Adjoint Simulation Research Articles

GIA imaging of 3-D mantle viscosity based on palaeo sea level observations - Part I: Sensitivity kernels for an Earth with laterally varying viscosity.

A key initial step in geophysical imaging is to devise an effective means of mapping the sensitivity of an observation to the model parameters, that is to compute its Fréchet derivatives or sensitivity kernel. In the absence of any simplifying assumptions and when faced with a large number of free parameters, the adjoint method can be an effective and efficient approach to calculating Fréchet derivatives and requires just two numerical simulations. In the Glacial Isostatic Adjustment problem, these consist of a forward simulation driven by changes in ice mass and an adjoint simulation driven by fictitious loads that are applied at the observation sites. The theoretical basis for this approach has seen considerable development over the last decade. Here, we present the final elements needed to image 3-D mantle viscosity using a dataset of palaeo sea-level observations. Developments include the calculation of viscosity Fréchet derivatives (i.e. sensitivity kernels) for relative sea-level observations, a modification to the numerical implementation of the forward and adjoint problem that permits application to 3-D viscosity structure, and a recalibration of initial sea level that ensures the forward simulation honours present-day topography. In the process of addressing these items, we build intuition concerning how absolute sea-level and relative sea-level observations sense Earth's viscosity structure and the physical processes involved. We discuss examples for potential observations located in the near field (Andenes, Norway), far field (Seychelles), and edge of the forebulge of the Laurentide ice sheet (Barbados). Examination of these kernels: (1) reveals why 1-D estimates of mantle viscosity from far-field relative sea-level observations can be biased; (2) hints at why an appropriate differential relative sea-level observation can provide a better constraint on local mantle viscosity and (3) demonstrates that sea-level observations have non-negligible 3-D sensitivity to deep mantle viscosity structure, which is counter to the intuition gained from 1-D radial viscosity Fréchet derivatives. Finally, we explore the influence of lateral variations in viscosity on relative sea-level observations in the Amundsen Sea Embayment and at Barbados. These predictions are based on a new global 3-D viscosity inference derived from the shear-wave speeds of GLAD-M25 and an inverse calibration scheme that ensures compatibility with certain fundamental geophysical observations. Use of the 3-D viscosity inference leads to: (1) generally greater complexity within the kernel; (2) an increase in sensitivity and presence of shorter length-scale features within lower viscosity regions; (3) a zeroing out of the sensitivity kernel within high-viscosity regions where elastic deformation dominates and (4) shifting of sensitivity at a given depth towards distal regions of weaker viscosity. The tools and intuition built here provide the necessary framework to explore inversions for 3-D mantle viscosity based on palaeo sea-level data.

Source-encoded waveform inversion in the Northern Hemisphere

SUMMARY We use source-encoded waveform inversion to image Earth’s Northern Hemisphere. The encoding method is based on measurements of Laplace coefficients of stationary wavefields. By assigning to each event a unique frequency, we compute Fréchet derivatives for all events simultaneously based on one ‘super’ forward and one ‘super’ adjoint simulation for a small fraction of the computational cost of classical waveform inversion with the same data set. No cross-talk noise is introduced in the process, and the method does not require all events to be recorded by all stations. Starting from global model GLAD_M25, we performed 100 conjugate gradient iterations using a data set consisting of 786 earthquakes recorded by 9846 stations. Synthetic inversion tests show that we achieve good convergence based on this data set, and we see a consistent misfit reduction during the inversion. The new model, named SE100, has much higher spatial resolution than GLAD_M25, revealing details of the Yellowstone and Iceland hotspots, subduction beneath the Western United States and the upper mantle structure beneath the Arctic Ocean.

