Adjoint Mode Research Articles

Identifying topological structures with adjoint mode filtering

We present new investigations on the Adjoint Filtering Method (AFM), a proposal for filtering gauge configurations by using adjoint zero modes. This method relies on the existence of the Supersymmetric Zero Mode (SZM), whose density follows the gauge action density for classical configurations. We review how a similar construction on the lattice is implemented based on the overlap operator and test the method against smooth configurations showing a remarkable agreement with the expected densities even when pairs of fractionals instanton/anti-instantons are present and noise is added. Then we explore the application of the method to Monte Carlo generated configurations based on SU(2) gauge group. The tuning of the parameters and quantitative results are explicitly shown for a T3 × R lattice. We show explicit examples comparing the AFM to the density obtained from the Gradient Flow. The agreement is remarkable for some specific configurations containing fractional instantons with the advantage that the AFM does not modify the underlying structures.

Deterministic and stochastic methods for sensitivity analysis of neutron noise

Neutron noise calculated from the neutron noise equation in the frequency domain is governed by the cross section and kinetic parameters. Deterministic and stochastic methods to obtain the sensitivity coefficient of neutron noise with respect to the abovementioned parameters are proposed. As a deterministic method, a diffusion equation for the first derivative of neutron noise with respect to a cross section or kinetic parameter is derived by differentiating the neutron noise diffusion equation. As a stochastic method, the differential operator sampling method, which is a well-established Monte Carlo technique, is applied to calculate the sensitivity coefficient. Neither method requires adjoint mode calculations and can be expanded to higher-order derivatives. Based on verifications performed in this study, it is discovered that these techniques yield accurate sensitivity coefficients. The methods developed in this study eliminates a large number of calculations that need to be performed in the random sampling method.

An efficient adjoint method for the aero-stealth shape optimization design

Sensitivity analysis is an important part of gradient-based aero-stealth shape optimization design. An efficient Radar Cross Section (RCS) gradient evaluation method based on the adjoint approach is proposed in this paper. The merits of this proposed method mainly include the following two aspects. First, the adjoint field will be obtained from a single solution of the scattering problem without solving the adjoint equation. Second, the residual derivatives computing code is generated through the adjoint mode of Automatic Differentiation (AD) technique and is optimized by changing the “two-sweeps” architecture into the “inner-loop two-sweeps” architecture. The time consumption and memory usage are tested and the results show that the AD adjoint mode with “inner-loop two-sweeps” architecture is high-efficiency while the memory consumption is friendly. In addition, the accuracy of the adjoint method is verified through the finite differences method. After that, a gradient-based aero-stealth optimization design method is developed and employed to study the shape requirements of aerodynamic and stealth performance. The cost function is composed of the drag coefficient and the RCS. The results show that several trade-off configurations would be obtained by selecting different weight factors and the pure stealth optimization design would lead to a screwy shape. Further studies illustrate that the RCS reduction comes from two aspects. On the one hand, the sharp leading edge could reduce the surface current magnitude and avoid the specular back-scattering. On the other hand, the undulations could change the phases and weaken the superimposed effect.

Radar Cross Section Reduction and Shape Optimization using Adjoint Method and Automatic Differentiation

An efficient Radar Cross Section (RCS) gradient evaluation method based on the adjoint method is presented. The Method of Moments is employed to solve the Combined Field Integral Equation (CFIE) and the corresponding derivatives computing routines are generated by the program transformation Automatic Differentiation (AD) technique. The differential code is developed using three kinds of AD mode: tangent mode, multidirectional tangent mode, and adjoint mode. The differential code in adjoint mode is modified and optimized by changing the “two-sweeps” architecture into the “inner-loop two-sweeps” architecture. Their efficiency and memory consumption are tested and the differential code using modified adjoint mode demonstrates the great advantages in both efficiency and memory consumption. A gradient-based shape optimization design method is established using the adjoint method and the mechanism of RCS reduction is studied. The results show that the sharp leading can avoid the specular back-scattering and the undulations of the surface could change the phases which result in a further RCS reduction.

