# Browsing by Author "Lu, Jianfeng"

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Item Open Access A Hybrid Global-local Numerical Method for Multiscale PDEs(2017-04-23) Huang, Y; Lu, Jianfeng; Ming, PWe present a new hybrid numerical method for multiscale partial differential equations, which simultaneously captures both the global macroscopic information and resolves the local microscopic events. The convergence of the proposed method is proved for problems with bounded and measurable coefficient, while the rate of convergence is established for problems with rapidly oscillating periodic or almost-periodic coefficients. Numerical results are reported to show the efficiency and accuracy of the proposed method.Item Open Access A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic EquationsLu, Jianfeng; Lu, Yulong; Wang, MinThis paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of two-layer neural networks in the framework of the DRM for solving two prototype elliptic PDEs: Poisson equation and static Schr\"odinger equation on the $d$-dimensional unit hypercube. Specifically, we prove that the convergence rates of generalization errors are independent of the dimension $d$, under the a priori assumption that the exact solutions of the PDEs lie in a suitable low-complexity space called spectral Barron space. Moreover, we give sufficient conditions on the forcing term and the potential function which guarantee that the solutions are spectral Barron functions. We achieve this by developing a new solution theory for the PDEs on the spectral Barron space, which can be viewed as an analog of the classical Sobolev regularity theory for PDEs.Item Open Access A Variation on the Donsker-Varadhan Inequality for the Principial Eigenvalue(2017-04-23) Lu, Jianfeng; Steinerberger, StefanThe purpose of this short note is to give a variation on the classical Donsker-Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain $\Omega$ by the largest mean first exit time of the associated drift-diffusion process via $$\lambda_1 \geq \frac{1}{\sup_{x \in \Omega} \mathbb{E}_x \tau_{\Omega^c}}.$$ Instead of looking at the mean of the first exist time, we study quantiles: let $d_{p, \partial \Omega}:\Omega \rightarrow \mathbb{R}_{\geq 0}$ be the smallest time $t$ such that the likelihood of exiting within that time is $p$, then $$\lambda_1 \geq \frac{\log{(1/p)}}{\sup_{x \in \Omega} d_{p,\partial \Omega}(x)}.$$ Moreover, as $p \rightarrow 0$, this lower bound converges to $\lambda_1$.Item Open Access Accelerated sampling by infinite swapping of path integral molecular dynamics with surface hopping(2017-11-30) Lu, Jianfeng; Zhou, ZhennanTo accelerate the thermal equilibrium sampling of multi-level quantum systems, the infinite swapping limit of a recently proposed multi-level ring polymer representation is investigated. In the infinite swapping limiting, the ring polymer evolves according to an averaged Hamiltonian with respect to all possible surface index configurations of the ring polymer. A multiscale integrator for the infinite swapping limit is also proposed to enable practical sampling based on the limiting dynamics, avoiding the enumeration of all possible surface index configurations, which grows exponentially with respect to the number of beads in the ring polymer. Numerical results demonstrate the huge improvement of sampling efficiency of the infinite swapping compared with the direct simulation of path integral molecular dynamics with surface hopping.Item Open Access Accelerating the Computation of Density Functional Theory's Correlation Energy under Random Phase Approximations(2019) Thicke, KyleWe propose novel algorithms for the fast computation of density functional theory's exchange-correlation energy in both the particle-hole and particle-particle random phase approximations (phRPA and ppRPA). For phRPA, we propose a new cubic scaling algorithm for the calculation of the RPA correlation energy. Our scheme splits up the dependence between the occupied and virtual orbitals in the density response function by use of Cauchy's integral formula. This introduces an additional integral to be carried out, for which we provide a geometrically convergent quadrature rule. Our scheme also uses the interpolative separable density fitting algorithm to further reduce the computational cost in a way analogous to that of the resolution of identity method.

For ppRPA, we propose an algorithm based on stochastic trace estimation. A contour integral is used to break up the dependence between orbitals. The logarithm is expanded into a polynomial, and a variant of the Hutchinson algorithm is proposed to find the trace of the polynomial. This modification of the Hutchinson algorithm allows us to use the structure of the problem to compute each Hutchinson iteration in only quadratic time. This is a large asymptotic improvement over the previous state-of-the-art quartic-scaling method and over the naive sextic-scaling method.

