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1
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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Hascoët, L., J. Utke, and U. Naumann. "Cheaper Adjoints by Reversing Address Computations." Scientific Programming 16, no. 1 (2008): 81–92. http://dx.doi.org/10.1155/2008/375243.
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The reverse mode of automatic differentiation is widely used in science and engineering. A severe bottleneck for the performance of the reverse mode, however, is the necessity to recover certain intermediate values of the program in reverse order. Among these values are computed addresses, which traditionally are recovered through forward recomputation and storage in memory. We propose an alternative approach for recovery that uses inverse computation based on dependency information. Address storage constitutes a significant portion of the overall storage requirements. An example illustrates substantial gains that the proposed approach yields, and we show use cases in practical applications.
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BISCHOF, CHRISTIAN H., H. MARTIN BÜCKER, and PO-TING WU. "TIME-PARALLEL COMPUTATION OF PSEUDO-ADJOINTS FOR A LEAPFROG SCHEME." International Journal of High Speed Computing 12, no. 01 (June 2004): 1–27. http://dx.doi.org/10.1142/s0129053304000219.
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Kohn, Kathlén, and Kristian Ranestad. "Projective Geometry of Wachspress Coordinates." Foundations of Computational Mathematics 20, no. 5 (November 11, 2019): 1135–73. http://dx.doi.org/10.1007/s10208-019-09441-z.
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Abstract We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress (A rational finite element basis, Academic Press, New York, 1975). The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren (Adv Comput Math 6:97–108, 1996). This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic.
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Cacuci, Dan Gabriel. "First-Order Comprehensive Adjoint Method for Computing Operator-Valued Response Sensitivities to Imprecisely Known Parameters, Internal Interfaces and Boundaries of Coupled Nonlinear Systems: II. Application to a Nuclear Reactor Heat Removal Benchmark." Journal of Nuclear Engineering 1, no. 1 (September 9, 2020): 18–45. http://dx.doi.org/10.3390/jne1010003.
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This work illustrates the application of a comprehensive first-order adjoint sensitivity analysis methodology (1st-CASAM) to a heat conduction and convection analytical benchmark problem which simulates heat removal from a nuclear reactor fuel rod. This analytical benchmark problem can be used to verify the accuracy of numerical solutions provided by software modeling heat transport and fluid flow systems. This illustrative heat transport benchmark shows that collocation methods require one adjoint computation for every collocation point while spectral expansion methods require one adjoint computation for each cardinal function appearing in the respective expansion when recursion relations cannot be developed between the corresponding adjoint functions. However, it is also shown that spectral methods are much more efficient when recursion relations provided by orthogonal polynomials make it possible to develop recursion relations for computing the corresponding adjoint functions. When recursion relations cannot be developed for the adjoint functions, the collocation method is probably more efficient than the spectral expansion method, since the sources for the corresponding adjoint systems are just Dirac delta functions (which makes the respective computation equivalent to the computation of a Green’s function), rather than the more elaborated sources involving high-order Fourier basis functions or orthogonal polynomials. For systems involving many independent variables, it is likely that a hybrid combination of spectral expansions in some independent variables and collocation in the remaining independent variables would provide the most efficient computational outcome.
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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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Ito, Shin-ichi, Takeru Matsuda, and Yuto Miyatake. "Adjoint-based exact Hessian computation." BIT Numerical Mathematics 61, no. 2 (February 17, 2021): 503–22. http://dx.doi.org/10.1007/s10543-020-00833-0.
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AbstractWe consider a scalar function depending on a numerical solution of an initial value problem, and its second-derivative (Hessian) matrix for the initial value. The need to extract the information of the Hessian or to solve a linear system having the Hessian as a coefficient matrix arises in many research fields such as optimization, Bayesian estimation, and uncertainty quantification. From the perspective of memory efficiency, these tasks often employ a Krylov subspace method that does not need to hold the Hessian matrix explicitly and only requires computing the multiplication of the Hessian and a given vector. One of the ways to obtain an approximation of such Hessian-vector multiplication is to integrate the so-called second-order adjoint system numerically. However, the error in the approximation could be significant even if the numerical integration to the second-order adjoint system is sufficiently accurate. This paper presents a novel algorithm that computes the intended Hessian-vector multiplication exactly and efficiently. For this aim, we give a new concise derivation of the second-order adjoint system and show that the intended multiplication can be computed exactly by applying a particular numerical method to the second-order adjoint system. In the discussion, symplectic partitioned Runge–Kutta methods play an essential role.
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Boehm, Christian, Mauricio Hanzich, Josep de la Puente, and Andreas Fichtner. "Wavefield compression for adjoint methods in full-waveform inversion." GEOPHYSICS 81, no. 6 (November 2016): R385—R397. http://dx.doi.org/10.1190/geo2015-0653.1.
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Adjoint methods are a key ingredient of gradient-based full-waveform inversion schemes. While being conceptually elegant, they face the challenge of massive memory requirements caused by the opposite time directions of forward and adjoint simulations and the necessity to access both wavefields simultaneously for the computation of the sensitivity kernel. To overcome this bottleneck, we have developed lossy compression techniques that significantly reduce the memory requirements with only a small computational overhead. Our approach is tailored to adjoint methods and uses the fact that the computation of a sufficiently accurate sensitivity kernel does not require the fully resolved forward wavefield. The collection of methods comprises reinterpolation with a coarse temporal grid as well as adaptively chosen polynomial degree and floating-point precision to represent spatial snapshots of the forward wavefield on hierarchical grids. Furthermore, the first arrivals of adjoint waves are used to identify “shadow zones” that do not contribute to the sensitivity kernel. Numerical experiments show the high potential of this approach achieving an effective compression factor of three orders of magnitude with only a minor reduction in the rate of convergence. Moreover, it is computationally cheap and straightforward to integrate in finite-element wave propagation codes with possible extensions to finite-difference methods.