Adjoint-Based Boundary Condition Sensitivity Analysis

Turbine performance in modern aeroengines is heavily affected by the nonuniform temperature distribution generated in the combustor. The conventional method to study the effect of nonuniform boundary conditions (BCs) is to compare flowfields and performance metrics with the nonuniformity at several typical positions. A much more efficient and intuitive method to analyze the BC sensitivity is proposed in this paper. An inlet guide vane is analyzed as a demonstration. The sensitivity analysis is achieved by pointwise derivatives of objectives with respect to inlet BCs, which are obtained by discrete adjoint method. Furthermore, to study the variation of gradients, a random sampling method that is based on the Gaussian process is proposed to explore the high-dimensional parameter space. When the gradient is invariant within the parameter space, the relationship between objectives and BCs is linear, and the effect of any perturbation of BCs can be directly predicted. In this case, the computational cost of the sensitivity analysis is drastically reduced to one computational fluid dynamics and one adjoint simulation. Interestingly, some gradients are nearly invariant across the parameter space. When the gradient varies, the active subspace method is introduced to reduce the number of dimensions. Then, several models, including the kriging, gradient-enhanced kriging (GEK), and Taylor expansions, are compared to predict objectives. The ordinary kriging model is recommended for its accuracy and stability. The GEK model can also predict the gradient and is a good choice for small BC variations. The method presented here offers a paradigm that extends the gradient-based sensitivity analysis to the global sense, which is suitable for high-dimensional problems such as BCs and spatially varying model parameters.

Efficient adjoint-based well-placement optimization using flow diagnostics proxies

Model-based optimization of placement and trajectories of wells in petroleum reservoirs by the means of reservoir simulation forecasts is computationally demanding due to the high number of simulations typically required to achieve a local optimum. In this work, we develop an efficient flow-diagnostics proxy for net-present-value (NPV) with adjoint capabilities for efficient computation of well control gradients and approximate sensitivities with respect to placement/trajectory parameters. The suggested flow-diagnostic proxy consists of numerically solving a single pressure equation for the given scenario and the solution of a few inter-well time-of-flight and steady-state tracer equations, typically achieved in a few seconds for a reservoir model of medium size. Although the proxy may not be a particularly good approximation for the full reservoir simulation response, we find that for the cases considered, the correlation is very good and hence the proxy is suitable for use in an optimization loop. The adjoint simulation for the proxy model which provides control gradients and placement sensitivities is of similar computational complexity as the forward proxy model (a few seconds). We employ a version of the generalized reduced gradient for handling individual well constraints (e.g., bottom-hole-pressures and rates). As a result, the individual well constraints are enforced within the flow-diagnostics computations, and hence every parameter update becomes feasible without sacrificing gradient information. We present two numerical experiments illustrating the efficiency and performance of the approach for well placement problems involving trajectories and simulation models of realistic complexity. The suggested placements are evaluated using full simulations. We conclude by discussing limitations and possible enhancements of the methodology.

On-the-Fly Full Hessian Kernel Calculations Based upon Seismic-Wave Simulations

AbstractFull waveform inversion or adjoint tomography has routinely been performed to image the internal structure of the Earth at high resolution. This is typically done using the Fréchet kernels and the approximate Hessian or the approximate inverse Hessian because of the high-computational cost of computing and storing the full Hessian. Alternatively, the full Hessian kernels can be used to improve inversion resolutions and convergence rates, as well as possibly to mitigate interparameter trade-offs. The storage requirements of the full Hessian kernel calculations can be reduced by compression methods, but often at a price of accuracy depending on the compression factor. Here, we present open-source codes to compute both Fréchet and full Hessian kernels on the fly in the computer random access memory (RAM) through simultaneously solving four wave equations, which we call Quad Spectral-Element Method (QuadSEM). By recomputing two forward fields at the same time that two adjoint fields are calculated during the adjoint simulation, QuadSEM constructs the full Hessian kernels using the exact forward and adjoint fields. In addition, we also implement an alternative approach based on the classical wavefield storage method (WSM), which stores forward wavefields every kth (k≥1) timestep during the forward simulation and reads required fields back into memory during the adjoint simulation for kernel construction. Both Fréchet and full Hessian kernels can be computed simultaneously through the QuadSEM or the WSM code, only doubling the computational cost compared with the computation of Fréchet kernels alone. Compared with WSM, QuadSEM can reduce the disk space and input/output cost by three orders of magnitude in the presented examples that use 15,000 timesteps. Numerical examples are presented to demonstrate the functionality of the methods, and the computer codes are provided with this contribution.

