Adjoint Variables Research Articles

Mean-Field Pontryagin Maximum Principle

We derive a maximum principle for optimal control problems with constraints given by the coupling of a system of ordinary differential equations and a partial differential equation of Vlasov type with smooth interaction kernel. Such problems arise naturally as Gamma-limits of optimal control problems constrained by ordinary differential equations, modeling, for instance, external interventions on crowd dynamics by means of leaders. We obtain these first-order optimality conditions in the form of Hamiltonian flows in the Wasserstein space of probability measures with forward–backward boundary conditions with respect to the first and second marginals, respectively. In particular, we recover the equations and their solutions by means of a constructive procedure, which can be seen as the mean-field limit of the Pontryagin Maximum Principle applied to the optimal control problem for the discretized density, under a suitable scaling of the adjoint variables.

Efficient adjoint sensitivity analysis of isotropic hardening elastoplasticity via load steps reduction approximation

Sensitivity analysis plays a key role in gradient-based shape optimization. In this paper, a highly efficient procedure for design sensitivity analysis of elastoplastic material with isotropic hardening rule is presented. Here, an adjoint variable method is employed, where the adjoint sensitivity formulation is derived from the discrete master equation, yield condition, and consistency condition. Due to history dependence of plasticity, adjoint variables must be solved via a stepwise backward procedure. Therefore, the computational cost increases in proportion to the number of load steps. To address this cost and improve efficiency, condensation rules are proposed to reduce the number of load steps in the sensitivity analysis based on properties of adjoint variables. The properties of adjoint variables and proposed condensation rules are all theoretically proven in this paper. The accuracy and efficiency of the proposed techniques are demonstrated using bar truss structures and solid structures in various complex load cases. The results show that the techniques significantly reduce the number of load steps required in the sensitivity analysis while keeping satisfying accuracy.

Investigation into Aerodynamic Shape Optimization of Planar and Nonplanar Wings

Problems in three-dimensional aerodynamic shape optimization can produce complex design spaces due to the nonlinear physics of the Navier–Stokes equations and the large number of design variables used. In this paper, a Newton–Krylov optimization algorithm is applied to a set of complex aerodynamic optimization problems in order to investigate its behaviour and performance. The methodology solves the Reynolds-averaged Navier–Stokes equations with a parallel Newton–Krylov algorithm. Aerodynamic geometries are meshed using structured multiblock grids, which are then fitted with B-spline control volumes for mesh deformation and geometry control. A gradient-based optimization method is used, with adjoint variables calculated using a Krylov method. The optimization of the Common Research Model (CRM) wing is revisited, with a focus on the effect of varying geometric constraints and on the possibility of multimodality. In addition, several cases are presented that involve a high degree of shape change: two planform optimizations starting from a rectangular wing, and investigation of various wingtip treatments. The results characterize the methodology, demonstrating its robustness and ability to address optimization problems with substantial geometric freedom.

Heterogeneous multiscale method for optimal control problem governed by parabolic equation with highly oscillatory coefficients

In this paper, we investigate the heterogeneous multiscale method (HMM) for the optimal control problem governed by the parabolic equation with highly oscillatory coefficients. The state variable and adjoint state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements, while the control variable is discretized by the piecewise constants. By applying the well-known Lions’ Lemma to the discretized optimal control problem, we obtain the necessary and sufficient optimality conditions. A priori error estimates are derived for the state, co-state and the control with uniform bounded constants. Finally, numerical results are presented to illustrate our theoretical findings.

A Full Multigrid Method for Distributed Control Problems Constrained by Stokes Equations

AbstractA full multigrid method with coarsening by a factor-of-three to distributed control problems constrained by Stokes equations is presented. An optimal control problem with cost functional of velocity and/or pressure tracking-type is considered with Dirichlet boundary conditions. The optimality system that results from a Lagrange multiplier framework, form a linear system connecting the state, adjoint, and control variables. We investigate multigrid methods with finite difference discretization on staggered grids. A coarsening by a factor-of-three is used on staggered grids that results nested hierarchy of staggered grids and simplified the inter-grid transfer operators. A distributive-Gauss-Seidel smoothing scheme is employed to update the state- and adjoint-variables and a gradient update step is used to update the control variables. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed multigrid framework to tracking-type optimal control problems.

New elliptic projections and a priori error estimates of H1-Galerkin mixed finite element methods for optimal control problems governed by parabolic integro-differential equations

In this paper, we discuss a priori error estimates of H1-Galerkin mixed finite element methods for optimal control problems governed by parabolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart–Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. Both semidiscrete and fully discrete schemes are considered. Based on some new elliptic projections, we derive a priori error estimates for the control variable, the state variables and the adjoint state variables. The related a priori error estimates for the new projections error are also established. A numerical example is given to demonstrate the theoretical results.