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Guerrette, J. J., and D. K. Henze. "Development and application of the WRFPLUS-Chem online chemistry adjoint and WRFDA-Chem assimilation system." Geoscientific Model Development Discussions 8, no. 2 (February 27, 2015): 2313–67. http://dx.doi.org/10.5194/gmdd-8-2313-2015.
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Abstract. Here we present the online meteorology and chemistry adjoint and tangent linear model, WRFPLUS-Chem, which incorporates modules to treat boundary layer mixing, emission, aging, dry deposition, and advection of black carbon aerosol. We also develop land surface and surface layer adjoints to account for coupling between radiation and vertical mixing. Model performance is verified against finite difference derivative approximations. A second order checkpointing scheme is created to reduce computational costs and enable simulations longer than six hours. The adjoint is coupled to WRFDA-Chem, in order to conduct a sensitivity study of anthropogenic and biomass burning sources throughout California during the 2008 Arctic Research of the Composition of the Troposphere from Aircraft and Satellites (ARCTAS) field campaign. A cost function weighting scheme was devised to increase adjoint sensitivity robustness in future inverse modeling studies. Results of the sensitivity study show that, for this domain and time period, anthropogenic emissions are over predicted, while wildfire emissions are under predicted. We consider the diurnal variation in emission sensitivities to determine at what time sources should be scaled up or down. Also, adjoint sensitivities for two choices of land surface model indicate that emission inversion results would be sensitive to forward model configuration. The tools described here are the first step in conducting four-dimensional variational data assimilation in a coupled meteorology-chemistry model, which will potentially provide new constraints on aerosol precursor emissions and their distributions. Such analyses will be invaluable to assessments of particulate matter health and climate impacts.
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Akbarzadeh, Siamak, Jan Hückelheim, and Jens-Dominik Müller. "Consistent treatment of incompletely converged iterative linear solvers in reverse-mode algorithmic differentiation." Computational Optimization and Applications 77, no. 2 (August 3, 2020): 597–616. http://dx.doi.org/10.1007/s10589-020-00214-x.
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Abstract Algorithmic differentiation (AD) is a widely-used approach to compute derivatives of numerical models. Many numerical models include an iterative process to solve non-linear systems of equations. To improve efficiency and numerical stability, AD is typically not applied to the linear solvers. Instead, the differentiated linear solver call is replaced with hand-produced derivative code that exploits the linearity of the original call. In practice, the iterative linear solvers are often stopped prematurely to recompute the linearisation of the non-linear outer loop. We show that in the reverse-mode of AD, the derivatives obtained with partial convergence become inconsistent with the original and the tangent-linear models, resulting in inaccurate adjoints. We present a correction term that restores consistency between adjoint and tangent-linear gradients if linear systems are only partially converged. We prove the consistency of this correction term and show in numerical experiments that the accuracy of adjoint gradients of an incompressible flow solver applied to an industrial test case is restored when the correction term is used.
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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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Endtmayer, Bernhard, and Thomas Wick. "A Partition-of-Unity Dual-Weighted Residual Approach for Multi-Objective Goal Functional Error Estimation Applied to Elliptic Problems." Computational Methods in Applied Mathematics 17, no. 4 (October 1, 2017): 575–99. http://dx.doi.org/10.1515/cmam-2017-0001.
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AbstractIn this work, we design a posteriori error estimation and mesh adaptivity for multiple goal functionals. Our method is based on a dual-weighted residual approach in which localization is achieved in a variational form using a partition-of-unity. The key advantage is that the method is simple to implement and backward integration by parts is not required. For treating multiple goal functionals we employ the adjoint to the adjoint problem (i.e., a discrete error problem) and suggest an alternative way for its computation. Our algorithmic developments are substantiated for elliptic problems in terms of four different numerical tests that cover various types of challenges, such as singularities, different boundary conditions, and diverse goal functionals. Moreover, several computations with higher-order finite elements are performed.
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Aupy, Guillaume, Julien Herrmann, Paul Hovland, and Yves Robert. "Optimal Multistage Algorithm for Adjoint Computation." SIAM Journal on Scientific Computing 38, no. 3 (January 2016): C232—C255. http://dx.doi.org/10.1137/15m1019222.
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Soci, Cornel, Claude Fischer, and András Horányi. "Sensitivity of High-Resolution Forecasts Using the Adjoint Technique at the 10-km Scale." Monthly Weather Review 134, no. 3 (March 1, 2006): 772–90. http://dx.doi.org/10.1175/mwr3091.1.
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Abstract This paper provides an experimental framework designed to assess the performance and the evolution of the diabatic Aire Limitée Adaptation Dynamique Développement International (ALADIN) adjoint model at 10-km grid size. Numerical experiments are carried out with the goal of evaluating the adjoint model solutions and the benefit of employing a complex linearized physical parameterization package in the gradient computation. Sensitivity studies with respect to initial conditions at high resolution on real meteorological events are performed. Numerical results obtained in the gradient computations show that, at high resolution, a strong nonlinear flow over complex orography might be a potential source of numerical instability in the absence of a robust dissipative physics employed in the adjoint model. Also, the scheme of the linearized large-scale precipitation is a source of noise in precipitating areas. The results on one particular case suggest that on the one hand the adjoint model is able to capture the dynamically sensitive area, but on the other hand the subsequent sensitivity forecast is more sensitive to the sign and the amplitude of the initial state perturbation rather than the structure of the gradient field.
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Hückelheim, Jan, Paul Hovland, Michelle Mills Strout, and Jens-Dominik Müller. "Reverse-mode algorithmic differentiation of an OpenMP-parallel compressible flow solver." International Journal of High Performance Computing Applications 33, no. 1 (June 29, 2017): 140–54. http://dx.doi.org/10.1177/1094342017712060.