Topology optimization of bi-material structures with frequency-domain objectives using time-domain simulation and sensitivity analysis

In this paper, we propose to use time-domain transient analysis to compute the response of structures in a wide frequency band by means of Fourier transform. A time-domain adjoint variable method is then developed to carry out the sensitivity analysis of frequency-domain objective functions. By using the concept of frequency response function, it turns out that both the objective function and its sensitivity information at multiple frequencies can be obtained by one original simulation and at most one adjoint simulation, respectively. It is also demonstrated that some commonly used performance indices, e.g., dynamic compliance and input power, are indeed self-adjoint; thus, no extra adjoint simulations are needed, which makes the sensitivity analysis extremely efficient. An obvious distinction between the proposed method and the traditional frequency domain methods is that in our method, the frequency response curves in a wide band can be obtained in each iteration with no extra costs. It follows that it is easy to track the evolution of the frequency response curve in our method, which is essential in both computational and engineering sense. Several numerical examples are tested to show the effectiveness of the proposed method.

Radiative backpropagation

Physically based differentiable rendering has recently evolved into a powerful tool for solving inverse problems involving light. Methods in this area perform a differentiable simulation of the physical process of light transport and scattering to estimate partial derivatives relating scene parameters to pixels in the rendered image. Together with gradient-based optimization, such algorithms have interesting applications in diverse disciplines, e.g., to improve the reconstruction of 3D scenes, while accounting for interreflection and transparency, or to design meta-materials with specified optical properties. The most versatile differentiable rendering algorithms rely on reverse-mode differentiation to compute all requested derivatives at once, enabling optimization of scene descriptions with millions of free parameters. However, a severe limitation of the reverse-mode approach is that it requires a detailed transcript of the computation that is subsequently replayed to back-propagate derivatives to the scene parameters. The transcript of typical renderings is extremely large, exceeding the available system memory by many orders of magnitude, hence current methods are limited to simple scenes rendered at low resolutions and sample counts. We introduce radiative backpropagation , a fundamentally different approach to differentiable rendering that does not require a transcript, greatly improving its scalability and efficiency. Our main insight is that reverse-mode propagation through a rendering algorithm can be interpreted as the solution of a continuous transport problem involving the partial derivative of radiance with respect to the optimization objective. This quantity is "emitted" by sensors, "scattered" by the scene, and eventually "received" by objects with differentiable parameters. Differentiable rendering then decomposes into two separate primal and adjoint simulation steps that scale to complex scenes rendered at high resolutions. We also investigated biased variants of this algorithm and find that they considerably improve both runtime and convergence speed. We showcase an efficient GPU implementation of radiative backpropagation and compare its performance and the quality of its gradients to prior work.

Adjoint method acceleration protocols for model maturation to update static models with time-lapse reservoir surveillance data

We develop an ensemble model-maturation method that is based on the Randomized Maximum Likelihood (RML) technique and adjoint-based computation of objective function gradients. The new approach is especially relevant for rich data sets with time-lapse information content. The inversion method that solves the model-maturation problem takes advantage of the adjoint-based computation of objective function gradients for a very large number of model parameters at the cost of a forward and a backward (adjoint) simulation. The inversion algorithm calibrates model parameters to arbitrary types of production data including time-lapse reservoir-pressure traces by use of a weighted and regularized objective function. We have also developed a new and effective multigrid preconditioning protocol for accelerated iterative linear solutions of the adjoint-simulation step for models with multiple levels of local grid refinement. The protocol is based on a geometric multigrid (GMG) preconditioning technique. Within the model-maturation workflow, a deep-learning technique is applied to establish links between the mesh-based inversion results (e.g., permeability-multiplier fields) and geologic modeling parameters inside a static model (e.g., object dimensions, etc.). Our workflow integrates the learnings from inversion back into the static model, and thereby, ensures the geologic consistency of the static model while improving the quality of the ensuing dynamic model in terms of honoring production and time-lapse data, and reducing forecast uncertainty. This use of deep learning (DL) to post-process the model-maturation outcome effectively converts the conventional continuous-parameter history-matching result into a discrete tomographic inversion result constrained to geological rules encoded in training images.We demonstrate the practical utilization of the adjoint-based model-maturation method on a large time-lapse reservoir-pressure data set using an ensemble of full-field models from a reservoir case study. The model-maturation technique effectively identifies the permeability modification zones that are consistent with alternative geological interpretations and proposes updates to the static model. Upon these updates, the model not only agrees better with the time-lapse reservoir-pressure data but also better honors the tubing-head pressure as well as production logging data. We also provide computational performance indicators that demonstrate the accelerated convergence characteristics of the new iterative linear solver for adjoint equations.