Optimal distributed control of a diffuse interface model of tumor growth**The financial support of the FP7-IDEAS-ERC-StG #256872 (EntroPhase) and of the project Fondazione Cariplo-Regione Lombardia MEGAsTAR ‘Matematica d’Eccellenza in biologia ed ingegneria come accelleratore di una nuona strateGia per l’ATtRattività dell’ateneo pavese’ is gratefully acknowledged. The paper also benefited from the support of the MIUR-PRIN Grant 2015PA5MP7 ‘Calculus of Variations’

In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed by Hawkins–Daruud et al in Hawkins–Daruud et al (2011 Int. J. Numer. Math. Biomed. Eng. 28 3–24). The model consists of a Cahn–Hilliard equation for the tumor cell fraction φ coupled to a reaction-diffusion equation for a function σ representing the nutrient-rich extracellular water volume fraction. The distributed control u monitors as a right-hand side of the equation for σ and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Fréchet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

Optimal switching between cash-flow streams

We consider the problem of finding the optimal time to switch between two measurable cash-flow streams. A complete characterization of the set of solutions is obtained in terms of adjoint variables which measure the available gain from deviations. An iterative procedure for the computation of the adjoint variables is provided. The results are generalized to multiple switching times, multiple cash-flow streams, switching costs, as well as switch-triggered cash-flow streams that arise in equipment-replacement problems.

Dynamics Response Control of a Mindlin-Type Beam

Optimal boundary control for damping the vibrations in a Mindlin-type beam is considered. Wellposedness and controllability of the system are investigated. A maximum principle is introduced and optimal control function is obtained by means of maximum principle. Also, by using maximum principle, control problem is reduced to solving a system of partial differential equations including state, adjoint variables, which are subject to initial, boundary and terminal conditions. The solution of the system is obtained by using MATLAB. Numerical results are presented in table and graphical forms.

Time domain viscoelastic full waveform inversion

Viscous attenuation can have a strong impact on seismic wave propagation, but it is rarely taken into account in full waveform inversion (FWI). When viscoelasticity is considered in time domain FWI, the displacement formulation of the wave equation is usually used instead of the popular velocity–stress formulation. However, inversion schemes rely on the adjoint equations, which are quite different for the velocity–stress formulation than for the displacement formulation. In this paper, we apply the adjoint state method to the isotropic viscoelastic wave equation in the velocity–stress formulation based on the generalized standard linear solid rheology. By applying linear transformations to the wave equation before deriving the adjoint state equations, we obtain two symmetric sets of partial differential equations for the forward and adjoint variables. The resulting sets of equations only differ by a sign change and can be solved by the same numerical implementation. We also investigate the crosstalk between parameter classes (velocity and attenuation) of the viscoelastic equation. More specifically, we show that the attenuation levels can be used to recover the quality factors of P and S waves, but that they are very sensitive to velocity errors. Finally, we present a synthetic example of viscoelastic FWI in the context of monitoring CO₂ geological sequestration. We show that FWI based on our formulation can indeed recover P- and S-wave velocities and their attenuation levels when attenuation is high enough. Both changes in velocity and attenuation levels recovered with FWI can be used to track the CO₂ plume during and after injection. Further studies are required to evaluate the performance of viscoelastic FWI on real data.

A nonlinear plate control without linearization

Abstract In this paper, an optimal vibration control problem for a nonlinear plate is considered. In order to obtain the optimal control function, wellposedness and controllability of the nonlinear system is investigated. The performance index functional of the system, to be minimized by minimum level of control, is chosen as the sum of the quadratic 10 functional of the displacement. The velocity of the plate and quadratic functional of the control function is added to the performance index functional as a penalty term. By using a maximum principle, the nonlinear control problem is transformed to solving a system of partial differential equations including state and adjoint variables linked by initial-boundary-terminal conditions. Hence, it is shown that optimal control of the nonlinear systems can be obtained without linearization of the nonlinear term and optimal control function can be obtained analytically for nonlinear systems without linearization.

Optimal Boundary Control of a Simplified Ericksen–Leslie System for Nematic Liquid Crystal Flows in 2D

In this paper, we investigate an optimal boundary control problem for a two dimensional simplified Ericksen--Leslie system modelling the incompressible nematic liquid crystal flows. The hydrodynamic system consists of the Navier--Stokes equations for the fluid velocity coupled with a convective Ginzburg--Landau type equation for the averaged molecular orientation. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the molecular orientation is subject to a time-dependent Dirichlet boundary condition that corresponds to the strong anchoring condition for liquid crystals. We first establish the existence of optimal boundary controls. Then we show that the control-to-state operator is Fr\'echet differentiable between appropriate Banach spaces and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.

