6. Optimization of eigenvalues and eigenmodes by using the adjoint method
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Anca-Maria Toader
Abstract
Application of the adjoint method has proven successful in shape optimization and topology optimization. In the present chapter the adjoint method is applied to the optimization of eigenvalues and eigenmodes (eigenvectors). The general case of an arbitrary cost function depending on the first n eigenvalues and eigenmodes is detailed. The direct problem does not involve a bilinear form and a linear form as usual in other applications. However, it is possible to follow the spirit of themethod and deduce n adjoint problems and obtain n adjoint states, where n is the number of eigenmodes taken into account for optimization. An optimization algorithm based on the derivative of the cost function is developed. This derivative depends on the derivatives of the eigenmodes and the adjoint method allows one to express it in terms of the the adjoint states and of the solutions of the direct eigenvalue problem. The formulas hold for the case when the eigenvalues are simple. A section is dedicated to discussions on the case when there are multiple eigenvalues. The same procedures are applied to optimization of microstructures, modeled by Bloch waves. The results obtained hold for general functionals depending on the eigenvalues and on the eigenmodes of the microstructure. However, the wave vector k⃗ is amore delicate case of optimization parameter. The derivative of a general functionalwith respect to k⃗ is obtained,which has interesting implications in band-gap maximization problems.
Abstract
Application of the adjoint method has proven successful in shape optimization and topology optimization. In the present chapter the adjoint method is applied to the optimization of eigenvalues and eigenmodes (eigenvectors). The general case of an arbitrary cost function depending on the first n eigenvalues and eigenmodes is detailed. The direct problem does not involve a bilinear form and a linear form as usual in other applications. However, it is possible to follow the spirit of themethod and deduce n adjoint problems and obtain n adjoint states, where n is the number of eigenmodes taken into account for optimization. An optimization algorithm based on the derivative of the cost function is developed. This derivative depends on the derivatives of the eigenmodes and the adjoint method allows one to express it in terms of the the adjoint states and of the solutions of the direct eigenvalue problem. The formulas hold for the case when the eigenvalues are simple. A section is dedicated to discussions on the case when there are multiple eigenvalues. The same procedures are applied to optimization of microstructures, modeled by Bloch waves. The results obtained hold for general functionals depending on the eigenvalues and on the eigenmodes of the microstructure. However, the wave vector k⃗ is amore delicate case of optimization parameter. The derivative of a general functionalwith respect to k⃗ is obtained,which has interesting implications in band-gap maximization problems.
Kapitel in diesem Buch
- Frontmatter I
- Contents V
-
Part I
- 1. Geometric issues in PDE problems related to the infinity Laplace operator 3
- 2. Solution of free boundary problems in the presence of geometric uncertainties 20
- 3. Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies 40
- 4. High-order topological expansions for Helmholtz problems in 2D 64
- 5. On a new phase field model for the approximation of interfacial energies of multiphase systems 123
- 6. Optimization of eigenvalues and eigenmodes by using the adjoint method 142
- 7. Discrete varifolds and surface approximation 159
-
Part II
- Preface 173
- 8. Weak Monge–Ampère solutions of the semi-discrete optimal transportation problem 175
- 9. Optimal transportation theory with repulsive costs 204
- 10. Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations 257
- 11. On the Lagrangian branched transport model and the equivalence with its Eulerian formulation 281
- 12. On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows 304
- 13. Pressureless Euler equations with maximal density constraint: a time-splitting scheme 333
- 14. Convergence of a fully discrete variational scheme for a thin-film equation 356
- 15. Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance 400
- Index 417
Kapitel in diesem Buch
- Frontmatter I
- Contents V
-
Part I
- 1. Geometric issues in PDE problems related to the infinity Laplace operator 3
- 2. Solution of free boundary problems in the presence of geometric uncertainties 20
- 3. Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies 40
- 4. High-order topological expansions for Helmholtz problems in 2D 64
- 5. On a new phase field model for the approximation of interfacial energies of multiphase systems 123
- 6. Optimization of eigenvalues and eigenmodes by using the adjoint method 142
- 7. Discrete varifolds and surface approximation 159
-
Part II
- Preface 173
- 8. Weak Monge–Ampère solutions of the semi-discrete optimal transportation problem 175
- 9. Optimal transportation theory with repulsive costs 204
- 10. Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations 257
- 11. On the Lagrangian branched transport model and the equivalence with its Eulerian formulation 281
- 12. On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows 304
- 13. Pressureless Euler equations with maximal density constraint: a time-splitting scheme 333
- 14. Convergence of a fully discrete variational scheme for a thin-film equation 356
- 15. Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance 400
- Index 417