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Reverse-mode algorithmic differentiation (AD) is an established method for obtaining adjoint derivatives of computer simulation applications. In computational fluid dynamics (CFD), adjoint derivatives of a cost function output such as drag or lift with respect to design parameters such as surface coordinates or geometry control points are a key ingredient for shape optimization, uncertainty quantification and flow control. The computational cost of CFD applications and their derivatives makes it essential to use high-performance computing hardware efficiently, including multi- and many-core architectures. Nevertheless, OpenMP is not supported in most AD tools, and previously shown methods achieve poor scalability of the derivative code. We present the AD of an OpenMP-parallelized finite volume compressible flow solver for unstructured meshes. Our approach enables us to reuse the parallelization of the original code in the computation of adjoint derivatives. The method works by identifying code segments that can be differentiated in reverse-mode without changing their memory access pattern. The OpenMP parallelization is integrated into the derivative code during the build process in a way that is robust to modifications of the original code and independent of the OpenMP support of the differentiation tool. We show the scalability of our adjoint CFD solver on test cases ranging from thousands to millions of finite volume mesh cells on CPUs with up to 16 threads as well as on an Intel XeonPhi card with 236 threads. We demonstrate that our approach is more practical to implement for production-sized CFD codes and produces more efficient adjoint derivative code than previously shown AD methods.
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Lee, Ciarán M., and Matty J. Hoban. "Bounds on the power of proofs and advice in general physical theories." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2190 (June 2016): 20160076. http://dx.doi.org/10.1098/rspa.2016.0076.
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Quantum theory presents us with the tools for computational and communication advantages over classical theory. One approach to uncovering the source of these advantages is to determine how computation and communication power vary as quantum theory is replaced by other operationally defined theories from a broad framework of such theories. Such investigations may reveal some of the key physical features required for powerful computation and communication. In this paper, we investigate how simple physical principles bound the power of two different computational paradigms which combine computation and communication in a non-trivial fashion: computation with advice and interactive proof systems. We show that the existence of non-trivial dynamics in a theory implies a bound on the power of computation with advice. Moreover, we provide an explicit example of a theory with no non-trivial dynamics in which the power of computation with advice is unbounded. Finally, we show that the power of simple interactive proof systems in theories where local measurements suffice for tomography is non-trivially bounded. This result provides a proof that Q M A is contained in P P , which does not make use of any uniquely quantum structure—such as the fact that observables correspond to self-adjoint operators—and thus may be of independent interest.
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Berry, Thomas, and Matt Visser. "Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions." Physics 3, no. 2 (May 13, 2021): 352–66. http://dx.doi.org/10.3390/physics3020024.
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In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. The Wigner rotation is subsequently related to the generic non-associativity of the composition of three 4-velocities, and a necessary and sufficient condition is developed for the associativity to hold. Finally, the authors relate the composition of 4-velocities to a specific implementation of the Baker–Campbell–Hausdorff theorem. As compared to ordinary 4×4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.
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Guerrette, J. J., and D. K. Henze. "Development and application of the WRFPLUS-Chem online chemistry adjoint and WRFDA-Chem assimilation system." Geoscientific Model Development 8, no. 6 (June 23, 2015): 1857–76. http://dx.doi.org/10.5194/gmd-8-1857-2015.
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Abstract. Here we present the online meteorology and chemistry adjoint and tangent linear model, WRFPLUS-Chem (Weather Research and Forecasting plus chemistry), which incorporates modules to treat boundary layer mixing, emission, aging, dry deposition, and advection of black carbon aerosol. We also develop land surface and surface layer adjoints to account for coupling between radiation and vertical mixing. Model performance is verified against finite difference derivative approximations. A second-order checkpointing scheme is created to reduce computational costs and enable simulations longer than 6 h. The adjoint is coupled to WRFDA-Chem, in order to conduct a sensitivity study of anthropogenic and biomass burning sources throughout California during the 2008 Arctic Research of the Composition of the Troposphere from Aircraft and Satellites (ARCTAS) field campaign. A cost-function weighting scheme was devised to reduce the impact of statistically insignificant residual errors in future inverse modeling studies. Results of the sensitivity study show that, for this domain and time period, anthropogenic emissions are overpredicted, while wildfire emission error signs vary spatially. We consider the diurnal variation in emission sensitivities to determine at what time sources should be scaled up or down. Also, adjoint sensitivities for two choices of land surface model (LSM) indicate that emission inversion results would be sensitive to forward model configuration. The tools described here are the first step in conducting four-dimensional variational data assimilation in a coupled meteorology–chemistry model, which will potentially provide new constraints on aerosol precursor emissions and their distributions. Such analyses will be invaluable to assessments of particulate matter health and climate impacts.
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Cacuci. "Towards Overcoming the Curse of Dimensionality: The Third-Order Adjoint Method for Sensitivity Analysis of Response-Coupled Linear Forward/Adjoint Systems, with Applications to Uncertainty Quantification and Predictive Modeling." Energies 12, no. 21 (November 5, 2019): 4216. http://dx.doi.org/10.3390/en12214216.
