Adjoint Problem Research Articles

Adjoint envelope tomography for scattering and absorption using radiative transfer theory

SUMMARY To investigate the small-scale elastic structure of the subsurface at length scales below the resolution limits of waveform tomography, envelopes of high-frequency scattered seismic waveforms have been used with a variety of approaches. However, a rigorous framework for the iterative inversion of seismogram envelopes to image heterogeneity and high-frequency attenuation comparable to full waveform inversion (FWI) is missing. We present the mathematical framework for an iterative full envelope inversion using forward and adjoint simulations of the radiative transfer equations, in full analogy to FWI that is based on the wave equation. The forward and adjoint problems are solved by modelling 2-D multiple non-isotropic scattering in a random elastic medium with spatially variable heterogeneity and attenuation using the Monte Carlo method. Sensitivity kernels are derived for the squared difference between the full observed and modelled envelopes which is iteratively minimized with the L-BFGS method. We apply this algorithm in numerical tests in the acoustic approximation and show that it is possible to image the spatial distribution of small-scale heterogeneity and attenuation in iterative inversions. Our analysis shows that the relative importance of scattering and attenuation anomalies needs to be considered when the model resolution is assessed. The inversions confirm that the early coda is important for imaging the distribution of heterogeneity while later coda waves are more sensitive to intrinsic attenuation and we show that this dependency can be used to cope with the trade-off that exists between both material properties.

Identification of source terms in heat equation with dynamic boundary conditions

We study an inverse parabolic problem of identifying two source terms in heat equation with dynamic boundary conditions from a final time overdetermination data. Using a weak solution approach by Hasanov, the associated cost functional is analyzed, especially a gradient formula of the functional is proved and given in terms of the solution of an adjoint problem. Next, the existence and uniqueness of a quasi‐solution are also investigated. Finally, the numerical reconstruction of some heat sources in a 1‐D equation is presented to show the efficiency of the proposed algorithm.

Estimation of transient boundary heat flux by using hybrid algorithm in a two dimensional laminar duct flow

Conjugate gradient method (CGM) with adjoint problem is a popular optimization algorithm in solving the inverse heat transfer problems. It starts with the initial guess solution of unknown parameters to be estimated which would be updated in an iterative procedure. The initial guess solution is one of the significant factors for the accuracy of estimation. In the current study, the Jaya algorithm has been developed to provide the initial guess solution to CGM. The resultant algorithm is called hybrid algorithm. The test problem considered here for the study is the estimation of transient boundary heat flux for two-dimensional hydrodynamically developed and thermally developing forced convective laminar duct flow. The hybrid algorithm is found to be robust and accurate than CGM.

Estimation of local heat flux for turbulent flow in a helical coil tube by conjugate gradient method

Estimation of local heat flux is challenging in a helical coil tube heat exchanger due to the complex flow field developed by tube curvature. The heat flux has uneven distribution in the angular direction of the tube cross-section. The current research aims to estimate the local heat flux at the fluid-solid interface for the turbulent flow of water in a helical coil tube by solving the inverse heat conduction problem (IHCP). Conjugate gradient method (CGM) with an adjoint problem is used as an inverse algorithm. First, the commercial CFD software ANSYS FLUENT is used for solving the governing equations of continuity, momentum, and energy for turbulent flow to obtain the heat flux at the fluid-solid interface. This heat flux is used to determine the temperature distribution at the outer surface of the tube. The heat flux is then considered unknown and it is estimated by CGM algorithm with the developed in-house code in MATLAB. The result shows that the estimation of heat flux by CGM is very accurate.

On the modeling of ultrasound wave propagation in the frame of inverse problem solution

In this paper we consider the inverse problem of detecting the inclusions inside the human tissue by using the acoustic sounding wave. The problem is considered in the form of coefficient inverse problem for first-order system of PDE and we use the gradient descent approach to recover the coefficients of that system. The important part of the sceme is the solution of the direct and adjoint problem on each iteration of the descent. We consider two finite-volume methods of solving the direct problem and study their Influence on the performance of recovering the coefficients.

