Adjoint Equations Research Articles

Time domain decomposition of parabolic control problems based on discontinuous Galerkin semi-discretization

A parallel-in-time method for the optimal control problem governed by linear parabolic equations is analyzed. The corresponding optimal system consists of a state equation (forward in time) and an adjoint equation (backward in time) which are coupled by the necessary condition. The whole time interval is divided into non-overlapping subdomains which are associated by the Robin-type transmission conditions. We apply discontinuous Galerkin method for time discretization in each subdomain and keep continuous in space. In order to prove the convergence of the algorithm, the semi-discretization errors at initial and final points are obtained based on nodal stability estimates. We complete the convergence analysis for iterative solution of the nonoverlapping Schwarz method through energy estimates for two subdomains and present the convergence rate affected by the transmission parameters. Finally, numerical experiments verify the theoretical results.

On numerical computation of sensitivity of response functions to system inputs in variational data assimilation problems

Abstract Prediction of pollution in water flow is a very important task. To this end, it is imperative to be able to define the uncertainty in a model prediction. This is the purpose of sensitivity analysis whose role is to identify what uncertainty in the model outputs is attributable to the model inputs (parameters in this case). Traditionally, this is achieved by running the model perturbed by many random samples in the parameter space to determine their impact on the model outputs. It provides information on how much of the output variance is controlled by each parameter of the inputs. The theoretical results related to the procedure based adjoint approach for computing a sensitivity of the response function (RF) to changes in the input source are presented in the paper. It is shown that this approach allows to compute, by one single integration of the adjoint equation over a given time interval, a sensitivity of the RF to any source located in the domain of interest. The proposed approach is applied to the 2D Saint-Venant flow equations for modelling the water pollution problem. A numerical experiment is formulated and implemented for the Thanh Nhan Lake in Hanoi for studying a sensitivity of some RF to observations. The numerical model is constructed by applying the well-known finite-volume method. Two appropriate optimization problems are introduced and solved on the basis of the BFGS algorithm. The numerical results show the efficiency of the proposed method and confirm the theoretical findings.

Evaluation of China's forest carbon sink service value.

A study on the value accounting of forest carbon sink services can promote the rapid development of the carbon sink market and help better understand the impact of forest carbon sinks on climate change and economic development. However, few studies have evaluated the value of China's current forest carbon sink services. Based on research on carbon peak and carbon neutrality, according to the characteristics of China's forest ecosystems and forest resource inventory data, the stock volume method was used to measure the amount and value of forest carbon sinks in China in 2009-2013 and 2014-2018. The results showed that: (1) the physical amount of forest carbon aggregates in China increased from 2009 to 2013 and from 2014 to 2018. The carbon storage of natural and plantation forests both showed an upward trend. Among them, the growth rate of the carbon storage of plantation forests was higher than that of natural forests. (2) The state, adjoint, and coupling equations of forest carbon sinks were employed to ascertain the best price for China's forest carbon sinks in 2020. The results showed that the price of China's forest carbon sinks was slightly higher than the internationally accepted carbon sink price, reflecting that the changes in the value of China's forest carbon sinks and international carbon sinks were roughly the same. (3) We obtained an appropriate accounting model for China's forest carbon sinks. (4) The value of China's forest carbon sinks increased from 2009 to 2013 and from 2014 to 2018. Although the price of carbon sinks has declined, the overall forest resource stock has increased, especially in plantation forests. The increase in the value of carbon sinks was as high as 24.7%, resulting in an overall increase in the value of forest carbon sinks, which was also in line with the physical amount of forest carbon sinks. The measurement conclusions were consistent. Several key points to note based on these findings are as follows: (1) China's current forest carbon sink transactions are all project-level certified emission reduction transactions, and diversified non-market means should be constructed to comprehensively promote carbon sink transactions. (2) China's current carbon sink transactions are mainly clean development mechanism projects, with few transactions between enterprises, and the carbon trading market situation is not optimistic. (3) The key to effective forest carbon sequestration trading is the accurate accounting of forest carbon storage and carbon sequestration value. Thus, it is of great significance to establish a forest carbon sequestration measurement method that is economical, simple, and accurate. (4) The physical amount and value of carbon sequestration of China's forest resources are rising, and the contribution rate is increasing year by year. However, there is still a gap in per capita forest area and storage compared with those in other countries worldwide. Thus, China must be vigilant in times of peace and further strengthen the protection and construction of forest resources.

