On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 5, p. 977-995

This paper considers Schrödinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. We then propose to use this formulation in the case of the so-called fixed node approximation of Fermion groundstates, defined by the bottom eigenelements of the Schrödinger operator of a Fermionic system with Dirichlet conditions on the nodes (the set of zeros) of an initially guessed skew-symmetric function. We show that shape derivatives of the fixed node energy vanishes if and only if either (i) the distribution on the nodes of the stopped random process is symmetric; or (ii) the nodes are exactly the zeros of a skew-symmetric eigenfunction of the operator. We propose an approximation of the shape derivative of the fixed node energy that can be computed with a Monte-Carlo algorithm, which can be referred to as Nodal Monte-Carlo (NMC). The latter approximation of the shape derivative also vanishes if and only if either (i) or (ii) holds.

DOI : https://doi.org/10.1051/m2an/2010049
Classification:  60H30,  65C35,  65C05,  35P99
Keywords: Schrödinger operator, groundstate, shape derivatives, Feynman-Kac formula, quantum Monte-Carlo methods, Fermion nodes, fixed node approximation
     author = {Rousset, Mathias},
     title = {On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {5},
     year = {2010},
     pages = {977-995},
     doi = {10.1051/m2an/2010049},
     zbl = {pre05798940},
     mrnumber = {2731400},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2010__44_5_977_0}
Rousset, Mathias. On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 5, pp. 977-995. doi : 10.1051/m2an/2010049. http://www.numdam.org/item/M2AN_2010__44_5_977_0/

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