Smoothing the Landscape: Moreau-Yosida Regularization in Modern Density-Functional Theory

In the standard convex‑analytic formulation of density‑functional theory the universal functional is non‑smooth, complicating the connection to a practical scheme. We apply Moreau–Yosida regularisation that adds a quadratic penalty term to obtain smooth approximants of the functional whose derivatives are Lipschitz‑continuous and converge to the exact functional as the regularisation parameter goes to zero. This regularity enables a framework for inversion to obtain the Kohn‑Sham "effective" potential utilizing the proximal map that is a crucial ingredient of Moreau–Yosida regularisation. This can also be done in practice using a plane‑wave basis.

References:
Herbst, M.F., Bakkestuen, V.H. and Laestadius, A., 2025. Kohn-Sham inversion with mathematical guarantees. Physical Review B, 111(20), p.205143.   
Penz, M. and Laestadius, A., 2026. Adaptation of Moreau–Yosida regularization to the modulus of convexity. Journal of Mathematical Analysis and Applications, 554(2), p.129956.

This event is jointly organized with the MATHICSE seminar.
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The electronic structure reading group brings together researchers and students interested in mathematical aspects of electronic structure problems and adjacent topics, including:

• Density Functional Theory
• Many-body Schrödinger equation for electrons
• Born-Oppenheimer Molecular Dynamics
• Numerical analysis and error control