Atef Elsherbeni

Second-order adjoint sensitivities

The authors show in this chapter, two different approaches for efficiently estimating the second-order derivatives (Hessian matrix) of a given objective function. The cost of evaluating the Hessian using classical finite difference approach is O(n2) where n is the number of parameters. The first adjoint approach reduces the cost of estimating all components of the Hessian matrix to only 2n extra simulations. This approach is simple, and it uses the algorithms developed in previous chapters. A second approach for estimating the complete Hessian is also presented. This approach is more complex than the first approach and requires extra memory storage. This approach requires only n + 1 extra simulations per Hessian evaluation. It follows that the computational cost is approximately one half of the first adjoint approach. This saving comes at the cost of a more complex algorithm and more extensive storage.

Adjoint sensitivity analysis of anisotropic structures

In this chapter, we present an algorithm for adjoint sensitivity analysis of anisotropic materials. We show that using only one extra adjoint simulation, the sensitivities of the objective function with respect to all parameters are estimated. The material property tensors of the adjoint problem are the transpose of those of the original problem. The computational cost of the adjoint problem is the same as that of the original simulation. The derivation considered in this chapter addresses the nondispersive case. The considered tensors are assumed to be independent of time. The formulation presented in this chapter is adapted. The anisotropic and dispersive case is still a subject of research.

Introduction to sensitivity analysis approaches

In this chapter, we give an initial introduction to adjoint techniques. We first review the classical approaches for derivative estimation and illustrate their computational cost. Approaches such as forward finite differences (FFD), backward finite differences (BFD), and central finite differences (CFD) are addressed. These approaches are utilized in later chapters as a reference to compare the estimated adjoint sensitivities against. We then present an adjoint sensitivity formulation that is suitable for analyzing electric circuits. The adjoint SA of electrical circuits and conductor transmission lines was addressed by several researchers. This analysis serves as a smooth introduction to the basic concepts involved in adjoint analysis of high-frequency structures. The same theorem is extended to high-frequency electromagnetic simulations as will be illustrated in the following chapters. We illustrate the theory presented in this chapter with circuit examples.

Transient adjoint sensitivity analysis

In this chapter, we address the transient time domain adjoint sensitivity analysis problem. Without loss of generalization, we will consider the desired transient response to be an electric field component at a point ro in the computational domain E(ro, kΔt). We show in this chapter that we can evaluate the derivatives ∂E(ro, kΔt)/ ∂pi, for all parameters, and for all time steps, using only one extra simulation. In other words, we predict how the complete transient response at the probe changes due to incremental changes in any of the parameters.

Adjoint sensitivity analysis with dispersive materials

In all the previously discussed cases, we assumed that the materials utilized in the FDTD simulations are nondispersive. This means that all material properties (permittivity, permeability, and conductivity) do not change with frequency. In many interesting applications, however, this is not the case. For example, in the emerging area of metamaterials, the effective permittivity and permeability show strong dependency on frequency [1]. In the area of plasmonics, all metals have dispersive properties [2-5]. The same applies to modeling materials in the THz and infrared frequency regimes [6,7]. It is thus of prime importance to be able to estimate the sensitivities of different responses with respect to all geometrical and material parameters of structures with dispersive material properties. In this chapter, we develop a general theory for adjoint sensitivity analysis of high frequency dispersive structures. This formulation applies to materials with commonly used types of dispersion profiles such as Lorentz [8,9], Drude [10-12], and Debye [13,14]. We show that only one dispersive adjoint simulation is required to estimate the sensitivities of the desired response with respect to all parameters. We illustrate this approach through one example.