We demonstrate how path integrals often used in problems of theoretical physics can be adapted to provide a machinery for performing Bayesian inference in function spaces. Such inference comes about naturally in the study of inverse problems of recovering continuous (infinite dimensional) coefficient functions from ordinary or partial differential equations, a problem which is typically ill-posed. Regularization of these problems using L2 function spaces (Tikhonov regularization) is equivalent to Bayesian probabilistic inference, using a Gaussian prior. The Bayesian interpretation of inverse problem regularization is useful since it allows one to quantify and characterize error and degree of precision in the solution of inverse problems, as well as examine assumptions made in solving the problem—namely whether the subjective choice of regularization is compatible with prior knowledge. Using path-integral formalism, Bayesian inference can be explored through various perturbative techniques, such as the semiclassical approximation, which we use in this manuscript. Perturbative path-integral approaches, while offering alternatives to computational approaches like Markov-Chain-Monte-Carlo (MCMC), also provide natural starting points for MCMC methods that can be used to refine approximations. In this manuscript, we illustrate a path-integral formulation for inverse problems and demonstrate it on an inverse problem in membrane biophysics as well as inverse problems in potential theories involving the Poisson equation.
The quantitative adiabatic condition (QAC), or quantitative condition, is a
convenient (a priori) tool for estimating the adiabaticity of quantum evolutions.
However, the range of the applicability of QAC is not well understood. It has
been shown that QAC can become insufficient for guaranteeing the validity of
the adiabatic approximation, but under what conditions the QAC would become
necessary has become controversial. Furthermore, it is believed that the inability
for the QAC to reveal quantum adiabaticity is due to induced resonant transitions.
However, it is not clear how to quantify these transitions in general. Here
we present a progress to this problem by finding an exact relation that can reveal
how transition amplitudes are related to QAC directly. As a posteriori condition
for quantum adiabaticity, our result is universally applicable to any (nondegenerate)
quantum system and gives a clear picture on how QAC could
become insufficient or unnecessary for the adiabatic approximation, which is a
problem that has gained considerable interest in the literature in recent years
We solve the entanglement classification under stochastic local operations and classical communication
(SLOCC) for general n-qubit states. For two arbitrary pure n-qubit states connected via local operations,
we establish an equation between the two coefficient matrices associated with the states. The rank of the
coefficient matrix is preserved under SLOCC and gives rise to a simple way of partitioning all the pure
states of n qubits into different families of entanglement classes, as exemplified here. When applied to the
symmetric states, this approach reveals that all the Dicke states |l,n> with l=1,..., [n/2] are
inequivalent under SLOCC.
Consider an age-dependent, single-species branching process defined by a progeny number distribution, and a lifetime distribution associated with each independent particle. In this paper, we focus on the associated inverse problem where one wishes to formally solve for the progeny number distribution or the lifetime distribution that defines the Bellman-Harris branching process. We derive results for the existence and uniqueness (the identifiability) of these two distributions given one of two types of information: the extinction time probability of the entire process (extinction time distribution), or the distribution of the total number of particles at one fixed time. We demonstrate that perfect knowledge of the distribution of extinction times allows us to formally determine either the progeny number distribution or the lifetime distribution. Furthermore, we show that these constructions are unique. We then consider “data” consisting of a perfectly known total number distribution given at one specific time. For a process with known progeny number distribution and exponentially distributed lifetimes, we show that the rate parameter is identifiable. For general lifetime distributions, we also show that the progeny distribution is globally unique. Our results are presented through four theorems, each describing the constructions in the four distinct cases.
Inhomogeneous percolation, for its closer relationship with real-life, can be more useful and reasonable than homogeneous percolation to illustrate the critical phenomena and dynamical behaviour of complex networks. However, due to its intricacy, the theoretical framework of inhomogeneous percolation is far from being complete and many challenging problems are still open. Due to this reasn, in this paper, we investigate inhomogeneous site percolation on Bethe Lattices with two occupation probabilities, and we also extend the result to percolation with m occupation probabilities. The critical behaviour of this inhomogeneous percolation is shown clearly by formulating the percolation probability with given occupation probability, the critical occupation probability, and the average cluster size. Moreover, using the above theory, we discuss in detail the diffusion behaviour of an infectious disease (SARS) and present specific disease-control strategies in consideration of groups with different infection probabilities.