Influence of shroud profiling on the compressor diffuser: Frozen-eddy approach and mode decomposition

The stability prediction and unsteady-flow characteristic explanations of centrifugal compressors mostly focus on radial vaneless diffusers with parallel walls (PWs). However, vaneless diffusers with curved shroud profiling are more often adopted in a practical compression system owing to their better performance or larger stability margin. In this work, the stability properties of the unsteady flow in the radial vaneless diffusers with parallel walls and curved shroud profiling were studied in the context of numerical simulations and global modes. First, the unsteady simulations of a full annular compressor, including vaneless diffusers with parallel walls and constant area walls, were carried out. The simulation results were validated against experimental data, and the dominant modes of the diffuser flow under stall conditions were captured using dynamic mode decomposition (DMD) analysis. Then, a frozen eddy stability analysis was performed on the vaneless diffusers, and the global mode associated with the flow perturbations was investigated. Finally, a continuous adjoint method was adopted to reveal the receptivity of the instability perturbations. By focusing on the effect of the curved shroud profiling on the stability characteristics, the global and adjoint eigenmodes corresponding to the onset of instability were used to provide a physical explanation for the stall mechanism. The simulation results indicated that the stability margin for diffuser with constant area walls was 2.15% wider than that with PWs. The DMD and global stability-analysis results showed that the diffuser with constant area walls had a stall frequency of 298 Hz, which was larger than that of diffuser with PWs. But both cases shared the same stall number of 4. The value of growth rate came to its minimum at 30% span and maximum at 95% span in both cases, indicating the dominant flow fields were near the shroud walls. For the vaneless diffuser with PWs, the reverse flow at 95% span within the radius range of r = 1.3–1.4 was the dominant reason for the occurrence of instability with large value of direct and adjoint mode. Meanwhile, flow mode indicated that flow instability of the diffuser with constant area walls was caused by the reverse flow at 90% span with the radius range of r = 1.02–1.1.

Development of the neutronics transport code based on the MOC method and adjoint-forward Coarse Mesh Finite Difference technique: PerMOC code for lattice and whole core calculations

In this research, the Coarse Mesh Finite Difference (CMFD) acceleration technique in both adjoint and forward mode is applied to the MOC kernel, to develop a new code called PerMOC. A new approach in the adjoint MOC kernel which is called thermal-up-scattering-like iteration scheme reduces the total number of adjoint iteration sweeps. In additions, more speed-up is achieved using a parallel transport sweeping in MOC kernel. Validation test for PerMOC C-Sharp code is carried out for four distinct criticality benchmarks in which C5G7 is one of those. In the case of C5G7 benchmark, the best-calculated multiplication factor has 51 pcm differences with the reference case. Furthermore, the newly proposed adjoint solver reduces about 30% the number of adjoint transport sweeps in comparison with the normal adjoint solver. The present article describes the PerMOC methodologies and illustrates its ability to model large problems with acceptable accuracy.

Solvability conditions in multi-scale dynamic analysis of one-dimensional structures with non-homogeneous boundaries: A general operator formulation

A general operator-based solvability condition is established for multi-scale analysis of continuous structures with non-homogeneous boundaries, extending the commonly used solvability conditions to account for the time varying/nonlinear boundary effects. The orthogonality relations between adjoint mode and resonant source terms of structure, formulated in the common solvability conditions, are found to be violated by the non-homogeneous boundaries. Explicitly, their inner product is found to be equal to non-zero boundary terms directly induced by the non-homogeneous boundary conditions, through a general operator-based derivation. Furthermore, the boundary terms are of a product-form between two boundary differential/integral operators, and an interesting reciprocity relation is fully presented based upon (virtual) work analysis. This general theoretical formulation is applied to four typical continuous structures, i.e., strings, cables, beams, and arches, and their corresponding solvability conditions are explicitly established for various non-homogeneous boundaries.

Guiding Actuator Designs for Active Flow Control of the Precessing Vortex Core by Adjoint Linear Stability Analysis

The fundamental impact of the precessing vortex core (PVC) as a dominant coherent flow structure in the flow field of swirl-stabilized gas turbine combustors has still not been investigated in depth. In order to do so, the PVC needs to be actively controlled to be able to set its parameters independently to any other of the combustion system. In this work, open-loop actuation is applied in the mixing section between the swirler and the generic combustion chamber of a nonreacting swirling jet setup to investigate the receptivity of the PVC with regard to its lock-in behavior at different streamwise positions. The mean flow in the mixing section as well as in the combustion chamber is measured by stereoscopic particle image velocimetry (SPIV), and the PVC is extracted from the snapshots using proper orthogonal decomposition (POD). The lock-in experiments reveal the axial position in the mixing section that is most suitable for actuation. Furthermore, a global linear stability analysis (LSA) is conducted to determine the adjoint mode of the PVC which reveals the regions of highest receptivity to periodic actuation based on mean flow input only. This theoretical receptivity model is compared with the experimentally obtained receptivity data, and the applicability of the adjoint-based model for the prediction of optimal actuator designs is discussed.