The Sharma–Tasso–Olver–Burgers equation: its conservation laws and kink solitons

The present paper deals with the Sharma–Tasso–Olver–Burgers equation (STOBE) and its conservation laws and kink solitons. More precisely, the formal Lagrangian, Lie symmetries, and adjoint equations of the STOBE are firstly constructed to retrieve its conservation laws. Kink solitons of the STOBE are then extracted through adopting a series of newly well-designed approaches such as Kudryashov and exponential methods. Diverse graphs in 2 and 3D postures are formally portrayed to reveal the dynamical features of kink solitons. According to the authors’ knowledge, the outcomes of the current investigation are new and have been listed for the first time.

Sufficient and necessary conditions of near-optimal controls for a diffusion dengue model with Lévy noise

In this paper, a dengue model, which is described by spatial diffusion and Lévy noise, is built, and the sufficient and necessary conditions for near-optimal controls of the stochastic dengue model are given. First, the existence and uniqueness of the global positive solution is proved by constructing a Lyapunov function. Then, to obtain the results, the first-order adjoint equation and several estimates of the variation of the state and adjoint processes are given. Finally, the sufficient and necessary conditions for near-optimal controls are derived by virtue of Pontryagin's stochastic maximum principle, and numerical simulations are introduced to verify the effect of diffusion and Lévy noise.

A topology optimization algorithm for magnetic structures based on a hybrid FEM\u2013BEM method utilizing the adjoint approach

A method to optimize the topology of hard as well as soft magnetic structures is implemented using the density approach for topology optimization. The stray field computation is performed by a hybrid finite element–boundary element method. Utilizing the adjoint approach the gradients necessary to perform the optimization can be calculated very efficiently. We derive the gradients using a “first optimize then discretize” scheme. Within this scheme, the stray field operator is self-adjoint allowing to solve the adjoint equation by the same means as the stray field calculation. The capabilities of the method are showcased by optimizing the topology of hard as well as soft magnetic thin film structures and the results are verified by comparison with an analytical solution.

A Hybrid Adjoint Network Model for Thermoacoustic Optimization

Abstract A method is proposed that allows the computation of the continuous adjoint of a thermoacoustic network model based on the discretized direct equations. This hybrid approach exploits the self-adjoint character of the duct element, which allows all jump conditions to be derived from the direct scattering matrix. In this way, the need to derive the adjoint equations for every element of the network model is eliminated. This methodology combines the advantages of the discrete and continuous adjoint, as the accuracy of the continuous adjoint is achieved while maintaining the flexibility of the discrete adjoint. It is demonstrated how the obtained adjoint system may be utilized to optimize a thermoacoustic configuration by determining the optimal damper setting for an annular combustor.

Learning the Parameters of a Differential Equation from its Trajectory Via the Adjoint Equation

Learning the Parameters of a Differential Equation from its Trajectory Via the Adjoint Equation

Simultaneous optimization of shape and topology for stiffener creation

In this paper, we propose a method for designing stiffeners on thin plate structures by shape and topology optimization. The compliance is minimized under the constraints of volume and equilibrium equations. The optimal stiffeners’ shape and placement are simultaneously determined. Shape optimization and topology optimization are formulated as distributed-parameter optimization problem, in which the shape gradient function and density gradient function are derived by using the adjoint equations and the equilibrium equation. The shape optimization is performed by applying the shape gradient function to the H1 gradient method for vector variable, and the topology optimization is performed by applying the density gradient function to the H1 gradient method for scalar deign variable using the SIMP method. Several numerical examples are presented to confirm the effectiveness of the proposed method for the design of stiffeners.

Gaseous Pollutent Source Term Estimation Based on Adjoint Probability and Regularization Method

Fast and accurate identification of source locations and release rates is particularly important for improving indoor air quality and ensuring the safety and health of people. Existing methods based on adjoint probability are difficult to distinguish the release rate of dynamic sources, and optimization algorithms based on regularization are limited to analysing only a small amount of potential pollutant source information. Therefore, this study proposed an algorithm combining adjoint equations and regularization models to identify the location and release intensity of pollutant sources in the entire computational domain of a room. Based on the validated indoor CFD computational model, we first obtained a series of response matrices corresponding to the sensor position by solving the adjoint equation, and then used the regularization method and Bayesian inference to extrapolate the release rate and location of dynamic pollutant source in the room. The results shown that the proposed algorithm is convenient and feasible to identify the location and intensity of the indoor pollutant source. Compared with the real source intensity, the identification of constant source intensity is lower than the error threshold (10%) in 97.4% of the time nodes, and the identification of periodic source is lower than the error threshold (10%) in 95.4% of the time nodes. This research provides a new method and perspective for the estimation of indoor pollutant source information.