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This work presents the Third-Order Adjoint Sensitivity Analysis Methodology (3rd-ASAM) for response-coupled forward and adjoint linear systems. The 3rd-ASAM enables the efficient computation of the exact expressions of the 3rd-order functional derivatives (“sensitivities”) of a general system response, which depends on both the forward and adjoint state functions, with respect to all of the parameters underlying the respective forward and adjoint systems. Such responses are often encountered when representing mathematically detector responses and reaction rates in reactor physics problems. The 3rd-ASAM extends the 2nd-ASAM in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling. This work also presents new formulas that incorporate the contributions of the 3rd-order sensitivities into the expressions of the first four cumulants of the response distribution in the phase-space of model parameters. Using these newly developed formulas, this work also presents a new mathematical formalism, called the 2nd/3rd-BERRU-PM “Second/Third-Order Best-Estimated Results with Reduced Uncertainties Predictive Modeling”) formalism, which combines experimental and computational information in the joint phase-space of responses and model parameters, including not only the 1st-order response sensitivities, but also the complete hessian matrix of 2nd-order second-sensitivities and also the 3rd-order sensitivities, all computed using the 3rd-ASAM. The 2nd/3rd-BERRU-PM uses the maximum entropy principle to eliminate the need for introducing and “minimizing” a user-chosen “cost functional quantifying the discrepancies between measurements and computations,” thus yielding results that are free of subjective user-interferences while generalizing and significantly extending the 4D-VAR data assimilation procedures. Incorporating correlations, including those between the imprecisely known model parameters and computed model responses, the 2nd/3rd-BERRU-PM also provides a quantitative metric, constructed from sensitivity and covariance matrices, for determining the degree of agreement among the various computational and experimental data while eliminating discrepant information. The mathematical framework of the 2nd/3rd-BERRU-PM formalism requires the inversion of a single matrix of size Nr Nr, where Nr denotes the number of considered responses. In the overwhelming majority of practical situations, the number of responses is much less than the number of model parameters. Thus, the 2nd-BERRU-PM methodology overcomes the curse of dimensionality which affects the inversion of hessian matrices in the parameter space.
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Beck, Tobias, and Josef Schicho. "Adjoint computation for hypersurfaces using formal desingularizations." Journal of Algebra 320, no. 11 (December 2008): 3984–96. http://dx.doi.org/10.1016/j.jalgebra.2008.08.002.
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Awotunde, A. A. A., and R. N. N. Horne. "An Improved Adjoint-Sensitivity Computation for Multiphase Flow Using Wavelets." SPE Journal 17, no. 02 (February 8, 2012): 402–17. http://dx.doi.org/10.2118/133866-pa.
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Summary In history matching, one of the challenges in the use of gradient-based Newton algorithms (e.g., Gauss-Newton and Leven-berg-Marquardt) in solving the inverse problem is the huge cost associated with the computation of the sensitivity matrix. Although the Newton type of algorithm gives faster convergence than most other gradient-based inverse solution algorithms, its use is limited to small- and medium-scale problems in which the sensitivity coefficients are easily and quickly computed. Modelers often use less-efficient algorithms (e.g., conjugate-gradient and quasi-Newton) to model large-scale problems because these algorithms avoid the direct computation of sensitivity coefficients. To find a direction of descent, such algorithms often use less-precise curvature information that would be contained in the gradient of the objective function. Using a sensitivity matrix gives more-complete information about the curvature of the function; however, this comes with a significant computational cost for large-scale problems. An improved adjoint-sensitivity computation is presented for time-dependent partial-differential equations describing multiphase flow in hydrocarbon reservoirs. The method combines the wavelet parameterization of data space with adjoint-sensitivity formulation to reduce the cost of computing sensitivities. This reduction in cost is achieved by reducing the size of the linear system of equations that are typically solved to obtain the sensitivities. This cost-saving technique makes solving an inverse problem with algorithms (e.g., Levenberg-Marquardt and Gauss-Newton) viable for large multiphase-flow history-matching problems. The effectiveness of this approach is demonstrated for two numerical examples involving multiphase flow in a reservoir with several production and injection wells.
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Boonyasiriwat, Chaiwoot, Paul Valasek, Partha Routh, Weiping Cao, Gerard T. Schuster, and Brian Macy. "An efficient multiscale method for time-domain waveform tomography." GEOPHYSICS 74, no. 6 (November 2009): WCC59—WCC68. http://dx.doi.org/10.1190/1.3151869.
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This efficient multiscale method for time-domain waveform tomography incorporates filters that are more efficient than Hamming-window filters. A strategy for choosing optimal frequency bands is proposed to achieve computational efficiency in the time domain. A staggered-grid, explicit finite-difference method with fourth-order accuracy in space and second-order accuracy in time is used for forward modeling and the adjoint calculation. The adjoint method is utilized in inverting for an efficient computation of the gradient directions. In the multiscale approach, multifrequency data and multiple grid sizes are used to overcome somewhat the severe local minima problem of waveform tomography. The method is applied successfully to 1D and 2D heterogeneous models; it can accurately recover low- and high-wavenumber components of the velocity models. The inversion result for the 2D model demonstrates that the multiscale method is computationally efficient and converges faster than a conventional, single-scale method.
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Liu, Quanhua, and Fuzhong Weng. "Advanced Doubling–Adding Method for Radiative Transfer in Planetary Atmospheres." Journal of the Atmospheric Sciences 63, no. 12 (December 2006): 3459–65. http://dx.doi.org/10.1175/jas3808.1.
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The doubling–adding method (DA) is one of the most accurate tools for detailed multiple-scattering calculations. The principle of the method goes back to the nineteenth century in a problem dealing with reflection and transmission by glass plates. Since then the doubling–adding method has been widely used as a reference tool for other radiative transfer models. The method has never been used in operational applications owing to tremendous demand on computational resources from the model. This study derives an analytical expression replacing the most complicated thermal source terms in the doubling–adding method. The new development is called the advanced doubling–adding (ADA) method. Thanks also to the efficiency of matrix and vector manipulations in FORTRAN 90/95, the advanced doubling–adding method is about 60 times faster than the doubling–adding method. The radiance (i.e., forward) computation code of ADA is easily translated into tangent linear and adjoint codes for radiance gradient calculations. The simplicity in forward and Jacobian computation codes is very useful for operational applications and for the consistency between the forward and adjoint calculations in satellite data assimilation. ADA is implemented into the Community Radiative Transfer Model (CRTM) developed at the U.S. Joint Center for Satellite Data Assimilation.
