Adjoint methods for computing derivatives of functions of eigenvectors using shift-and-invert preconditioning

The derivatives of functions of eigenvectors arise in numerous applications, including in the design of structural and thermal systems. The adjoint method can be used to compute these derivatives provided the eigenvalues are distinct. However, the computational cost of the adjoint method grows rapidly with the number of eigenvectors, since a unique adjoint equation is needed for each eigenvector, necessitating the solution of a sequence of linear systems. To address this challenge, three novel methods are presented that each utilize shift-and-invert preconditioning: an approximate method based on fixed Lanczos vectors, a method based on differentiating the Lanczos eigenvalue solution procedure and a preconditioned block Krylov method. An adjoint modification is proposed, which enables the evaluation of derivatives of functions of eigenvectors when numerically repeated eigenvalues occur, provided the function has regular derivatives. The accuracy and efficiency of these proposed methods are compared to conventional methods for computing the derivatives of eigenvector functions and demonstrated on thermal, natural frequency and buckling problems.