On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes

Mathias Rousset

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 5, page 977-995
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Rousset, Mathias. "On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes." ESAIM: Mathematical Modelling and Numerical Analysis 44.5 (2010): 977-995. <http://eudml.org/doc/250804>.

@article{Rousset2010,
abstract = { 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. },
author = {Rousset, Mathias},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Schrödinger operator; groundstate; shape derivatives; Feynman-Kac formula; quantum Monte-Carlo methods; Fermion nodes; fixed node approximation; Dirichlet groundstates; random processes; quantum Monte-Carlo method},
language = {eng},
month = {8},
number = {5},
pages = {977-995},
publisher = {EDP Sciences},
title = {On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes},
url = {http://eudml.org/doc/250804},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Rousset, Mathias
TI - On a probabilistic interpretation of shape derivatives of Dirichlet groundstates with application to Fermion nodes
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/8//
PB - EDP Sciences
VL - 44
IS - 5
SP - 977
EP - 995
AB - 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.
LA - eng
KW - Schrödinger operator; groundstate; shape derivatives; Feynman-Kac formula; quantum Monte-Carlo methods; Fermion nodes; fixed node approximation; Dirichlet groundstates; random processes; quantum Monte-Carlo method
UR - http://eudml.org/doc/250804
ER -

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