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.

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 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.

Stress-Constrained Topology Optimization Using Approximate Reanalysis with on-the-Fly Reduced Order Modeling

Most of the methods used today for handling local stress constraints in topology optimization, fail to directly address the non-self-adjointness of the stress-constrained topology optimization problem. This in turn could drastically raise the computational cost for an already large-scale problem. These problems involve both the equilibrium equations resulting from finite element analysis (FEA) in each iteration, as well as the adjoint equations from the sensitivity analysis of the stress constraints. In this work, we present a paradigm for large-scale stress-constrained topology optimization problems, where we build a multi-grid approach using an on-the-fly Reduced Order Model (ROM) and the p-norm aggregation function, in which the discrete reduced-order basis functions (modes) are adaptively constructed for both the primal and dual problems. In addition to reducing the computational savings due to the ROM, we also address the computational cost of the ROM learning and updating phases. Both reduced-order bases are enriched according to the residual threshold of the corresponding linear systems, and the grid resolution is adaptively selected based on the relative error in approximating the objective function and constraint values during the iteration. The tests on 2D and 3D benchmark problems demonstrate improved performance with acceptable objective and constraint violation errors. Finally, we thoroughly investigate the influence of relevant stress constraint parameters such as the coagulation factor, stress penalty factor, and the allowable stress value.

Stability analysis of functionals in variational data assimilation with respect to uncertainties of input data for a sea thermodynamics model

The problem of stability and sensitivity of functionals of the optimal solution of the variational data assimilation of sea surface temperature for the model of sea thermodynamics is considered. The variational data assimilation problem is formulated as an optimal control problem to find the initial state and the boundary heat flux. The sensitivity of the response functions as functionals of the optimal solution with respect to the observation data is studied. Computing the gradient of the response function reduces to the solution of a non-standard problem being a coupled system of direct and adjoint equations with mutually dependent initial and boundary values. The algorithm to compute the gradient of the response function is presented, based on the Hessian of the original cost functional. Stability analysis of the response function with respect to uncertainties of input data is given. Numerical examples are presented for the Black and Azov seas thermodynamics model.

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.

Sensitivity of functionals in variational data assimilation for a sea thermodynamics model

The sensitivity of functionals of the optimal solution to a variational data assimilation problem for the sea thermodynamics model is studied. The variational data assimilation problem is formulated as an optimal control problem to find the initial state and the boundary condition. The sensitivity of the response functions as functionals of the optimal solution with respect to the observation data is determined by the gradient of the response function and reduces to the solution of a non-standard problem being a coupled system of direct and adjoint equations with mutually dependent initial and boundary values. The algorithm to compute the gradient of the response function is presented, based on the Hessian of the original cost functional. The sensitivity analysis of the response function with respect to errors of observation data is carried out. Numerical examples are presented for the Black Sea thermodynamics model.

A posteriori error estimates for semilinear optimal control problems

In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we devise and analyze a reliable and efficient a posteriori error estimator for a semilinear optimal control problem; control constraints are also considered. We consider a fully discrete scheme that discretizes the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. The devised error estimator can be decomposed as the sum of three contributions which are associated to the discretization of the state and adjoint equations and the control variable. We extend our results to a scheme that approximates the control variable with piecewise linear functions and also to a scheme that approximates the solution to a nondifferentiable optimal control problem. We illustrate the theory with two and three-dimensional numerical examples.

Neural closure models for dynamical systems

Complex dynamical systems are used for predictions in many domains. Because of computational costs, models are truncated, coarsened or aggregated. As the neglected and unresolved terms become important, the utility of model predictions diminishes. We develop a novel, versatile and rigorous methodology to learn non-Markovian closure parametrizations for known-physics/low-fidelity models using data from high-fidelity simulations. The new neural closure models augment low-fidelity models with neural delay differential equations (nDDEs), motivated by the Mori–Zwanzig formulation and the inherent delays in complex dynamical systems. We demonstrate that neural closures efficiently account for truncated modes in reduced-order-models, capture the effects of subgrid-scale processes in coarse models and augment the simplification of complex biological and physical–biogeochemical models. We find that using non-Markovian over Markovian closures improves long-term prediction accuracy and requires smaller networks. We derive adjoint equations and network architectures needed to efficiently implement the new discrete and distributed nDDEs, for any time-integration schemes and allowing non-uniformly spaced temporal training data. The performance of discrete over distributed delays in closure models is explained using information theory, and we find an optimal amount of past information for a specified architecture. Finally, we analyse computational complexity and explain the limited additional cost due to neural closure models.

A Hybrid Adjoint Network Model for Thermoacoustic Optimization

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 whilst 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.

A Hybrid Adjoint Network Model for Thermoacoustic Optimization

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 whilst 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.