Grothendieck proved that if f : X −→ Y is a proper morphism ofnice schemes, then Rf∗ has a right adjoint, which is given as tensor productwith the relative canonical bundle. The original proof was by patching localdata. Deligne proved the existence of the adjoint by a global argument, andVerdier showed that this global adjoint may be computed locally.In this article we show ...

A new adjoint sensitivity analysis approach is presented for time-dependent partial differential equations with adaptive mesh refinement. The new approach, called ADDA, combines the best features of both the adjoint of the discretization (AD) and discretization of the adjoint (DA) approaches. It removes the obstacles of applying AD to adaptive methods and, in contrast to DA, requires for its us...

Let H be a Hilbert space. For a bounded operator A : H → H its Hilbert space adjoint is an operator A∗ : H → H such that 〈Ax, y〉 = 〈x,A∗y〉 for all x, y ∈ H. We say that A is bounded self adjoint if A = A∗. In this chapter we discussed several results about the spectrum of a bounded self adjoint operator on a Hilbert space. We emphasize that in this chapter A is bounded, there is also a notion o...

This paper presents an adjoint method for the calculation of non-collocated sensitivities in supersonic flow. The goal is to develop a set of discrete adjoint equations and their corresponding boundary conditions in order to quantify the influence of geometry modifications on the pressure distribution at an arbitrary location within the domain of interest. First, this paper presents the complet...

This paper is concerned with state-space realizations of the adjoints of the variationals of Hamiltonian control systems. It will be shown that the variational systems of a class of Hamiltonian systems have self-adjoint state-space realizations, that is, the variational system and its adjoint have the same state-space realizations. This implies that the inputoutput mapping of the adjoint of the...

This paper presents an adjoint method for the calculation of remote sensitivities in supersonic flow. The goal is to develop a set of adjoint equations and their corresponding boundary conditions in order to quantify the influence of geometry modifications on the pressure distribution at an arbitrary location within the domain of interest. First, this paper presents the formulation and discreti...

In this paper has been studied the wave equation in some non-classic cases. In the  rst case boundary conditions are non-local and non-periodic. At that case the associated spectral problem is a self-adjoint problem and consequently the eigenvalues are real. But the second case the associated spectral problem is non-self-adjoint and consequently the eigenvalues are complex numbers,in which two ...

We present an adjoint-based optimization for electromagnetic design. It embeds commercial Maxwell solvers within a steepest-descent inverse-design optimization algorithm. The adjoint approach calculates shape derivatives at all points in space, but requires only two "forward" simulations. Geometrical shape parameterization is by the level set method. Our adjoint design optimization is applied t...

Because detailed aerodynamic shape optimizations still suffer from high computational costs, efficient optimization strategies are required. Regarding the deterministic optimization methods, the adjoint approach is seen as a promising alternative to the classical finite difference approach [1, 2]. With the adjoint approach the sensitivities needed for the aerodynamic shape optimization can be e...

An observation sensitivity method to identify targeted observations is implemented in the context of four dimensional variational (4D-Var) data assimilation. This methodology is compared with the well-established adjoint sensitivity method using a nonlinear Burgers equation as a test model. Automatic differentiation software is used to implement the first order adjoint model to calculate the gr...