Optimal emission control and identification of an unknown pollution source

The advection-diffusion-reaction equation is used for describing the dispersion of a quasi-passive contaminant from industrial point sources in a limited area. The conditions established on the open boundary ensure that the problem is correct in the sense of Hadamard, that is, its solution exists, is unique, and is stable to initial perturbations. The Lagrange identity is used to construct the adjoint operator and formulate an adjoint problem. Equivalent direct and adjoint estimates are derived to assess the concentration of the pollutant at monitoring sites of the area. Formulas obtained on the basis of adjoint estimates are useful in analysing the sensitivity of the model to both variations in the intensity of pollution sources and variations in the initial distribution of the pollutant concentration in the area. New optimal emission control strategies based on using the adjoint estimates are developed in order to prevent violations of existing sanitary standards by timely reduction of emission rates of operating sources. Optimal control here lies in minimizing these reductions. In addition, this control is primarily aimed at reducing the intensity of emissions from sources that most pollute the monitoring site. Also, new methods are proposed for identifying the main parameters of an unknown point source that arose as a result of a dangerous incident (accident, explosion, etc.). These methods allow determining the location and intensity of a constant or non-stationary point source, as well as the moment of emission of a pollutant in the case of an instantaneous point source. This helps to quickly assess the scale of the incident and its consequences. Numerical results show the effectiveness of the methods.

Adjoint Numerical Method for a Multiphysical Inverse Problem of Two-Phase Well Testing in Petroleum Reservoirs

Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

Kronecker Product of matrices and their applications to self-adjoint two-point boundary value problems associated with first order matrix differential systems

In this paper, we shall be concerned with Kronecker product or Tensor product of matrices and develop their properties in a systematic way. The properties of the Kronecker product of matrices is used as a tool to establish existence and uniqueness of solutions to two-point boundary value problems associated with system of first order differential systems. A new approach is described to solve the Kronecker product linear systems and establish best least square solutions to the problem. Several interesting examples are given to highlight the importance of Kronecker product of matrices. We present adjoint boundary value problems and deduce a set of necessary and sufficient conditions for the Kronecker product boundary value problem to be self-adjoint.

Hamilton-Green solver for the forward and adjoint problems in photoacoustic tomography

The majority of the solvers for the acoustic problem in Photoacoustic Tomography (PAT) rely on full solution of the wave equation, which makes them less suitable for real-time and dynamic applications where only partial data is available. This is in contrast to other tomographic modalities, e.g. X-ray tomography, where partial data implies partial cost for the application of the forward and adjoint operators. In this work we present a novel solver for the forward and adjoint wave equations for the acoustic problem in PAT. We term the proposed solver Hamilton-Green as it approximates the fundamental solution to the respective wave equation along the trajectories of the Hamiltonian system resulting from the high frequency asymptotic approximate solution for the wave equation. This approach is flexible and scalable in the sense that it allows computing the solution for each sensor independently at a fraction of the cost of the full wave solution. The theoretical foundations of our approach are rooted in results available in seismics and ocean acoustics. To demonstrate the feasibility of our approach we present results for 2D domains with homogeneous and heterogeneous sound speeds and evaluate them against a full wave solution obtained with a pseudospectral finite difference method implemented in the k-Wave toolbox [1].

Topology optimization in quasi-static plasticity with hardening using a level-set method

We study topology optimization in quasi-static plasticity with linear kinematic and linear isotropic hardening using a level-set method. We consider the primal variational formulation for the plasticity problem. This formulation is subjected to penalization and regularization, resulting in an approximate problem that is shape-differentiable. The shape derivative for the approximate problem is computed using the adjoint method. Thanks to the proposed penalization and regularization, the time discretization of the adjoint problem is proved to be well-posed. For comparison purposes, the shape derivative for the original problem is computed in a formal manner. Finally, shape and topology optimization is performed numerically using the level-set method, and 2D and 3D case studies are presented. Shapes are captured exactly using a body-fitted mesh at every iteration of the optimization algorithm.