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Cacuci, Dan Gabriel. "Fourth-Order Comprehensive Adjoint Sensitivity Analysis (4th-CASAM) of Response-Coupled Linear Forward/Adjoint Systems: I. Theoretical Framework." Energies 14, no. 11 (June 6, 2021): 3335. http://dx.doi.org/10.3390/en14113335.
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The most general quantities of interest (called “responses”) produced by the computational model of a linear physical system can depend on both the forward and adjoint state functions that describe the respective system. This work presents the Fourth-Order Comprehensive Adjoint Sensitivity Analysis Methodology (4th-CASAM) for linear systems, which enables the efficient computation of the exact expressions of the 1st-, 2nd-, 3rd- and 4th-order sensitivities of a generic system response, which can depend on both the forward and adjoint state functions, with respect to all of the parameters underlying the respective forward/adjoint systems. Among the best known such system responses are various Lagrangians, including the Schwinger and Roussopoulos functionals, for analyzing ratios of reaction rates, the Rayleigh quotient for analyzing eigenvalues and/or separation constants, etc., which require the simultaneous consideration of both the forward and adjoint systems when computing them and/or their sensitivities (i.e., functional derivatives) with respect to the model parameters. Evidently, such responses encompass, as particular cases, responses that may depend just on the forward or just on the adjoint state functions pertaining to the linear system under consideration. This work also compares the CPU-times needed by the 4th-CASAM versus other deterministic methods (e.g., finite-difference schemes) for computing response sensitivities These comparisons underscore the fact that the 4th-CASAM is the only practically implementable methodology for obtaining and subsequently computing the exact expressions (i.e., free of methodologically-introduced approximations) of the 1st-, 2nd, 3rd- and 4th-order sensitivities (i.e., functional derivatives) of responses to system parameters, for coupled forward/adjoint linear systems. By enabling the practical computation of any and all of the 1st-, 2nd, 3rd- and 4th-order response sensitivities to model parameters, the 4th-CASAM makes it possible to compare the relative values of the sensitivities of various order, in order to assess which sensitivities are important and which may actually be neglected, thus enabling future investigations of the convergence of the (multivariate) Taylor series expansion of the response in terms of parameter variations, as well as investigating the range of validity of other important quantities (e.g., response variances/covariance, skewness, kurtosis, etc.) that are derived from Taylor-expansion of the response as a function of the model’s parameters. The 4th-CASAM presented in this work provides the basis for significant future advances towards overcoming the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling.
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Pattanaik, S. N., and S. P. Mudur. "Adjoint equations and random walks for illumination computation." ACM Transactions on Graphics 14, no. 1 (January 1995): 77–102. http://dx.doi.org/10.1145/200972.200985.
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Morales Betancourt, R., and A. Nenes. "Understanding the contributions of aerosol properties and parameterization discrepancies to droplet number variability in a Global Climate Model." Atmospheric Chemistry and Physics Discussions 13, no. 12 (December 2, 2013): 31479–526. http://dx.doi.org/10.5194/acpd-13-31479-2013.
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Abstract. Aerosol indirect effects in climate models strongly depend on the representation of the aerosol activation process. In this study, we assess the process level differences across activation parameterizations that contribute to droplet number uncertainty by using the adjoints of the Abdul-Razzak and Ghan (2000) and Fountoukis and Nenes (2005) droplet activation parameterizations in the framework of the Community Atmospheric Model version 5.1 (CAM5.1). The adjoint sensitivities of Nd to relevant input parameters are used to: (i) unravel the spatially resolved contribution of aerosol number, mass, and chemical composition to changes in $N_\\mathrm{d}$ between present day and pre-industrial simulations; (ii) identify the key variables responsible for the differences in Nd fields and aerosol indirect effect estimates when different activation schemes are used within the same modeling framework. The sensitivities are computed online at minimal computational cost. Changes in aerosol number and aerosol mass concentrations were found to contribute to Nd differences much more strongly than chemical composition effects. The main sources of discrepancy between the activation parameterization considered were the treatment of the water uptake by coarse mode particles, and the sensitivity of the parameterized Nd accumulation mode aerosol geometric mean diameter. These two factors explain the different predictions of Nd over land and over oceans when these parameterizations are employed. Discrepancies in the sensitivity to aerosol size are responsible for an exaggerated response to aerosol volume changes over heavily polluted regions. Because these regions are collocated with areas of deep clouds their impact on short wave cloud forcing is amplified through liquid water path changes. Application of the adjoint-sensitivities illustrated the importance of primary organic matter emissions in controlling the droplet number concentration changes in several areas. The same framework is also utilized to efficiently explore droplet number uncertainty attributable to hygroscopicity parameter of organic aerosol (primary and secondary). Comparisons between the parameterization-derived sensitivities of droplet number against predictions with detailed numerical simulations of the activation process were performed to validate the physical consistency of the adjoint sensitivities.
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Enting, Ian Graham. "Tangents, adjoints and computational complexity in terrestrial carbon modelling." ANZIAM Journal 52 (October 10, 2011): 806. http://dx.doi.org/10.21914/anziamj.v52i0.3871.
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Wang, Qiqi. "Forward and adjoint sensitivity computation of chaotic dynamical systems." Journal of Computational Physics 235 (February 2013): 1–13. http://dx.doi.org/10.1016/j.jcp.2012.09.007.
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Ashok, Akshay, and Steven R. H. Barrett. "Adjoint-based computation of U.S. nationwide ozone exposure isopleths." Atmospheric Environment 133 (May 2016): 68–80. http://dx.doi.org/10.1016/j.atmosenv.2016.03.025.
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Silva, Paulo S. D. da, Gudélia Morales, and Príscila H. G. Oliveira. "Variational data assimilation: mathematical foundations and one application." Revista Brasileira de Meteorologia 26, no. 3 (September 2011): 433–42. http://dx.doi.org/10.1590/s0102-77862011000300009.