Single Scattering Adjoint Monte Carlo Electron Transport in FRENSIE

A new single scattering adjoint transport capability was implemented in Framework for REsearch in Nuclear ScIence and Engineering (FRENSIE). The Evaluated Electron Data Library (EEDL) was used to generate new tabulated adjoint data. All adjoint data were generated using refined EEDL data and a unit-base grid policy. Verification and validation tests were performed for the adjoint electron transport in FRENSIE. Adjoint simulation results were compared with forward simulation results for a self-adjoint infinite medium problem as well as experimental results for electron low-energy backscattering coefficients. Only a refined unit-base grid policy and coupled elastic scattering were tested for adjoint tests. The adjoint transport capability shows good agreement with the forward transport capability. The adjoint atomic excitation physics were unable to model a discrete forward source. For the self-adjoint infinite medium problems, the adjoint results matched the forward results to within 2% except near the cutoff energy. For backscattering coefficients, the adjoint results matched the forward results to within 5% for all converged bins. Overall, the adjoint transport capability was in good agreement with the forward transport capability validating the adjoint transport scheme.

Adjoint-Based Design Optimization of Nonlinear Switched Reluctance Motors

This work investigates the application of the adjoint variable method (AVM) to switched reluctance motors (SRMs). A MATLAB toolbox developed by the authors estimates the sensitivities of the required objective function with respect to different geometric design parameters using at most one adjoint simulation. In this work, the AVM evaluates the sensitivities of the x and y components of the magnetic flux density, the phase flux linkage, and the electromagnetic torque of switched reluctance motors with respect to teeth height, yoke thickness, teeth pole arc angle, and teeth taper angle of both stator and rotor. The nonlinearity of the motor magnetic material is taken into consideration. The estimated sensitivities using AVM are compared with those obtained using the more accurate but time intensive central finite differences (CFD). An interior-point optimization algorithm utilizes the sensitivities of the electromagnetic torque of an SRM to maximize the motor static torque profile. Structural mapping technique is used to control the geometric design parameters through the optimization process.

Adjoint Sensitivity Analysis of Switched Reluctance Motors

In this work, we report on the application of the adjoint variable method (AVM) to sensitivity analysis of switched reluctance motors (SRMs). Using at most one adjoint simulation, the sensitivities of the desired response with respect to all geometrical and material parameters are estimated. In this work, the AVM evaluates the sensitivities of the air region magnetic energy and the electromagnetic torque of a SRM with respect to all parameters. Good match is achieved between the AVM sensitivities and those estimated using the more accurate but computationally expensive central finite differences (CFD).

A variational approach for parameter estimation based on balanced proper orthogonal decomposition

The adjoint method has been used significantly for parameter estimation. This requires a significant programming effort to build adjoint model code and is computationally expensive as cost of one adjoint simulation often exceeds several original model runs. The work proposed here is variational data assimilation based on balanced proper orthogonal decomposition (BPOD) to identify uncertain parameters in numerical models and avoids the implementation of the adjoint with respect to model input parameters. An ensemble of model simulations (forward and backward) is used to determine the model subspace while considering both inputs and outputs of the system. By projecting the original model onto this subspace an approximate linear reduced model is obtained. The adjoint of the tangent linear model is replaced by the adjoint of linear reduced model and the minimization problem is then solved in the reduced space at very low computational cost. The performance of the method are illustrated with a number of data assimilation experiments in a 2D-advection diffusion model. The results demonstrate that the BPOD based estimation approach successfully estimates the diffusion coefficient for both advection and diffusion dominated problems. The paper also proposes an efficient method for computing the observable subspace when the number of observations is large is also proposed.