An Adjoint Sensitivity Model for Radionuclide Transport in Heterogeneous Discrete Fractured Porous Formation in Steady‐State Regime

The development of the adjoint state method for the system of partial differential equations (PDEs) describing radionuclide transport in heterogeneous discrete fractured porous media in steady-state regime is presented in this study for both continuous and discrete adjoint approaches. The continuous adjoint approach has its own advantage in the sense that the adjoint system can be derived in terms of PDEs which can be solved analytically for specific performance measures. These PDEs can then be used to obtain a physical interpretation of the adjoint state functions. The discrete adjoint approach permits a simpler mathematical treatment based on the use of the same matrix (or its transpose) of the linear system of the discretized transport equations referring to from hereon as the forward problem. Analytical solutions for both the forward problem and its associated adjoint problem have been derived for radionuclide transport in layered formations including a fracture using an analytical inversion of tridiagonal matrices. A set of 14 formulas of sensitivity coefficients with respect to pertinent system parameters has been derived and presented in integral forms. Analytical results have been compared with numerical results and have shown excellent agreements. The discrete adjoint approach has been coupled with the differential sensitivity analysis method and applied to a real field case involving the performance assessment of a site currently being considered for underground nuclear storage in northern Switzerland, Zürich Nordost. The results show that when a fracture is present in the vicinity of waste repository, the state parameter controlling flow through the fracture, referred to herein as fracture transmissivity, is among the important parameters which should be considered in sensitivity analysis studies.

Fast multi-resolution 3D inversion of potential fields with application to high-resolution gravity and magnetic anomaly data from the Eastern Goldfields in Western Australia

We present a fast inversion algorithm tailored for high-resolution potential field (gravity or magnetic) anomaly maps. The algorithm design objectives are to optimize performance with respect to the size of problem that can be solved and computation speed. It is based on a finite element method (FEM) discretization of both the inversion and forward problems using unstructured tetrahedral meshes with coarsening at the far field. This allows a significant reduction in computational costs from a structured mesh with the same ground level resolution, as well as supporting geometrical features such as topography. Minimization of the cost function uses the Lagrange minimization approach and the resulting system of partial differential equations is solved using the integral preconditioned conjugate gradient method (I-PCG). I-PCG works naturally with an unstructured mesh FEM and can effectively precondition with the Hessian of the regularization term. In contrast to established approaches, minimization of the cost function occurs prior to discretization and using I-PCG avoids the need to invert a large dense sensitivity matrix, a major obstacle to solving large 3D problems using parallel computers. Fast solvers for the forward problem, the adjoint problem and the preconditioner make use of the smoothed aggregation algebraic multi-grid method for the preconditioner in another PCG iteration (AMG-PCG). Parallel implementation uses a domain decomposition approach to support scalability with spatial extent. The proposed method is applied to a synthetic example and tohigh-resolution data set of rastered gravity and magnetic anomaly maps from the Eastern Goldfields in Western Australia with the finest resolution of over 1 million data points and about 60 million cells for the 3D inversion. For this field application we demonstrate that the proposed algorithm achieves computation time scaling with grid size, an indication of optimal algorithm performance and delivers strong scalability with number of cores. The gravity inversion result is compared to the inversion result using the Broyden Fletcher Goldfarb Shanno (BFGS) method on a structured grid.

Linear perturbations of the Linet–Tian metrics with a positive cosmological constant

The Linet–Tian metrics are solutions of the Einstein equations with a cosmological constant, Λ, that can be positive or negative. The linear instability of these metrics in the case Λ < 0, has already been established (Gleiser 2017 Class. Quantum Grav. 34 065010). In the case Λ > 0, it was found in a recent analysis that the perturbation equations admit unstable modes (Gleiser 2018 arXiv:1810.07296v2[gr-qc]). The analysis was based on the construction of a gauge invariant function of the metric perturbation coefficients, called here W(y). This function satisfied a linear second order equation that could be used to set up a boundary value problem determining the allowed, real or purely imaginary frequencies for the perturbations. Nevertheless, the relation of these solutions to the full spectrum of perturbations, and, therefore, to the evolution of arbitrary perturbations, remained open. In this paper we consider again the perturbations of the Linet–Tian metric with Λ > 0, and show, using a form of the Darboux transformation, that one can associate with the perturbation equations a self adjoint problem that provides a solution to the completeness and spectrum of the perturbations. This is also used to construct the explicit relation between the solutions of the gauge invariant equation for W(y), and the evolution of arbitrary initial data, thus solving the problem that remained open in the previous study. Numerical methods are then used to confirm the existence of unstable modes as a part of the complete spectrum of the perturbations, thus establishing the linear gravitational instability of the Linet–Tian metrics with Λ > 0.