Item Open Access An isoperimetric problem with Coulomb repulsion and attraction to a background nucleus(2017-04-23) Lu, Jianfeng; Otto, FelixWe study an isoperimetric problem the energy of which contains the perimeter of a set, Coulomb repulsion of the set with itself, and attraction of the set to a background nucleus as a point charge with charge $Z$. For the variational problem with constrained volume $V$, our main result is that the minimizer does not exist if $V - Z$ is larger than a constant multiple of $\max(Z^{2/3}, 1)$. The main technical ingredients of our proof are a uniform density lemma and electrostatic screening arguments.Item Open Access Analysis of Score-based Generative Models(2024) Tan, YixinIn this thesis, we study the convergence of diffusion models and related flow-based methods, which are highly successful approaches for learning a probability distribution from data and generating further samples. For diffusion models, we established the first convergence result applying to data distributions satisfying the log-sobolev inequality without suffering the curse of dimensionality. Our analysis gives theoretical grounding to the observation that an annealed procedure is required in practice to generate good samples, as our proof depends essentially on using annealing to obtain a warm start at each step. Moreover, we show that a predictor-corrector algorithm gives better convergence than using either portion alone. Then we generalized the results to any distribution with bounded 2nd moment, relying only on a $L^2$-accurate score estimates, with polynomial dependence on all parameters and no reliance on smoothness or functional inequalities. We also provide a theoretical guarantee of generating data distribution by a progressive flow model, the so-called JKO flow model, which implements the Jordan-Kinderleherer-Otto (JKO) scheme in a normalizing flow network. Leveraging the exponential convergence of the proximal gradient descent (GD) in Wasserstein space, we provethe Kullback-Leibler (KL) guarantee of data generation by a JKO flow model where the assumption on data density is merely a finite second moment

Item Open Access Analytical and Numerical Study of Lindblad Equations(2020) Cao, YuLindblad equations, since introduced in 1976 by Lindblad, and by Gorini, Kossakowski, and Sudarshan, have received much attention in many areas of scientific research. Around the past fifty years, many properties and structures of Lindblad equations have been discovered and identified. In this dissertation, we study Lindblad equations from three aspects: (I) physical perspective; (II) numerical perspective; and (III) information theory perspective.

In Chp. 2, we study Lindblad equations from the physical perspective. More specifically, we derive a Lindblad equation for a simplified Anderson-Holstein model arising from quantum chemistry. Though we consider the classical approach (i.e., the weak coupling limit), we provide more explicit scaling for parameters when the approximations are made. Moreover, we derive a classical master equation based on the Lindbladian formalism.

In Chp. 3, we consider numerical aspects of Lindblad equations. Motivated by the dynamical low-rank approximation method for matrix ODEs and stochastic unraveling for Lindblad equations, we are curious about the relation between the action of dynamical low-rank approximation and the action of stochastic unraveling. To address this, we propose a stochastic dynamical low-rank approximation method. In the context of Lindblad equations, we illustrate a commuting relation between the dynamical low-rank approximation and the stochastic unraveling.

In Chp. 4, we investigate Lindblad equations from the information theory perspective. We consider a particular family of Lindblad equations: primitive Lindblad equations with GNS-detailed balance. We identify Riemannian manifolds in which these Lindblad equations are gradient flow dynamics of sandwiched Rényi divergences. The necessary condition for such a geometric structure is also studied. Moreover, we study the exponential convergence behavior of these Lindblad equations to their equilibria, quantified by the whole family of sandwiched Rényi divergences.

Item Open Access Asymptotic Analysis and Rare Event Simulation for Failure Probabilities in Discrete Random Media(2019) LaComb, Jeffrey MichaelThe problem of material failure is of considerable importance in several applications. We will analyze a discrete atom chain model as a means of studying a material failure problem in a random medium. For different assumptions on the atomistic interaction potential, we determine the conditions necessary for material failure, and conclude failure may only occur in the event of a large deviation in the random model parameters. This observation is then used to derive asymptotic bounds on the probability of failure. Furthermore, we use our theoretical results to motivate the development of an importance sampling algorithm to calculate rare failure probabilities with greater efficiency than standard Monte Carlo methods.