Stability and Sensitivity Analysis of Hydrodynamic Instabilities in Industrial Swirled Injection Systems

The hydrodynamic instability in an industrial, two-staged, counter-rotative, swirled injector of highly complex geometry is under investigation. Large eddy simulations (LES) show that the complicated and strongly nonparallel flow field in the injector is superimposed by a strong precessing vortex core (PVC). Mean flow fields of LES, validated by experimental particle image velocimetry (PIV) measurements, are used as input for both local and global linear stability analysis (LSA). It is shown that the origin of the instability is located at the exit plane of the primary injector. Mode shapes of both global and local LSA are compared to dynamic mode decomposition (DMD) based on LES snapshots, showing good agreement. The estimated frequencies for the instability are in good agreement with both the experiment and the simulation. Furthermore, the adjoint mode shapes retrieved by the global approach are used to find the best location for periodic forcing in order to control the PVC.

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

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).

Adjoint optimization of natural convection problems: differentially heated cavity

Optimization of natural convection-driven flows may provide significant improvements to the performance of cooling devices, but a theoretical investigation of such flows has been rarely done. The present paper illustrates an efficient gradient-based optimization method for analyzing such systems. We consider numerically the natural convection-driven flow in a differentially heated cavity with three Prandtl numbers (\(Pr=0.15{-}7\)) at super-critical conditions. All results and implementations were done with the spectral element code Nek5000. The flow is analyzed using linear direct and adjoint computations about a nonlinear base flow, extracting in particular optimal initial conditions using power iteration and the solution of the full adjoint direct eigenproblem. The cost function for both temperature and velocity is based on the kinetic energy and the concept of entransy, which yields a quadratic functional. Results are presented as a function of Prandtl number, time horizons and weights between kinetic energy and entransy. In particular, it is shown that the maximum transient growth is achieved at time horizons on the order of 5 time units for all cases, whereas for larger time horizons the adjoint mode is recovered as optimal initial condition. For smaller time horizons, the influence of the weights leads either to a concentric temperature distribution or to an initial condition pattern that opposes the mean shear and grows according to the Orr mechanism. For specific cases, it could also been shown that the computation of optimal initial conditions leads to a degenerate problem, with a potential loss of symmetry. In these situations, it turns out that any initial condition lying in a specific span of the eigenfunctions will yield exactly the same transient amplification. As a consequence, the power iteration converges very slowly and fails to extract all possible optimal initial conditions. According to the authors’ knowledge, this behavior is illustrated here for the first time.

Crow instability: nonlinear response to the linear optimal perturbation

The potential for anticipated destruction of a counter-rotating vortex pair using the linear optimal perturbation of the Crow instability is assessed. Direct numerical simulation is used to study the development of the Crow instability and the subsequent evolution of the flow up to 30 characteristic times at a circulation-based Reynolds number of 1000. The conventional development of the instability leads to multiple contortions of the vortices including the linear growth of sinusoidal deformation, vortex linking and the formation of vortex rings. A new evolution stage is identified, succeeding this well-established sequence: the vortex rings undergo periodic oscillation. Two complete periods are simulated during which the vortical system is hardly altered, thereby demonstrating the extraordinary resilience of the vortices. The possibility of preventing these dynamics using the linear optimal perturbation of the Crow instability, the adjoint mode, is analysed. By appropriately setting the forcing amplitude, the lifetime of the vortices until their loss of coherence is reduced to approximately 13 characteristic times, which is less than half that of the natural Crow behaviour observed with infinitesimal forcing. The dynamics of the flow induced by the linear optimal perturbation that enable this result are connected to processes already known to efficiently alter vortical flows, in particular transient growth and four-vortex dynamics.