A multigrid solver for Stokes control problems

ABSTRACTMultigrid solvers for distributed optimal control problems constrained by Stokes equations are presented. The distributed velocity tracking problem is considered with Dirichlet boundary conditions. The optimality system of the control problem that results from a Lagrange multiplier framework, forms a linear system connecting the state, adjoint, and control variables. We investigate multigrid methods on staggered grids. A coarsening by a factor of three is introduced that results in a nested hierarchy of staggered grids and simplified the intergrid transfer operators. On these grids a distributive Gauss–Seidel smoothing scheme is employed. Numerical experiments are performed to validate the effectiveness and efficiency of the proposed multigrid staggered grid framework.

The Gradient Discretization Method for Optimal Control Problems, with Superconvergence for Nonconforming Finite Elements and Mixed-Hybrid Mimetic Finite Differences

In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretisation method. Gradient schemes are defined for the optimality system of the control problem. Error estimates for state, adjoint and control variables are derived. Superconvergence results for gradient schemes under realistic regularity assumptions on the exact solution is discussed. These super-convergence results are shown to apply to non-conforming $\mathbb{P}_1$ finite elements, and to the mixed/hybrid mimetic finite differences. Results of numerical experiments are demonstrated for the conforming, nonconforming and mixed/hybrid mimetic finite difference schemes.

A NEW IMAGING ALGORITHM FOR ELECTRIC CAPACITANCE TOMOGRAPHY

A new imaging algorithm of permittivity for Electrical Capacitance Tomography (ECT) was proposed in this paper. Some aspects of sensitivity analysis and the application of adjoint variables were discussed. In the final section of this paper some advantages and disadvantages were shown.

A level-set-based topology optimisation for acoustic–elastic coupled problems with a fast BEM–FEM solver

This paper presents a structural optimisation method in three-dimensional acoustic–elastic coupled problems. The proposed optimisation method finds an optimal allocation of elastic materials which reduces the sound level on some fixed observation points. In the process of the optimisation, configuration of the elastic materials is expressed with a level set function, and the distribution of the level set function is iteratively updated with the help of the topological derivative. The topological derivative is associated with state and adjoint variables which are the solutions of the acoustic–elastic coupled problems. In this paper, the acoustic–elastic coupled problems are solved by a BEM–FEM coupled solver, in which the fast multipole method (FMM) and a multi-frontal solver for sparse matrices are efficiently combined. Along with the detailed formulations for the topological derivative and the BEM–FEM coupled solver, we present some numerical examples of optimal designs of elastic sound scatterer to manipulate sound waves, from which we confirm the effectiveness of the present method.

A convergent adaptive finite element method for electrical impedance tomography

In this work we develop and analyze an adaptive finite element method for efficiently solving electrical impedance tomography -- a severely ill-posed nonlinear inverse problem for recovering the conductivity from boundary voltage measurements. The reconstruction technique is based on Tikhonov regularization with a Sobolev smoothness penalty and discretizing the forward model using continuous piecewise linear finite elements. We derive an adaptive finite element algorithm with an a posteriori error estimator involving the concerned state and adjoint variables and the recovered conductivity. The convergence of the algorithm is established, in the sense that the sequence of discrete solutions contains a convergent subsequence to a solution of the optimality system for the continuous formulation. Numerical results are presented to verify the convergence and efficiency of the algorithm.

Global Optimization on an Interval

This paper provides expressions for solutions of a one-dimensional global optimization problem using an adjoint variable which represents the available one-sided improvements up to the interval “horizon.” Interpreting the problem in terms of optimal stopping or optimal starting, the solution characterization yields two-point boundary problems as in dynamic optimization. Results also include a procedure for computing the adjoint variable, as well as necessary and sufficient global optimality conditions.

A Nonconforming Finite Element Approximation for Optimal Control of an Obstacle Problem

Abstract The article deals with the analysis of a nonconforming finite element method for the discretization of optimization problems governed by variational inequalities. The state and adjoint variables are discretized using Crouzeix–Raviart nonconforming finite elements, and the control is discretized using a variational discretization approach. Error estimates have been established for the state and control variables. The results of numerical experiments are presented.

Adaptive finite element method for elliptic optimal control problems: convergence and optimality

In this paper we consider the convergence analysis of adaptive finite element method for elliptic optimal control problems with pointwise control constraints. We use variational discretization concept to discretize the control variable and piecewise linear and continuous finite elements to approximate the state variable. Based on the well-established convergence theory of AFEM for elliptic boundary value problems, we rigorously prove the convergence and quasi-optimality of AFEM for optimal control problems with respect to the state and adjoint state variables, by using the so-called perturbation argument. Numerical experiments confirm our theoretical analysis.