Error estimation and control are critical ingredients for improving the reliability of computational simulations. Adjoint-based techniques can be used to both estimate the error in chosen solution outputs and to provide local indicators for adaptive refinement. This article reviews recent work on these techniques for computational fluid dynamics applications in aerospace engineering. The defini...

Integrating factors and adjoint equations are determined for linear and nonlinear differential equations of an arbitrary order. The new concept of an adjoint equation is used for construction of a Lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables. The method is illustrated by...

Adjoint method is widely used in aerodynamic design because only once solution of flow field required for it to obtain the gradients all variables. However, computational cost adjoint vector approximately equal that computation. In order accelerate and improve efficiency adjoint-based optimization, machine learning modeling presented. Deep neural network (DNN) employed construct mapping between...

Adjoint methods form a class of importance sampling methods that are used to accelerate Monte Carlo (MC) simulations of transport equations. Ideally, adjoint methods allow for zero-variance MC estimators provided that the solution to an adjoint transport equation is known. Hybrid methods aim at (i) approximately solving the adjoint transport equation with a deterministic method; and (ii) use th...

This paper presents a systematic procedure based on the adjoint method for solving a class of inverse directional alloy solidification design problems in which a desired growth velocity v f is achieved under stable growth conditions. To the best of our knowledge, this is the first time that a continuum adjoint formulation is proposed for the solution of an inverse problem with simultaneous heat...

— This paper proposes a method for optimal dynamic control of nonlinear time-invariant systems with a quadratic performance criterion by using the adjoint formulation of the system. From the Lagrangian the adjoint system is derived in terms of the Lagrange Multipliers, which are also called costate variables. Our approach makes the elimination of the costate possible and provides a dynamic cont...

We consider parabolic SPDEs driven by purely spatial noise, and show the existence of solutions with random initial data and forcing terms. We perform error analysis for the semi-discrete stochastic finite element method applied to a class of equations with self-adjoint differential operators that are independent of time. The analysis employs the formal stochastic adjoint problem and the corres...

This article is concerned with geometric integrators which are linearization-preserving, i.e. numerical integrators which preserve the exact linearization at every fixed point of an arbitrary system of ODEs. For a canonical Hamiltonian system, we propose a new symplectic and self-adjoint B-series method which is also linearization-preserving. In a similar fashion, we show that it is possible to...