arXiv subjects

J. C. Lemm

Publications and source records attributed to J. C. Lemm.

11 recordsLinked to original sources

arXiv2003

Bayesian Approach to Inverse Quantum Statistics: Reconstruction of Potentials in the Feynman Path Integral Representation of Quantum Theory

The Feynman path integral representation of quantum theory is used in a non--parametric Bayesian approach to determine quantum potentials from measurements on a canonical ensemble. This representation allows to study explicitly the classical and semiclassical limits and provides a unified description in terms of functional integrals: the Feynman path integral for the statistical operator, and the integration over the space of potentials for calculating the predictive density. The latter is treated in maximum a posteriori approximation, and various approximation schemes for the former are developed and discussed. A simple numerical example shows the applicability of the method.

quant-ph↗

arXiv2000

Bayesian Inverse Quantum Theory

A Bayesian approach is developed to determine quantum mechanical potentials from empirical data. Bayesian methods, combining empirical measurements and "a priori" information, provide flexible tools for such empirical learning problems. The paper presents the basic theory, concentrating in particular on measurements of particle coordinates in quantum mechanical systems at finite temperature. The computational feasibility of the approach is demonstrated by numerical case studies. Finally, it is shown how the approach can be generalized to such many-body and few-body systems for which a mean field description is appropriate. This is done by means of a Bayesian inverse Hartree-Fock approximation.

quant-ph↗

arXiv2000

Bayesian Reconstruction of Approximately Periodic Potentials at Finite Temperature

The paper discusses the reconstruction of potentials for quantum systems at finite temperatures from observational data. A nonparametric approach is developed, based on the framework of Bayesian statistics, to solve such inverse problems. Besides the specific model of quantum statistics giving the probability of observational data, a Bayesian approach is essentially based on "a priori" information available for the potential. Different possibilities to implement "a priori" information are discussed in detail, including hyperparameters, hyperfields, and non--Gaussian auxiliary fields. Special emphasis is put on the reconstruction of potentials with approximate periodicity. The feasibility of the approach is demonstrated for a numerical model.

quant-ph↗

arXiv2000

Bayesian Field Theory: Nonparametric Approaches to Density Estimation, Regression, Classification, and Inverse Quantum Problems

Bayesian field theory denotes a nonparametric Bayesian approach for learning functions from observational data. Based on the principles of Bayesian statistics, a particular Bayesian field theory is defined by combining two models: a likelihood model, providing a probabilistic description of the measurement process, and a prior model, providing the information necessary to generalize from training to non-training data. The particular likelihood models discussed in the paper are those of general density estimation, Gaussian regression, clustering, classification, and models specific for inverse quantum problems. Besides problem typical hard constraints, like normalization and positivity for probabilities, prior models have to implement all the specific, and often vague, "a priori" knowledge available for a specific task. Nonparametric prior models discussed in the paper are Gaussian processes, mixtures of Gaussian processes, and non-quadratic potentials. Prior models are made flexible by including hyperparameters. In particular, the adaption of mean functions and covariance operators of Gaussian process components is discussed in detail. Even if constructed using Gaussian process building blocks, Bayesian field theories are typically non-Gaussian and have thus to be solved numerically. According to increasing computational resources the class of non-Gaussian Bayesian field theories of practical interest which are numerically feasible is steadily growing. Models which turn out to be computationally too demanding can serve as starting point to construct easier to solve parametric approaches, using for example variational techniques.

physics.data-an↗

arXiv2000

Inverse Time-Dependent Quantum Mechanics

Using a new Bayesian method for solving inverse quantum problems, potentials of quantum systems are reconstructed from coordinate measurements in non-stationary states. The approach is based on two basic inputs: 1. a likelihood model, providing the probabilistic description of the measurement process as given by the axioms of quantum mechanics, and 2. additional "a priori" information implemented in form of stochastic processes over potentials.