Item Open Access Bloch dynamics with second order Berry phase correction(2017-04-23) Lu, Jianfeng; Zhang, Zihang; Zhou, ZhennanWe derive the semiclassical Bloch dynamics with the second order Berry phase correction, based on a two-scale WKB asymptotic analysis. For uniform external electric field, the bi-characteristics system after a positional shift introduced by Berry connections agrees with the recent result in the physics literature.Item Open Access Bold Diagrammatic Monte Carlo in the Lens of Stochastic Iterative Methods(2017-11-30) Li, Yingzhou; Lu, JianfengThis work aims at understanding of bold diagrammatic Monte Carlo (BDMC) methods for stochastic summation of Feynman diagrams from the angle of stochastic iterative methods. The convergence enhancement trick of the BDMC is investigated from the analysis of condition number and convergence of the stochastic iterative methods. Numerical experiments are carried out for model systems to compare the BDMC with related stochastic iterative approaches.Item Open Access Complexity of zigzag sampling algorithm for strongly log-concave distributionsLu, Jianfeng; Wang, LihanWe study the computational complexity of zigzag sampling algorithm for strongly log-concave distributions. The zigzag process has the advantage of not requiring time discretization for implementation, and that each proposed bouncing event requires only one evaluation of partial derivative of the potential, while its convergence rate is dimension independent. Using these properties, we prove that the zigzag sampling algorithm achieves $\varepsilon$ error in chi-square divergence with a computational cost equivalent to $O\bigl(\kappa^2 d^\frac{1}{2}(\log\frac{1}{\varepsilon})^{\frac{3}{2}}\bigr)$ gradient evaluations in the regime $\kappa \ll \frac{d}{\log d}$ under a warm start assumption, where $\kappa$ is the condition number and $d$ is the dimension.Item Open Access Deep Learning Method for Partial Differential Equations and Optimal Problems(2023) Zhou, MoScientific computing problems in high dimensions are difficult to solve with traditional methods due to the curse of dimensionality. The recently fast developing machine learning techniques provide us a promising way to resolve this problem, elevating the field of scientific computing to new heights. This thesis collects my works on machine learning to solve traditional scientific computing problems during my Ph.D. studies, which include partial differential equation (PDE) problems and optimal control problems. The numerical algorithms in the works demonstrate significant advantage over traditional methods. Moreover, the theoretical analysis of the algorithms enhances our understanding of machine learning, providing guarantees that enable us to avoid treating it as a black box.

Item Open Access Defect resonances of truncated crystal structuresLu, Jianfeng; Marzuola, Jeremy L; Watson, Alexander BDefects in the atomic structure of crystalline materials may spawn electronic bound states, known as \emph{defect states}, which decay rapidly away from the defect. Simplified models of defect states typically assume the defect is surrounded on all sides by an infinite perfectly crystalline material. In reality the surrounding structure must be finite, and in certain contexts the structure can be small enough that edge effects are significant. In this work we investigate these edge effects and prove the following result. Suppose that a one-dimensional infinite crystalline material hosting a positive energy defect state is truncated a distance $M$ from the defect. Then, for sufficiently large $M$, there exists a resonance \emph{exponentially close} (in $M$) to the bound state eigenvalue. It follows that the truncated structure hosts a metastable state with an exponentially long lifetime. Our methods allow both the resonance frequency and associated resonant state to be computed to all orders in $e^{-M}$. We expect this result to be of particular interest in the context of photonic crystals, where defect states are used for wave-guiding and structures are relatively small. Finally, under a mild additional assumption we prove that if the defect state has negative energy then the truncated structure hosts a bound state with exponentially-close energy.Item Open Access Efficient Algorithms for High-dimensional Eigenvalue Problems(2020) Wang, ZheThe eigenvalue problem is a traditional mathematical problem and has a wide applications. Although there are many algorithms and theories, it is still challenging to solve the leading eigenvalue problem of extreme high dimension. Full configuration interaction (FCI) problem in quantum chemistry is such a problem. This thesis tries to understand some existing algorithms of FCI problem and propose new efficient algorithms for the high-dimensional eigenvalue problem. In more details, we first establish a general framework of inexact power iteration and establish the convergence theorem of full configuration interaction quantum Monte Carlo (FCIQMC) and fast randomized iteration (FRI). Second, we reformulate the leading eigenvalue problem as an optimization problem, then compare the show the convergence of several coordinate descent methods (CDM) to solve the leading eigenvalue problem. Third, we propose a new efficient algorithm named Coordinate Descent Full Configuration Interaction (CDFCI) based on coordinate descent methods to solve the FCI problem, which produces some state-of-the-art results. Finally, we conduct various numerical experiments to fully test the algorithms.

