Projected Langevin Monte Carlo algorithms in non-convex and super-linear setting

It is of significant interest in many applications to sample from a high-dimensional target distribution $\pi$ with the density $\pi(\text{d} x) \propto e^{-U(x)} (\text{d} x)$, based on the temporal discretization of the Langevin stochastic differential equations (SDEs). In this paper, we propose an explicit projected Langevin Monte Carlo (PLMC) algorithm with non-convex potential $U$ and super-linear gradient of $U$ and investigate the non-asymptotic analysis of its sampling error in total variation distance. Equipped with time-independent regularity estimates for the corresponding Kolmogorov equation, we derive the non-asymptotic bounds on the total variation distance between the target distribution of the Langevin SDEs and the law induced by the PLMC scheme with order $\mathcal{O}(h |\ln h|)$. Moreover, for a given precision $\epsilon$, the smallest number of iterations of the classical Langevin Monte Carlo (LMC) scheme with the non-convex potential $U$ and the globally Lipshitz gradient of $U$ can be guaranteed by order ${\mathcal{O}}\big(\tfrac{d^{3/2}}{\epsilon} \cdot \ln (\tfrac{d}{\epsilon}) \cdot \ln (\tfrac{1}{\epsilon}) \big)$. Numerical experiments are provided to confirm the theoretical findings.Comment: 31 pages, 6 figure

Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients

This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with an order one. Numerical experiments are provided to verify our theoretical findings.Comment: 45 pages, 7 figure

Synthesis, properties, and optical applications of noble metal nanoparticle-biomolecule conjugates

Noble metal nanoparticles, such as gold or silver nanoparticles and nanorods, exhibit unique photonic, electronic and catalytic properties. Functionalization of noble metal nanoparticles with biomolecules (e. g., protein and DNA) produces systems that possess numerous applications in catalysis, delivery, therapy, imaging, sensing, constructing nanostructures and controlling the structure of biomolecules. In this paper, the recent development of noble metal nanoparticle-biomolecule conjugates is reviewed from the following three aspects: (1) synthesis of noble metal nanoparticle-biomolecule systems by electrostatic adsorption, direct chemisorption of thiol derivatives, covalent binding through bifunctional linkers and specific affinity interactions; (2) the photonic properties and bioactivation of noble metal nanoparticle-biomolecule conjugates; and (3) the optical applications of such systems in biosensors, and medical imaging, diagnosis, and therapy. The conjugation of Au and Ag nanoparticles with biomolecules and the most recent optical applications of the resulting systems have been focused on

Projected Langevin Monte Carlo algorithms in non-convex and super-linear setting

It is of significant interest in many applications to sample from a high-dimensional target distribution π with the density π(dx)∝e−U(x)(dx), based on the temporal discretization of the Langevin stochastic differential equations (SDEs). In this paper, we propose an explicit projected Langevin Monte Carlo (PLMC) algorithm with non-convex potential U and super-linear gradient of U and investigate the non-asymptotic analysis of its sampling error in total variation distance. Equipped with time-independent regularity estimates for the corresponding Kolmogorov equation, we derive the non-asymptotic bounds on the total variation distance between the target distribution of the Langevin SDEs and the law induced by the PLMC scheme with order O(h| ln h|). Moreover, for a given precision ϵ, the smallest number of iterations of the classical Langevin Monte Carlo (LMC) scheme with the non-convex potential U and the globally Lipshitz gradient of U can be guaranteed by order O(d3/2ϵ⋅ln(dϵ)⋅ln(1ϵ)). Numerical experiments are provided to confirm the theoretical findings

Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients

This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings

Synthesis, properties, and optical applications of noble metal nanoparticle-biomolecule conjugates

China-MOST [2008DFA51230]; National Basic Research Program of China [2007CB936603]; National Natural Science Foundation of China [11074207, 60776007]Noble metal nanoparticles, such as gold or silver nanoparticles and nanorods, exhibit unique photonic, electronic and catalytic properties. Functionalization of noble metal nanoparticles with biomolecules (e. g., protein and DNA) produces systems that possess numerous applications in catalysis, delivery, therapy, imaging, sensing, constructing nanostructures and controlling the structure of biomolecules. In this paper, the recent development of noble metal nanoparticle-biomolecule conjugates is reviewed from the following three aspects: (1) synthesis of noble metal nanoparticle-biomolecule systems by electrostatic adsorption, direct chemisorption of thiol derivatives, covalent binding through bifunctional linkers and specific affinity interactions; (2) the photonic properties and bioactivation of noble metal nanoparticle-biomolecule conjugates; and (3) the optical applications of such systems in biosensors, and medical imaging, diagnosis, and therapy. The conjugation of Au and Ag nanoparticles with biomolecules and the most recent optical applications of the resulting systems have been focused on

Wave-Packet Surface Propagation for Light-Induced Molecular Dissociation

Recent advances in laser technology have enabled tremendous progress in photochemistry, at the heart of which is the breaking and formation of chemical bonds. Such progress has been greatly facilitated by the development of accurate quantum-mechanical simulation method, which, however, does not necessarily accompany clear dynamical scenarios and is rather often a black box, other than being computationally heavy. Here, we develop a wave-packet surface propagation (WASP) approach to describe the molecular bond-breaking dynamics from a hybrid quantum-classical perspective. Via the introduction of quantum elements including state transitions and phase accumulations to the Newtonian propagation of the nuclear wave-packet, the WASP approach naturally comes with intuitive physical scenarios and accuracies. It is carefully benchmarked with the H2+ molecule and is shown to be capable of precisely reproducing experimental observations. The WASP method is promising for the intuitive visualization of strong-field molecular dynamics and is straightforwardly extensible toward complex molecules.Comment: 24 pages, 4 figure