quant-ph↗

arXiv1999

Mixtures of Gaussian process priors

Nonparametric Bayesian approaches based on Gaussian processes have recently become popular in the empirical learning community. They encompass many classical methods of statistics, like Radial Basis Functions or various splines, and are technically convenient because Gaussian integrals can be calculated analytically. Restricting to Gaussian processes, however, forbids for example the implemention of genuine nonconcave priors. Mixtures of Gaussian process priors, on the other hand, allow the flexible implementation of complex and situation specific, also nonconcave "a priori" information. This is essential for tasks with, compared to their complexity, a small number of available training data. The paper concentrates on the formalism for Gaussian regression problems where prior mixture models provide a generalisation of classical quadratic, typically smoothness related, regularisation approaches being more flexible without having a much larger computational complexity.

physics.data-an↗

arXiv1999

Beyond the time independent mean field theory for nuclear and atomic reactions: Inclusion of particle-hole correlations in a generalized random phase approximation

The time independent mean field method (TIMF) for scattering defines biorthonormal sets of single-particle wave functions and corresponding creation and annihilation operators. 2p - 2h correlations can be introduced through a generalized random phase approximation; 1p - 1h contributions vanish (Brillouin theorem). While the general variational method for scattering by Giraud and Nagarajan solves inhomogeneous Euler equations by inversion of the standard, hermitean Hamiltonian, the present approach diagonalizes a non-hermitean Hamiltonian, which carries the information about entrance and exit channels.

nucl-th↗

arXiv1999

Hartree-Fock Approximation for Inverse Many-Body Problems

A new method is presented to reconstruct the potential of a quantum mechanical many-body system from observational data, combining a nonparametric Bayesian approach with a Hartree-Fock approximation. A priori information is implemented as a stochastic process, defined on the space of potentials. The method is computationally feasible and provides a general framework to treat inverse problems for quantum mechanical many-body systems.

nucl-th↗

arXiv1999

A Bayesian Approach to Inverse Quantum Statistics

A nonparametric Bayesian approach is developed to determine quantum potentials from empirical data for quantum systems at finite temperature. The approach combines the likelihood model of quantum mechanics with a priori information over potentials implemented in form of stochastic processes. Its specific advantages are the possibilities to deal with heterogeneous data and to express a priori information explicitly, i.e., directly in terms of the potential of interest. A numerical solution in maximum a posteriori approximation was feasible for one--dimensional problems. Using correct a priori information turned out to be essential.

cond-mat.stat-mech↗

arXiv1998

Mean Field Methods for Atomic and Nuclear Reactions: The Link between Time--Dependent and Time--Independent Approaches

Three variants of mean field methods for atomic and nuclear reactions are compared with respect to both conception and applicability: The time--dependent Hartree--Fock method solves the equation of motion for a Hermitian density operator as initial value problem, with the colliding fragments in a continuum state of relative motion. With no specification of the final state, the method is restricted to inclusive reactions. The time--dependent mean field method, as developed by Kerman, Levit and Negele as well as by Reinhardt, calculates the density for specific transitions and thus applies to exclusive reactions. It uses the Hubbard--Stratonovich transformation to express the full time--development operator with two--body interactions as functional integral over one--body densities. In stationary phase approximation and with Slater determinants as initial and final states, it defines non--Hermitian, time--dependent mean field equations to be solved self--consistently as boundary value problem in time. The time--independent mean field method of Giraud and Nagarajan is based on a Schwinger--type variational principle for the resolvent. It leads to a set of inhomogeneous, non--Hermitian equations of Hartree--Fock type to be solved for given total energy. All information about initial and final channels is contained in the inhomogeneities, hence the method is designed for exclusive reactions. A direct link is established between the time--dependent and time--independent versions. Their relation is non--trivial due to the non--linear nature of mean field methods.

nucl-th↗

arXiv1994

Antisymmetrization of a Mean Field Calculation of the T-Matrix

The usual definition of the prior(post) interaction $V(V^\prime )$ between projectile and target (resp. ejectile and residual target) being contradictory with full antisymmetrization between nucleons, an explicit antisymmetrization projector ${\cal A}$ must be included in the definition of the transition operator, $ T\equiv V^\prime{\cal A}+V^\prime{\cal A}GV. $ We derive the suitably antisymmetrized mean field equations leading to a non perturbative estimate of $T$. The theory is illustrated by a calculation of forward $α$-$α$ scattering, making use of self consistent symmetries.

nucl-th↗