A posteriori goal-oriented bounds for the Poisson problem using potential and equilibrated flux reconstructions: Application to the hybridizable discontinuous Galerkin method

We present a general framework to compute upper and lower bounds for linear-functional outputs of the exact solutions of the Poisson equation based on reconstructions of the field variable and flux for both the primal and adjoint problems. The method is devised from a generalization of the complementary energy principle and the duality theory. Using duality theory, the computation of bounds is reduced to finding independent potential and equilibrated flux reconstructions. A generalization of this result is also introduced allowing to derive alternative guaranteed bounds from nearly-arbitrary H(div;Ω) flux reconstructions (only zero-order equilibration is required). This approach is applicable to any numerical method used to compute the solution. In this work, the proposed approach is applied to derive bounds for the hybridizable discontinuous Galerkin (HDG) method. An attractive feature of the proposed approach is that superconvergence on the bound gap is achieved, yielding accurate bounds even for very coarse meshes. Numerical experiments are presented to illustrate the performance and convergence of the bounds for the HDG method in both uniform and adaptive mesh refinements.

Adjoint‐Based Inversion of Geodetic Data for Sources of Deformation and Strain

AbstractAn adjoint‐based formulation leads to a particularly efficient approach for inverting geodetic measurements for the source of the deformation. Specifically, the quantities necessary to iteratively improve the fit to the observations can be computed with just three forward calculations, one to obtain the current residuals, another to solve the adjoint problem, and a third to compute the step length. An inversion algorithm utilizing the adjoint‐based gradient is applied to a set of Interferometric Synthetic Aperture Radar (InSAR) data gathered between 2016 and 2018 over the Tulare Basin in California's Central Valley. Because the measured deformation is due to groundwater withdrawal, a penalty function is included in the inversion to avoid placing aquifer volume change in locations that are far from any documented wells. The solution of the inverse problem provides estimates of aquifer compaction that provide a match to the observed range changes while honoring the well data. The solution indicates an average aquifer volume loss of 2.17 /year over the two year period from January 2016 to January 2018, encompassing one drought year (2016) and one wet year (2017). This magnitude of lost volume is compatible with the 3.1 /year decrease in water volume for the entire Central Valley, estimated from GRACE satellite gravity data.

On the cavity detection in a heat conductive medium from time-average boundary temperature measurement

Consider the shape identification of an inclusion in heat conductive medium from the time average measurement, which is modeled by an initial boundary value problem for a parabolic system with extra nonlocal measurement data specified on the outer boundary. For this nonlocal and nonlinear inverse problem for the two-dimensioned parabolic equation in a doubly-connected domain, the radius function describing the shape of inner boundary to be identified is defined as the minimizer of a regularizing cost functional. The existence of this minimizer is firstly proven in a suitable admissible set. Then we establish the convergence rate of the regularizing solution under a-posteriori choice strategy for the regularizing parameter. Finally the differentiability of the cost functional is proven, which provides a fundamental basis for gradient type iteration scheme. Based on the adjoint and sensitivity problem of the original problem which give the gradient of the cost functional, we propose a steepest descent iteration algorithm for finding the minimizer approximately. Numerical examples are presented to show the validity of our algorithm.

