| Abstract: |
| We study a class of PDE-constrained optimisation problems in which the control is restricted to take two values under a global constraint, leading naturally to a topological optimisation framework. Such problems arise in applications including material design, population dynamics, and spectral optimisation. From an analytical perspective, optimal solutions exhibit a bang--bang structure and can be characterised using rearrangement ttheory and symmetrisation techniques, providing insight into the geometry of optimal configurations, although explicit solutions are typically available only in special cases. On the computational side, rearrangement methods offer an efficient iterative approach based on thresholding of state or adjoint variables. These methods are known to converge and, in certain settings, achieve linear convergence rates; however, their performance may still be limited in practice. Motivated by acceleration techniques in optimisation, we introduce an accelerated rearrangement method (ARM) for two-phase problems. The method incorporates momentum-type updates through extrapolation of Fr\`echet derivatives while preserving admissibility of the control. Numerical results demonstrate significantly improved convergence compared to classical rearrangement schemes. |
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