Method for computationally efficient design of dielectric laser accelerator structures.

Dielectric microstructures have generated much interest in recent years as a means of accelerating charged particles when powered by solid state lasers. The acceleration gradient (or particle energy gain per unit length) is an important figure of merit. To design structures with high acceleration gradients, we explore the adjoint variable method, a highly efficient technique used to compute the sensitivity of an objective with respect to a large number of parameters. With this formalism, the sensitivity of the acceleration gradient of a dielectric structure with respect to its entire spatial permittivity distribution is calculated by the use of only two full-field electromagnetic simulations, the original and 'adjoint'. The adjoint simulation corresponds physically to the reciprocal situation of a point charge moving through the accelerator gap and radiating. Using this formalism, we perform numerical optimizations aimed at maximizing acceleration gradients, which generate fabricable structures of greatly improved performance in comparison to previously examined geometries.

Closed-Loop Predictions in Reservoir Management Under Uncertainty

Summary Uncertainty is a major challenge in reservoir management. To take the uncertainty into consideration, optimization can be carried out over a set of scenarios. Most approaches on reservoir management under uncertainty optimize a sequence of control inputs applied to all scenarios over the prediction horizon; hence, they are open-loop predictions. In this paper, we optimize over control policies, as opposed to a sequence of control inputs, to obtain closed-loop predictions. The policies are specified as a set of implicit algebraic equations, allowing for efficient gradient calculation by an adjoint simulation. The method is compared with the more traditional open-loop approach in a case study, indicating a significant potential for reservoir optimization by use of closed-loop predictions.

Wideband Cloaking of Objects With Arbitrary Shapes Exploiting Adjoint Sensitivities

We propose an effective approach for optimal wideband cloaking of objects with arbitrary shapes exploiting adjoint sensitivity analysis. Our method optimizes the voxel-by-voxel constitutive parameters of an anisotropic cloak resulting in a large number of optimizable parameters. The associated sensitivities of a wideband cloaking objective function are efficiently estimated using our in-house anisotropic adjoint variable method (AVM) simulator. Only one adjoint simulation is needed regardless of the number of required sensitivities. A gradient-based optimization algorithm utilizes the available sensitivity information to iteratively improve the wideband cloaking objective function and to determine the constitutive parameters of the optimal cloak.

Three-Dimensional Flow Optimization of a Nozzle with a Continuous Adjoint

Abstract The article presents results of multi-criteria optimization of air nozzle topology. The optimization in the CFD has been recently developed since equations of flow in porous media were applied among others governing equations. Optimization is a seeking for extremum of an objective function with respect to the function constraints. With this definition in mind, the optimization by using a continuous adjoint for the current cases is a finding such channel topology which minimizes for example pressure or energy loss when the constraints of objective function are in the form of flow governing equations of momentum and continuity. This methodology of optimization makes a design process faster comparing to the methods related to Design of Experiments (DoE). However, for the sake of flow governing equations nonlinearity, the continuous adjoint method can be successfully applied only in relatively simply and steady-state cases. The reason is of possibility of finding the global extremum of the objective function only for that kind of cases. The results of optimization of two selected cases are presented in the article and show advantages and limitations of the method applied. The continuous adjoint simulation results indicate the nozzles design directions and can be applied in industry with limited reliability. The object of research reported in the article is the nozzle which is augmented equipment used with a pneumatic pulsator. The pulsators are devices that utilize an air stream to destruct vaults created in loose material structure. The pulsator productivity equipped with a nozzle depends on outlet pressure. Therefore, the optimization problem was stated so that pressure loss is to be as low as possible.

