 3 (July 26, 2022): 777–88. http://dx.doi.org/10.37193/cjm.2022.03.21.

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"This paper considers nonuniform exponential stability and nonuniform exponential instability concepts for one-sided discrete-time random dynamical systems. These concepts are generalizations from the deterministic case. Using this, characterizations in terms of Lyapunov functions respectively Lyapunov norms are presented. Also, an approach in terms of considered concepts for the inverse and adjoint random discrete- time systems is derived."

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50

Zhang, Lixin, Jean W. Zu, and Zhichao Hou. "Complex Modal Analysis of Non-Self-Adjoint Hybrid Serpentine Belt Drive Systems." Journal of Vibration and Acoustics 123, no. 2 (November 1, 2000): 150–56. http://dx.doi.org/10.1115/1.1356697.

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A linear damped hybrid (continuous/discrete components) model is developed in this paper to characterize the dynamic behavior of serpentine belt drive systems. Both internal material damping and external tensioner arm damping are considered. The complex modal analysis method is developed to perform dynamic analysis of linear non-self-adjoint hybrid serpentine belt-drive systems. The adjoint eigenfunctions are acquired in terms of the mode shapes of an auxiliary hybrid system. The closed-form characteristic equation of eigenvalues and the exact closed-form solution for dynamic response of the non-self-adjoint hybrid model are obtained. Numerical simulations are performed to demonstrate the method of analysis. It is shown that there exists an optimum damping value for each vibration mode at which vibration decays the fastest.

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