Estimation of Functional Form of Time-Dependent Heat Transfer Coefficient Using an Accurate and Robust Parameter Estimation Approach: An Inverse Analysis

This paper presents a numerical method to address function estimation problems in inverse heat transfer problems using parameter estimation approach without prior information on the functional form of the variable to be estimated. Using an inverse analysis, the functional form of a time-dependent heat transfer coefficient is estimated efficiently and accurately. The functional form of the heat transfer coefficient is assumed unknown and the inverse heat transfer problem should be treated using a function estimation approach by solving sensitivity and adjoint problems during the minimization process. Based on proposing a new sensitivity matrix, however, the functional form can be estimated in an accurate and very efficient manner using a parameter estimation approach without the need for solving the sensitivity and adjoint problems and imposing extra computational cost, mathematical complexity, and implementation efforts. In the proposed sensitivity analysis scheme, all sensitivity coefficients can be computed in only one direct problem solution at each iteration. In this inverse heat transfer problem, the body shape is irregular and meshed using a body-fitted grid generation method. The direct heat conduction problem is solved using the finite-difference method. The steepest-descent method is used as a minimization algorithm to minimize the defined objective function and the termination of the minimization process is carried out based on the discrepancy principle. A test case with three different functional forms and two different measurement errors is considered to show the accuracy and efficiency of the used inverse analysis.

An Adaptive Material Interpolation for the Reconstruction of P-Wave Velocity Models with Sharp Interfaces using the Topology Optimization Method

A topology optimization (TO) approach is used to reconstruct P-wave velocity models with sharp interfaces. The concept of material model (interpolation), usually applied in TO to design structures and multi-physics devices, is explored in this work to solve this inverse problem. An adaptive interpolation rule is proposed to deal with the reconstruction problem as a transition from a single- to a multi-material approach combining the Solid Isotropic Material with Penalization (SIMP) and peak function material models. Data collected during the optimization process is used to find material candidates by means of a curve fitting strategy based on generalized simulated annealing (GSA), if this information is not available. The numerical analysis is carried out using a finite element (FE) approach in the frequency domain. Both forward and adjoint problems are solved aided by an open source Domain-Specific Language (DSL) framework and automated derivation tool, while the optimization problem is solved by using a BFGS algorithm. Numerical results for 2D examples demonstrated that proposed material interpolation can lead to solutions with sharper interfaces and improved resolution without including any type of regularization or extra constraint in the optimization problem.

A note on the exact boundary controllability for an imperfect transmission problem

In this note, we consider a hyperbolic system of equations in a domain made up of two components. We prescribe a homogeneous Dirichlet condition on the exterior boundary and a jump of the displacement proportional to the conormal derivatives on the interface. This last condition is the mathematical interpretation of an imperfect interface. We apply a control on the external boundary and, by means of the Hilbert Uniqueness Method, introduced by J. L. Lions, we study the related boundary exact controllability problem. The key point is to derive an observability inequality by using the so called Lagrange multipliers method, and then to construct the exact control through the solution of an adjoint problem. Eventually, we prove a lower bound for the control time which depends on the geometry of the domain, on the coefficients matrix and on the proportionality between the jump of the solution and the conormal derivatives on the interface.

Steady‐state estimation of thermal contact conductance between sliding disk and stationary cylinder with similar/dissimilar materials under the isothermally heated boundary condition

AbstractAn accurate understanding of the thermal contact conductance (TCC) is imperative for the enhancement of the performance and service life of metallic cylindrical joints. However, the evaluation of the TCC between conforming cylindrical solids is quite intricate, as it depends on the temperature, pressure, roughness, relative sliding speed, and thermophysical properties of solids. Instead of generating contacting surfaces stochastically, in this communication, an experimental setup in the lab‐scale consisting of a stationary cylinder and a sliding hot disk has been fabricated in which temporal evolutions of the temperature on both solids are recorded by thermocouples. Effects of the disk temperature and relative sliding speed on the steady‐state TCC have been investigated. There is a cylindrical interface between two conforming smooth solids of similar/dissimilar materials. As there is the differential expansion of cylindrical solids due to the temperature gradient, the diameter of sliding disks has been truncated to fix the radial gap (contact pressure). Inverse solution with the conjugate gradient method along with the adjoint problem has been applied to estimate the steady‐state TCC. One‐dimensional heat transfer analysis shows that frictional heating is negligible. The steady‐state TCC is almost constant longitudinally along the axis of the steel cylinder. Under identical initial and isothermally heated boundary conditions, the TCC between the aluminum disk and cylinder is higher than that between steel disk and cylinder. Also, the TCC has been increased with a rising slope when the working temperature is enhanced. However, the augmentation of the relative sliding speed engenders a decreasing slope in the TCC graph.