Algorithmic Differentiation of Numerical Methods: Second-order Adjoint Solvers for Parameterized Systems of Nonlinear Equations

Adjoint mode algorithmic (also know as automatic) differentiation (AD) transforms implementations of multivariate vector functions as computer programs into first-order adjoint code. Its reapplication or combinations with tangent mode AD yields higher-order adjoint code. Second derivatives play an important role in nonlinear programming. For example, second-order (Newton-type) nonlinear optimization methods promise faster convergence in the neighborhood of the minimum through taking into account second derivative information. The adjoint mode is of particular interest in large-scale gradient-based nonlinear optimization due to the independence of its computational cost on the number of free variables. Part of the objective function may be given implicitly as the solution of a system of n parameterized nonlinear equations. If the system parameters depend on the free variables of the objective, then second derivatives of the nonlinear system's solution with respect to those parameters are required. The local computational overhead as well as the additional memory requirement for the computation of second-order adjoints of the solution vector with respect to the parameters by AD depends on the number of iterations performed by the nonlinear solver. This dependence can be eliminated by taking a symbolic approach to the differentiation of the nonlinear system.

Modal and non-modal evolution of perturbations for parallel round jets

The present work investigates the local modal and non-modal stability of round jets for varying aspect ratios α = R/θ, where R is the jet radius and θ the shear layer momentum thickness, for Reynolds numbers ranging from 10 to 10 000. The competition between axisymmetric (azimuthal wavenumber m = 0) and helical (m = 1) perturbations depending on the aspect ratio, α, is quantified at different time horizons. Three different techniques have been used, namely, a classical temporal stability analysis in order to characterize the unstable modes of the jet; an optimal excitation analysis, based on the resolution of the adjoint problem, to quantify the potential for non-modal perturbation dynamics; and finally an optimal perturbation analysis, focused on the very short time transient dynamics, to complement the adjoint-based study. Besides providing with the determination of the critical aspect ratio below which the most unstable perturbations switch from m = 0 to m = 1 depending on the Reynolds number, the study shows that perturbations can undergo a rapid transient growth. It is found that helical perturbations always experience the highest transient growth, although for large values of aspect ratio, this transient domination can be overcome by the eventual emergence of axisymmetric perturbation when more exponentially unstable. Furthermore, the adjoint mode, which excites optimally the most unstable mode of the flow, is found to coincide with the optimal perturbation even for short time horizons, and to drive the transient dynamics for finite times. Therefore, the adjoint-based analysis is found to characterize adequately the transient dynamics of jets, showing that a mechanism equivalent to the Orr one takes place for moderate to small wavelengths. However, in the long wavelength limit, a specific mechanism is found to shift the jet as a whole in a way that resembles the classical lift-up effect active in wall shear flows.

Pricing Interest Rate Derivatives and Computing Pathwise Greeks in the Extended LIBOR Market Model

Classical interest rate models were formulated under the no-arbitrage assumption of a unique yield curve. However, in the outbreak of the 2007 global financial crisis, market interest rates witnessed unprecedented segmentation, and this instigated the modelling of a basis spread, whose non-negligible widening could no longer be neglected. One extension of the LIBOR market model (LMM) to incorporate basis spread was proposed by Mercurio (2010a), and in this paper, we examine how his theoretical description is translated into practical implementation to price interest rate derivatives and compute pathwise Greeks. Specifically, our contribution is to develop and implement the theory of the extended LMM in the framework of a displaced diffusion process in the spot LIBOR measure. The adjoint mode from algorithmic differentiation is premiered in this new implementation to compute pathwise deltas and vegas before the accumulated knowledge is applied in the pricing of interest rate caps and one-way floaters.

The Tapenade automatic differentiation tool

Tapenade is an Automatic Differentiation (AD) tool which, given a Fortran or C code that computes a function, creates a new code that computes its tangent or adjoint derivatives. Tapenade puts particular emphasis on adjoint differentiation, which computes gradients at a remarkably low cost. This article describes the principles of Tapenade, a subset of the general principles of AD. We motivate and illustrate with examples the AD model of Tapenade, that is, the structure of differentiated codes and the strategies used to make them more efficient. Along with this informal description, we formally specify this model by means of data-flow equations and rules of Operational Semantics, making this the reference specification of the tangent and adjoint modes of Tapenade. One benefit we expect from this formal specification is the capacity to formally study the AD model itself, especially for the adjoint mode and its sophisticated strategies. This article also describes the architectural choices of the implementation of Tapenade. We describe the current performance of Tapenade on a set of codes that include industrial-size applications. We present the extensions of the tool that are planned in a foreseeable future, deriving from our ongoing research on AD.