ICEEM07: Three-Dimensional Flow Optimization of a Pneumatic Pulsator Nozzle with a Continuous Adjoint

Abstract The article presents results of multi-criteria optimization of air nozzle topology. The optimization in the Computational Fluid Dynamics (CFD) has been recently developed since equations of flow in porous media were applied among others governing equations. Optimization is a seeking for extremum of an objective function with respect to the function constraints. With this definition in mind, the optimization by using a continuous adjoint for the current cases is a finding such channel topology which minimizes for example pressure or energy loss when the constraints of objective function are in the form of flow governing equations of momentum and continuity. This methodology of optimization makes a design process faster comparing to the methods related to Design of Experiments (DoE). However, for the sake of flow governing equations nonlinearity, the continuous adjoint method can be successfully applied only in relatively simply and steady-state cases. The method consists in seeking of a global extremum of objective function. This extremum can sometimes be find in only simple and stationary events. The results of optimization of two selected cases are presented in the article and show advantages and limitations of the method applied. The continuous adjoint simulation results indicate the nozzles design directions and can be applied in industry with limited reliability. The object of research reported in the article is the nozzle which is augmented equipment used with a pneumatic pulsator. The pulsators are devices that utilize an air stream to destruct vaults created in loose material structure. The pulsator productivity equipped with a nozzle depends on outlet pressure. Therefore, the optimization problem was stated so that pressure loss is to be as low as possible.

Adjoint Analysis of Guidance Systems with Nonstandard Inputs

The adjoint simulation method is a very efficient tool for the preliminary analysis and design of conceptual guidance systems. A pragmatic approach to adjoint simulation is based on the application of a set of rules to a block diagram representation of the system. These rules were first proposed by Laning and Battin. Knowledge and implementation of these rules on the guidance system block diagram allow the construction of an associated adjoint block diagram. This latter block diagram may then be used as a basis for generating, in a rapid and efficient manner, the desired performance data associated with the guidance system. Inherent in this process are rules stipulating how the guidance system inputs are to be treated to properly derive adjoint system outputs. However, the success of this approach is contingent on the nature and complexity of these inputs. Using numerical examples, we establish the basic requirements that must be met by the inputs to facilitate efficient adjoint analysis. For those input signals not meeting these requirements, we propose a new approach. As a practical illustration of the new approach, we investigate the miss distance performance of a generic interceptor against a weaving ballistic missile entering the atmosphere.

Time domain adjoint sensitivity analysis of electromagnetic problems with nonlinear media.

In this paper, we propose a theory for wideband adjoint sensitivity analysis of problems with nonlinear media. We show that the sensitivities of the desired response with respect to all shape and material parameters are obtained through one extra adjoint simulation. Unlike linear problems, the system matrices of this adjoint simulation are time varying. Their values are determined during the original simulation. The proposed theory exploits the time-domain transmission line modeling (TLM) and provides an efficient AVM approach for sensitivity analysis of general time domain objective functions. The theory has been illustrated through a number of examples.

Time-reversal checkpointing methods for RTM and FWI

Time-domain seismic simulation can form the basis of reverse time depth migration and full-waveform inversion. These applications need to temporally crosscorrelate a forward simulation state with an adjoint simulation state and therefore need to be able to access each time step of a forward simulation in time-reverse order. This requires saving all forward states for all times (which can require more memory than is typically available on a computer system for many problems of interest), or the ability to checkpoint information and rapidly recompute forward simulation states as needed. Prior work has suggested how to do the latter by optimally choosing which forward simulation time steps to checkpoint, thereby enabling the most efficient reuse of memory buffers and minimizing recomputation. The optimal trade-off between memory usage and recomputation can be further improved under the assumption that the information needed to do temporal crosscorrelation is smaller than the information required to restart a simulation from a given time step. This assumption is true for many geophysical problems of interest. The modification can yield a reduction in the memory requirement and recomputation time. The tested examples applied to isotropic elastic reverse time migration and anisotropic viscoelastic full-waveform inversion.