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Variational Data Assimilation and Adjoint Equation Method are presented here as a general methodology designed to improve the quality of computational simulation when are given the dynamics and a set of observation of the system under study. The mathematical foundations the procedures to obtain the adjoint of a given computational program, a fundamental task in order to apply the methodology, are carefully examined.
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Kenway, Gaetan K. W., Charles A. Mader, Ping He, and Joaquim R. R. A. Martins. "Effective adjoint approaches for computational fluid dynamics." Progress in Aerospace Sciences 110 (October 2019): 100542. http://dx.doi.org/10.1016/j.paerosci.2019.05.002.
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Cacuci, Dan. "First-Order Comprehensive Adjoint Method for Computing Operator-Valued Response Sensitivities to Imprecisely Known Parameters, Internal Interfaces and Boundaries of Coupled Nonlinear Systems: I. Mathematical Framework." Journal of Nuclear Engineering 1, no. 1 (September 8, 2020): 3–17. http://dx.doi.org/10.3390/jne1010002.
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This work presents the first-order comprehensive adjoint sensitivity analysis methodology (1st-CASAM) for computing efficiently the first-order sensitivities (i.e., functional derivatives) of operator-valued responses (i.e., model results) of general models of coupled nonlinear physical systems characterized by imprecisely known or and/or uncertain parameters, external boundaries, and internal interfaces between the coupled systems. The explicit mathematical formalism developed within the 1st-CASAM for computing the first-order sensitivities of operator-valued response to uncertain internal interfaces and external boundaries in the models’ phase–space enables this methodology to generalize all of the previously published methodologies for computing first-order response sensitivities. The computational resources needed for using forward versus adjoint operators in conjunction with spectral versus collocation methods for computing the response sensitivities are analyzed in detail. By enabling the exact computations of operator-valued response sensitivities to internal interfaces and external boundary parameters and conditions, the 1st-CASAM presented in this work makes it possible, inter alia, to quantify the effects of manufacturing tolerances on operator-valued responses of physical and engineering systems.
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Cacuci, Dan Gabriel. "First-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Critical Points in Coupled Nonlinear Systems. I: Mathematical Framework." Fluids 6, no. 1 (January 10, 2021): 33. http://dx.doi.org/10.3390/fluids6010033.
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This work presents the novel first-order comprehensive adjoint sensitivity analysis methodology for critical points (1st-CASAM-CP), which enables the exact and efficient computation of the first-order sensitivities of responses defined at critical points (maxima, minima, saddle points) of coupled nonlinear models of physical systems characterized by imprecisely known parameters underlying the models, boundaries, and interfaces between the coupled systems. Responses defined at critical points are important in many applications, including system optimization, safety analyses and licensing. For the design and licensing of nuclear reactors, such essentially important responses include the maximum temperatures of the fuel and cladding in hot channels. The 1st-CASAM-CP presented in this work makes it possible to determine, using a single large-scale “adjoint” computation, the first-order sensitivities of the magnitude of a response defined at a critical point of a function in the phase-space of the systems’ independent variables. In addition, the 1st-CASAM-CP enables the computation of the sensitivities of the location in phase-space of the critical point at which the respective response is located: one “adjoint” computation is required for each component of the respective critical point in the phase-space of independent variables. By enabling the exact and efficient computation of the sensitivities of responses and of their critical locations to imprecisely known model parameters, boundaries, and interfaces, the 1st-CASAM-CP significantly extends the practicality of analyzing crucially important responses for large-scale systems involving many uncertain parameters, interfaces, and boundaries.
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Cacuci, Dan Gabriel. "First-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Critical Points in Coupled Nonlinear Systems. I: Mathematical Framework." Fluids 6, no. 1 (January 10, 2021): 33. http://dx.doi.org/10.3390/fluids6010033.
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This work presents the novel first-order comprehensive adjoint sensitivity analysis methodology for critical points (1st-CASAM-CP), which enables the exact and efficient computation of the first-order sensitivities of responses defined at critical points (maxima, minima, saddle points) of coupled nonlinear models of physical systems characterized by imprecisely known parameters underlying the models, boundaries, and interfaces between the coupled systems. Responses defined at critical points are important in many applications, including system optimization, safety analyses and licensing. For the design and licensing of nuclear reactors, such essentially important responses include the maximum temperatures of the fuel and cladding in hot channels. The 1st-CASAM-CP presented in this work makes it possible to determine, using a single large-scale “adjoint” computation, the first-order sensitivities of the magnitude of a response defined at a critical point of a function in the phase-space of the systems’ independent variables. In addition, the 1st-CASAM-CP enables the computation of the sensitivities of the location in phase-space of the critical point at which the respective response is located: one “adjoint” computation is required for each component of the respective critical point in the phase-space of independent variables. By enabling the exact and efficient computation of the sensitivities of responses and of their critical locations to imprecisely known model parameters, boundaries, and interfaces, the 1st-CASAM-CP significantly extends the practicality of analyzing crucially important responses for large-scale systems involving many uncertain parameters, interfaces, and boundaries.
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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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Güler, Fatma. "The adjoint trajectory of robot end effector using the curvature theory of ruled surface." Filomat 34, no. 12 (2020): 4061–69. http://dx.doi.org/10.2298/fil2012061g.
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The ruled surface is formed by the movement of a director based on a curve. The point P not on the director vector at fixed frame o-ijk draws a curve. However, each position of this point on the curve always corresponds to position of director on the ruled surface, or this point is adjoint to director vector. Thus, the curve is adjoint to the ruled surface. In this study, we expressed the adjoint trajectory of robot end effector. We can change the trajectory of the robot movement by defining the adjoint trajectory when it may not be physically achievable and not re-computation of the robot trajectory. We investigated the angular acceleration and angular velocity of adjoint trajectory of the robot end effector. Also, we obtained the condition that moving point is a fixed point.