Transient growth in Rayleigh-Bénard-Poiseuille/Couette convection

An investigation of the effect of a destabilizing cross-stream temperature gradient on the transient growth phenomenon of plane Poiseuille flow and plane Couette flow is presented. Only the streamwise-uniform and nearly streamwise-uniform disturbances are highly influenced by the Rayleigh number Ra and Prandtl number Pr. The maximum optimal transient growth Gmax of streamwise-uniform disturbances increases slowly with increasing Ra and decreasing Pr. For all Ra and Pr, at moderately large Reynolds numbers Re, the supremum of Gmax is always attained for streamwise-uniform perturbations (or nearly streamwise-uniform perturbations, in the case of plane Couette flow) which produce large streamwise streaks and Rayleigh-Bénard convection rolls (RB). The optimal growth curves retain the same large-Reynolds-number scaling as in pure shear flow. A 3D vector model of the governing equations demonstrates that the short-time behavior is governed by the classical lift-up mechanism and that the influence of Ra on this mechanism is secondary and negligible. The optimal input for the largest long-time response is given by the adjoint of the dominant eigenmode with respect to the energy scalar product: the RB eigenmode without its streamwise velocity component. These short-time and long-time responses depict, to leading order, the optimal transient growth G(t). At moderately large Ra (or small Pr at a fixed Ra), the dominant adjoint mode is a good approximation to the optimal initial condition for all time. Over a general class of norms that can be considered as growth functions, the results remain qualitatively similar, for example, the dominant adjoint eigenmode still approximates the maximum optimal response.

Rigorous Evaluation of Biological Dose for Fusion Systems and Comparison with Approximate Contact Dose Approach

Every few years, maintenance will be required to replace the plasma facing components of any fusion power plant. To come up with a realistic maintenance scheme, an accurate method of evaluating the biological dose rate is needed. In some studies, the simple and quick approach of the contact dose rate for a specific component was used to estimate the biological dose rate. This method doesn’t take radiation from nearby components into account and the accuracy of its methodology is questionable. The more accurate multi-step method, which involves transporting the delayed gammas from induced activation in the forward or adjoint mode, is able to take radiation from all surrounding components into account. In this report, the biological dose rate was evaluated with the adjoint method at selected radial locations of ARIES-CS, and then compared to the contact dose rate of the nearest component to determine the accuracy of the contact dose method. Our results indicate that the approximate contact dose rate could be off by an order of magnitude.

Adjoint-Weighted Tallies for k-Eigenvalue Calculations with Continuous-Energy Monte Carlo

A Monte Carlo method is developed that performs adjoint-weighted tallies in continuous-energy k-eigenvalue calculations. Each contribution to a tally score is weighted by an estimate of the relative magnitude of the fundamental adjoint mode, by way of the iterated fission probability, at the phase-space location of the contribution. The method is designed around the power iteration method such that no additional random walks are necessary, resulting in a minimal increase in computational time. The method is implemented in the Monte Carlo N-Particle (MCNP) code. These adjoint-weighted tallies are used to calculate adjoint-weighted fluxes, point reactor kinetics parameters, and reactivity changes from first-order perturbation theory. The results are benchmarked against discrete ordinates calculations, experimental measurements, and direct Monte Carlo calculations.

Fast Greeks by algorithmic differentiation

We show how algorithmic differentiation can be used to efficiently implement the pathwise derivative method for the calculation of option sensitivities using Monte Carlosimulations.Themainpracticaldifficultyofthepathwisederivativemethod is that it requires the differentiation of the payout function. For the type of structured options for which Monte Carlo simulations are usually employed, these derivatives are typically cumbersome to calculate analytically, and too time consuming to evaluate with standard finite-difference approaches. In this paper we address this problem and show how algorithmic differentiation can be employed to calculate these derivatives very efficiently and with machine-precision accuracy.Weillustratethebasicworkingsofthiscomputationaltechniquebymeansof simple examples, and we demonstrate with several numerical tests how the pathwise derivative method combined with algorithmic differentiation ‐ especially in the adjoint mode ‐ can provide speed-ups of several orders of magnitude with respect to standard methods.