Item Open Access Emergence of step flow from an atomistic scheme of epitaxial growth in 1+1 dimensions.(Phys Rev E Stat Nonlin Soft Matter Phys, 2015-03) Lu, Jianfeng; Liu, Jian-Guo; Margetis, DionisiosThe Burton-Cabrera-Frank (BCF) model for the flow of line defects (steps) on crystal surfaces has offered useful insights into nanostructure evolution. This model has rested on phenomenological grounds. Our goal is to show via scaling arguments the emergence of the BCF theory for noninteracting steps from a stochastic atomistic scheme of a kinetic restricted solid-on-solid model in one spatial dimension. Our main assumptions are: adsorbed atoms (adatoms) form a dilute system, and elastic effects of the crystal lattice are absent. The step edge is treated as a front that propagates via probabilistic rules for atom attachment and detachment at the step. We formally derive a quasistatic step flow description by averaging out the stochastic scheme when terrace diffusion, adatom desorption, and deposition from above are present.Item Open Access Existence and computation of generalized Wannier functions for non-periodic systems in two dimensions and higherLu, Jianfeng; Stubbs, Kevin D; Watson, Alexander BExponentially-localized Wannier functions (ELWFs) are a basis of the Fermi projection of a material consisting of functions which decay exponentially fast away from their maxima. When the material is insulating and crystalline, conditions which guarantee existence of ELWFs in dimensions one, two, and three are well-known, and methods for constructing the ELWFs numerically are well-developed. We consider the case where the material is insulating but not necessarily crystalline, where much less is known. In one spatial dimension, Kivelson and Nenciu-Nenciu have proved ELWFs can be constructed as the eigenfunctions of a self-adjoint operator acting on the Fermi projection. In this work, we identify an assumption under which we can generalize the Kivelson-Nenciu-Nenciu result to two dimensions and higher. Under this assumption, we prove that ELWFs can be constructed as the eigenfunctions of a sequence of self-adjoint operators acting on the Fermi projection. We conjecture that the assumption we make is equivalent to vanishing of topological obstructions to the existence of ELWFs in the special case where the material is crystalline. We numerically verify that our construction yields ELWFs in various cases where our assumption holds and provide numerical evidence for our conjecture.Item Open Access Fast algorithm for periodic density fitting for Bloch waves(2017-04-23) Lu, Jianfeng; Ying, LexingWe propose an efficient algorithm for density fitting of Bloch waves for Hamiltonian operators with periodic potential. The algorithm is based on column selection and random Fourier projection of the orbital functions. The computational cost of the algorithm scales as $\mathcal{O}\bigl(N_{\text{grid}} N^2 + N_{\text{grid}} NK \log (NK)\bigr)$, where $N_{\text{grid}}$ is number of spatial grid points, $K$ is the number of sampling $k$-points in first Brillouin zone, and $N$ is the number of bands under consideration. We validate the algorithm by numerical examples in both two and three dimensions.Item Metadata only Fractional stochastic differential equations satisfying fluctuation-dissipation theorem(2017-04-23) Li, L; Liu, J-G; Lu, JianfengWe consider in this work stochastic differential equation (SDE) model for particles in contact with a heat bath when the memory effects are non-negligible. As a result of the fluctuation-dissipation theorem, the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives and based on this we consider fractional stochastic differential equations (FSDEs), which should be understood in an integral form. We establish the existence of strong solutions for such equations. In the linear forcing regime, we compute the solutions explicitly and analyze the asymptotic behavior, through which we verify that satisfying fluctuation-dissipation indeed leads to the correct physical behavior. We further discuss possible extensions to nonlinear forcing regime, while leave the rigorous analysis for future works.Item Open Access Global optimality of softmax policy gradient with single hidden layer neural networks in the mean-field regimeAgazzi, Andrea; Lu, JianfengWe study the problem of policy optimization for infinite-horizon discounted Markov Decision Processes with softmax policy and nonlinear function approximation trained with policy gradient algorithms. We concentrate on the training dynamics in the mean-field regime, modeling e.g., the behavior of wide single hidden layer neural networks, when exploration is encouraged through entropy regularization. The dynamics of these models is established as a Wasserstein gradient flow of distributions in parameter space. We further prove global optimality of the fixed points of this dynamics under mild conditions on their initialization.