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Morales Betancourt, R., and A. Nenes. "Understanding the contributions of aerosol properties and parameterization discrepancies to droplet number variability in a global climate model." Atmospheric Chemistry and Physics 14, no. 9 (May 14, 2014): 4809–26. http://dx.doi.org/10.5194/acp-14-4809-2014.
Full textAbstract:
Abstract. Aerosol indirect effects in climate models strongly depend on the representation of the aerosol activation process. In this study, we assess the process-level differences across activation parameterizations that contribute to droplet number uncertainty by using the adjoints of the Abdul-Razzak and Ghan (2000) and Fountoukis and Nenes (2005) droplet activation parameterizations in the framework of the Community Atmospheric Model version 5.1 (CAM5.1). The adjoint sensitivities of Nd to relevant input parameters are used to (i) unravel the spatially resolved contribution of aerosol number, mass, and chemical composition to changes in Nd between present-day and pre-industrial simulations and (ii) identify the key variables responsible for the differences in Nd fields and aerosol indirect effect estimates when different activation schemes are used within the same modeling framework. The sensitivities are computed online at minimal computational cost. Changes in aerosol number and aerosol mass concentrations were found to contribute to Nd differences much more strongly than chemical composition effects. The main sources of discrepancy between the activation parameterizations considered were the treatment of the water uptake by coarse mode particles, and the sensitivity of the parameterized Nd accumulation mode aerosol geometric mean diameter. These two factors explain the different predictions of Nd over land and over oceans when these parameterizations are employed. Discrepancies in the sensitivity to aerosol size are responsible for an exaggerated response to aerosol volume changes over heavily polluted regions. Because these regions are collocated with areas of deep clouds, their impact on shortwave cloud forcing is amplified through liquid water path changes. The same framework is also utilized to efficiently explore droplet number uncertainty attributable to hygroscopicity parameter of organic aerosol (primary and secondary). Comparisons between the parameterization-derived sensitivities of droplet number against predictions with detailed numerical simulations of the activation process were performed to validate the physical consistency of the adjoint sensitivities.
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Martin, N., and J. Monnier. "Of the gradient accuracy in Full-Stokes ice flow model: basal slipperiness inference." Cryosphere Discussions 7, no. 4 (August 5, 2013): 3853–97. http://dx.doi.org/10.5194/tcd-7-3853-2013.
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Abstract. This work focuses on the numerical assessment of the accuracy of an adjoint-based gradient in the perspective of variational data assimilation and parameter identification in glaciology. We quantify the ability to identify the basal slipperiness for such methods with a non-linear friction law. The complete adjoint problem is solved and a comparison with the so called "self-adjoint" method, neglecting the viscosity dependency to the velocity, common in glaciology, is carried out. A lower bound of identifiable wavelengths of 10 ice thickness in the friction coefficient is established, when using the full adjoint method, while the "self-adjoint" method is limited to a maximum of 20 ice thickness wavelengths. In addition, the full adjoint method demonstrates a better robustness and reliability for the parameter identification process. The derivation of the adjoint model using algorithmic differentiation leads to formulate a generalization of the "self-adjoint" approximation towards an incomplete adjoint method, adjustable in precision and computational burden.
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Villard, Gilles. "Kaltofen’s division-free determinant algorithm differentiated for matrix adjoint computation." Journal of Symbolic Computation 46, no. 7 (July 2011): 773–90. http://dx.doi.org/10.1016/j.jsc.2010.08.012.
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曹, 小群. "Optimization Techniques for Adjoint Sensitivity Computation in Variational Data Assimilation." Advances in Geosciences 10, no. 08 (2020): 675–86. http://dx.doi.org/10.12677/ag.2020.108067.
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Ledger, P. D., and K. Morgan. "An Adjoint Enhanced Reduced-Order Model for Monostatic RCS Computation." Electromagnetics 28, no. 1-2 (February 21, 2008): 54–76. http://dx.doi.org/10.1080/02726340701818063.
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Folberth, James, and Stephen Becker. "Efficient Adjoint Computation for Wavelet and Convolution Operators [Lecture Notes]." IEEE Signal Processing Magazine 33, no. 6 (November 2016): 135–47. http://dx.doi.org/10.1109/msp.2016.2594277.
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TAM, CHRISTOPHER K. W., and LAURENT AURIAULT. "Mean flow refraction effects on sound radiated from localized sources in a jet." Journal of Fluid Mechanics 370 (September 10, 1998): 149–74. http://dx.doi.org/10.1017/s0022112098001852.
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It is well-known that sound generated by localized sources embedded in a jet undergoes refraction as the acoustic waves propagate through the jet mean flow. For isothermal or hot jets, the effect of refraction causes the deflection of the radiated sound waves away from the jet flow direction. This gives rise to a cone of silence around the jet axis where there is a significant reduction in the radiated sound intensity. In this work, the mean flow refraction problem is investigated through the use of the reciprocity principle. Instead of the direct source Green's function, the adjoint Green's function with the source and observation points interchanged is used to quantify the effect of mean flow on sound radiation. One advantage of the adjoint Green's function is that the Green's functions for all the source locations in the jet radiating to a given direction in the far field can be obtained in a single calculation. This provides great savings in computational effort. Another advantage of the adjoint Green's function is that there is no singularity in the jet flow so that the problem can be solved numerically with axial as well as radial mean flow gradients included in a fairly straightforward manner. Extensive numerical computations have been carried out for realistic jet flow profiles with and without exercising the locally parallel flow approximation. It is concluded that the locally parallel flow approximation is valid as long as the direction of radiation is outside the cone of silence.