Optimal control of SDEs with expected path constraints and related constrained FBSDEs

In this paper, we consider optimal control of stochastic differential equations subject to an expected path constraint. The stochastic maximum principle is given for a general optimal stochastic control in terms of constrained FBSDEs. In particular, the compensated process in our adjoint equation is deterministic, which seems to be new in the literature. For the typical case of linear stochastic systems and quadratic cost functionals (i.e., the so-called LQ optimal stochastic control), a verification theorem is established, and the existence and uniqueness of the constrained reflected FBSDEs are also given.

An improved adjoint-based ocean wave reconstruction and prediction method

We propose an improved adjoint-based method for the reconstruction and prediction of the nonlinear wave field from coarse-resolution measurement data. We adopt the data assimilation framework using an adjoint equation to search for the optimal initial wave field to match the wave field simulation result at later times with the given measurement data. Compared with the conventional approach where the optimised initial surface elevation and velocity potential are independent of each other, our method features an additional constraint to dynamically connect these two control variables based on the dispersion relation of waves. The performance of our new method and the conventional method is assessed with the nonlinear wave data generated from phase-resolved nonlinear wave simulations using the high-order spectral method. We consider a variety of wave steepness and noise levels for the nonlinear irregular waves. It is found that the conventional method tends to overestimate the surface elevation in the high-frequency region and underestimate the velocity potential. In comparison, our new method shows significantly improved performance in the reconstruction and prediction of instantaneous surface elevation, surface velocity potential and high-order wave statistics, including the skewness and kurtosis.

Backward iterations for solving integral equations with polynomial nonlinearity

The theory of adjoint operators is widely used in solving applied multidimensional problems with Monte Carlo method. Efficient algorithms are constructed using the duality principle for many problems described in linear integral equations of the second kind. On the other hand, important applications of adjoint equations for designing experiments were suggested by G.Marchuk and his colleagues in their respective works. Some results obtained in these fields are also generalized to the case of nonlinear operators. Linearization methods were mostly used for that purpose. Results for Lyapunov-Schmidt nonlinear polynomial equations were obtained in the theory of Monte Carlo methods. However, many interesting questions in this subject area are remained open. New results about dual processes used for solving polynomial equations with Monte Carlo method are presented. In particular, an adjoint Markov process for the branching process and the corresponding unbiased estimate of the functional of the solution to the equation are constructed in a general form. The possibility of constructing an adjoint operator to a nonlinear one is discussed.

Sensor configuration optimization for statistical source estimation: method based on the entropy of adjoint equations and its application to building group model

Sensor configuration optimization for statistical source estimation: method based on the entropy of adjoint equations and its application to building group model

On-the-fly dual reduction for time-dependent topology optimization

A new model reduction approach is presented for time-dependent topology optimization of elastodynamic problems. Two salient features of the model reduction approach are on-the-fly and dual reduction. That is, the snapshots and reduced basis are obtained during the optimization rather than through off-line training; primal and adjoint equations are reduced independently. For the on-the-fly reduction, we use two complementary dynamic sampling strategies for snapshots: nearest nf full-order solutions and rolling transient ensemble of nt solutions. For the dual reduction, we apply the proper orthogonal decomposition and Galerkin procedure independently to the primal equations and to the adjoint equations for reduction. Central to our dual reduction strategy is the use of continuous adjoint to obtain the adjoint equations so that the adjoint equations can be reduced independently and along side the primal equation reduction. The residuals of the governing primal and adjoint equations are used to determine whether full-order model (FOM) based solutions should be used. Two-dimensional and three-dimensional numerical examples on topology optimization of elastodynamics problems are presented. Our numerical results demonstrate that the on-the-fly dual-reduction approach leads to two orders of magnitude improvement in FOM reduction (in terms of the number of FOM solution procedures being invoked) and data storage (in terms of storage of primal solutions required to solve the backward time-dependent adjoint equations and compute the sensitivity).

