Browsing by Author "Lu, Jianfeng"

We 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.

In 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

Lindblad 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.

The 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.

Scientific 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.

The 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.