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Hosseinzadegan, Samar, Andreas Fhager, Mikael Persson, Shireen Geimer, and Paul Meaney. "Expansion of the Nodal-Adjoint Method for Simple and Efficient Computation of the 2D Tomographic Imaging Jacobian Matrix." Sensors 21, no. 3 (January 22, 2021): 729. http://dx.doi.org/10.3390/s21030729.
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This paper focuses on the construction of the Jacobian matrix required in tomographic reconstruction algorithms. In microwave tomography, computing the forward solutions during the iterative reconstruction process impacts the accuracy and computational efficiency. Towards this end, we have applied the discrete dipole approximation for the forward solutions with significant time savings. However, while we have discovered that the imaging problem configuration can dramatically impact the computation time required for the forward solver, it can be equally beneficial in constructing the Jacobian matrix calculated in iterative image reconstruction algorithms. Key to this implementation, we propose to use the same simulation grid for both the forward and imaging domain discretizations for the discrete dipole approximation solutions and report in detail the theoretical aspects for this localization. In this way, the computational cost of the nodal adjoint method decreases by several orders of magnitude. Our investigations show that this expansion is a significant enhancement compared to previous implementations and results in a rapid calculation of the Jacobian matrix with a high level of accuracy. The discrete dipole approximation and the newly efficient Jacobian matrices are effectively implemented to produce quantitative images of the simplified breast phantom from the microwave imaging system.
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Aupy, Guillaume, and Julien Herrmann. "Periodicity in optimal hierarchical checkpointing schemes for adjoint computations." Optimization Methods and Software 32, no. 3 (September 29, 2016): 594–624. http://dx.doi.org/10.1080/10556788.2016.1230612.
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Cheng, Xiaoliang, Guangliang Lin, Ye Zhang, Rongfang Gong, and Mårten Gulliksson. "A modified coupled complex boundary method for an inverse chromatography problem." Journal of Inverse and Ill-posed Problems 26, no. 1 (February 1, 2018): 33–49. http://dx.doi.org/10.1515/jiip-2016-0057.
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AbstractAdsorption isotherms are the most important parameters in rigorous models of chromatographic processes. In this paper, in order to recover adsorption isotherms, we consider a coupled complex boundary method (CCBM), which was previously proposed for solving an inverse source problem [2]. With CCBM, the original boundary fitting problem is transferred to a domain fitting problem. Thus, this method has advantages regarding robustness and computation in reconstruction. In contrast to the traditional CCBM, for the sake of the reduction of computational complexity and computational cost, the recovered adsorption isotherm only corresponds to the real part of the solution of a forward complex initial boundary value problem. Furthermore, we take into account the position of the profiles and apply the momentum criterion to improve the optimization progress. Using Tikhonov regularization, the well-posedness, convergence properties and regularization parameter selection methods are studied. Based on an adjoint technique, we derive the exact Jacobian of the objective function and give an algorithm to reconstruct the adsorption isotherm. Finally, numerical simulations are given to show the feasibility and efficiency of the proposed regularization method.
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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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Fahsi, Adil, Azzeddine Soulaïmani, and Georges W. Tchamen. "Application of reliability techniques for the estimation of uncertainties in fluvial hydraulics simulationsThis article is one of a selection of papers published in this Special Issue on Hydrotechnical Engineering." Canadian Journal of Civil Engineering 37, no. 7 (July 2010): 991–1002. http://dx.doi.org/10.1139/l10-011.
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This work forms part of a general research effort into developing a methodology for reliability estimates of results obtained in numerical hydrodynamic computations. We consider the possibility of overtopping at the crest of a dyke of height h0 positioned on a river with several uncertain parameters, such as discharge, Manning coefficient, bathymetry, etc. The estimation model is based either on the first-order reliability method (FORM), on the multi-form method, or on the importance sampling methods and is coupled with algorithms for the resolution of an adjoint optimization problem. Numerical tests are carried out on flows over a channel with a bump and on a river. The results obtained with our algorithms are compared with those obtained with the commercial software Nessus® and with the Monte Carlo method. The proposed multi-form approach combined with a robust optimization algorithm provides reliable results within reasonable computation times.
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Colbrook, Matthew J. "Computing Spectral Measures and Spectral Types." Communications in Mathematical Physics 384, no. 1 (April 11, 2021): 433–501. http://dx.doi.org/10.1007/s00220-021-04072-4.
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AbstractSpectral measures arise in numerous applications such as quantum mechanics, signal processing, resonance phenomena, and fluid stability analysis. Similarly, spectral decompositions (into pure point, absolutely continuous and singular continuous parts) often characterise relevant physical properties such as the long-time dynamics of quantum systems. Despite new results on computing spectra, there remains no general method able to compute spectral measures or spectral decompositions of infinite-dimensional normal operators. Previous efforts have focused on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators that carry a lot of structure. Hence the general computational problem is predominantly open. We solve this problem by providing the first set of general algorithms that compute spectral measures and decompositions of a wide class of operators. Given a matrix representation of a self-adjoint or unitary operator, such that each column decays at infinity at a known asymptotic rate, we show how to compute spectral measures and decompositions. We discuss how these methods allow the computation of objects such as the functional calculus, and how they generalise to a large class of partial differential operators, allowing, for example, solutions to evolution PDEs such as the linear Schrödinger equation on $$L^2({\mathbb {R}}^d)$$ L 2 ( R d ) . Computational spectral problems in infinite dimensions have led to the Solvability Complexity Index (SCI) hierarchy, which classifies the difficulty of computational problems. We classify the computation of measures, measure decompositions, types of spectra, functional calculus, and Radon–Nikodym derivatives in the SCI hierarchy. The new algorithms are demonstrated to be efficient on examples taken from orthogonal polynomials on the real line and the unit circle (giving, for example, computational realisations of Favard’s theorem and Verblunsky’s theorem, respectively), and are applied to evolution equations on a two-dimensional quasicrystal.
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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.