Research on the uniqueness and algorithm of multi-parameter identification in coupling thermal acoustic and solid problem

<p indent="0mm">In the coupled thermal-acoustic solid field, we consider the identification problem of initial boundary value from the boundary measurements of temperature and ultrasonic propagation time to reconstruct the boundary heat flux and thickness of a solid structure simultaneously. Through rigorous theoretical analysis, the uniqueness of the simultaneous identification is demonstrated, which provides theoretical support for engineering applications. By reformulating an inverse problem as an optimization problem and deriving the adjoint equation, we propose an efficient alternating iteration algorithm for solving the inverse problem. Rigorous convergence analysis of the proposed algorithm is performed. Finally, the reliability and feasibility of the algorithm are verified via numerical examples, and the algorithm’s accuracy is verified by comparing it with a single-parameter identification algorithm that only identifies heat flux.

Finite‐difference‐based simulation and adjoint optimization of gas networks

The stable operation of gas networks is an important optimization target. While for this task commonly finite volume methods are used, we introduce a new finite difference approach. With a summation by part formulation for the spatial discretization, we get well‐defined fluxes between the pipes. This allows a simple and explicit formulation of the coupling conditions at the node. From that, we derive the adjoint equations for the network simply and transparently. The resulting direct and adjoint equations are numerically efficient and easy to implement.

Solute Transport Control at Channel Junctions Using Adjoint Sensitivity

Water quality control and the control of contaminant spill in water in particular are becoming a primary need today. Gradient descent sensitivity methods based on the adjoint formulation have proved to be encouraging techniques in this context for river and channel flows. Taking into account that most channels and rivers include junctions with other branches, the objective of this study is to explore the adjoint technique on a channel network to reconstruct the upstream boundary condition of the convection-reaction equation. For this purpose, the one-dimensional shallow water equations and the transport equation for a reactive solute are considered. The control is formulated through the gradient-descent technique supplied with a first-order iterative process. Both the physical and the adjoint equations are supplied with suitable internal boundary conditions at the junction and are numerically solved using a finite volume upwind scheme. The results reveal that the adjoint technique is capable of reconstructing the inlet solute concentration boundary condition in an acceptable number of iterations for both steady state and transient configurations using a downstream measurement location. It was also observed that the reconstruction of the boundary condition tends to be less effective the further away the measurement station is from the target.

Parsimonious truncated Newton method for time-domain full-waveform inversion based on the Fourier-domain full-scattered-field approximation

To exploit Hessian information in full-waveform inversion (FWI), the matrix-free truncated Newton method can be used. In such a method, Hessian-vector product computation is one of the major concerns due to the huge memory requirements and demanding computational cost. Using the adjoint-state method, the Hessian-vector product can be estimated by zero-lag crosscorrelation of the first-/second-order incident wavefields and the second-/first-order adjoint wavefields. Different from the implementation in frequency-domain FWI, Hessian-vector product construction in the time domain becomes much more challenging because it is not affordable to store all of the time-dependent wavefields. The widely used wavefield recomputation strategy leads to computationally intensive tasks. We have developed an efficient alternative approach to computing the Hessian-vector product for time-domain FWI. In our method, discrete Fourier transform is applied to extract frequency-domain components of involved wavefields, which are used to compute wavefield crosscorrelation in the frequency domain. This makes it possible to avoid reconstructing the first- and second-order incident wavefields. In addition, a full-scattered-field approximation is proposed to efficiently simplify the second-order incident and adjoint wavefield computation, which enables us to refrain from repeatedly solving the first-order incident and adjoint equations for the second-order incident and adjoint wavefields (re)computation. With our method, the computational time can be reduced by 70% and 80% in viscous media for Gauss-Newton and full-Newton Hessian-vector product construction, respectively. The effectiveness of our method is also verified in the frame of a 2D multiparameter inversion, in which our method almost reaches the same iterative convergence of the conventional time-domain implementation.

Adjoint-based optimal control of incompressible flows with convective-like energy-stable open boundary conditions

This paper describes an open-loop optimal control strategy for flow problems governed by the incompressible Navier-Stokes equations with convective-like energy-stable boundary conditions. A quasi-Newton optimization procedure is employed, and the required objective sensitivities are computed using the continuous adjoint method. The adjoint equations for the corresponding system are derived and discussed. Both the primal and adjoint systems are solved using the least-squares finite element method. This choice circumvents the LBB condition and leads to symmetric positive-definite algebraic systems. Numerical examples are provided to show the stability and accuracy of the proposed approach.