arXiv subjects

Qiang Du

Publications and source records attributed to Qiang Du.

At least 37 records · Page 2Linked to original sources

arXiv2025

A Stochastic Algorithm for Searching Saddle Points with Convergence Guarantee

Saddle points provide a hierarchical view of the energy landscape, revealing transition pathways and interconnected basins of attraction, and offering insight into the global structure, metastability, and possible collective mechanisms of the underlying system. In this work, we propose a stochastic saddle-search algorithm to circumvent exact derivative and Hessian evaluations that have been used in implementing traditional and deterministic saddle dynamics. At each iteration, the algorithm uses a stochastic eigenvector-search method, based on a stochastic Hessian, to approximate the unstable directions, followed by a stochastic gradient update with reflections in the approximate unstable direction to advance toward the saddle point. We carry out rigorous numerical analysis to establish the almost sure convergence for the stochastic eigenvector search and local almost sure convergence with an $O(1/n)$ rate for the saddle search, and present a theoretical guarantee to ensure the high-probability identification of the saddle point when the initial point is sufficiently close. Numerical experiments, including the application to a neural network loss landscape and a Landau-de Gennes type model for nematic liquid crystal, demonstrate the practical applicability and the ability for escaping from "bad" areas of the algorithm.

math.NA↗

arXiv2025

Digital Low-Level RF system for the Linac Electronics Modernization Plan at LCLS

The LCLS began operations in 2009, utilizing SLAC's normal-conducting (NC) LINAC, which features control equipment dating back to the 1960s and 1980s. The Linac Electronics Modernization Plan (LEMP) aims to replace the legacy control equipment with a system based on the open-source Marble carrier board and Zest+ digitizer board, both of which are used in the LCLS-II HE LLRF system. Adaptation of the LLRF system developed for the continuous-wave (CW) superconducting RF (SRF) LCLS-II to the short-RF pulse NC LCLS includes leveraging the knowledge and experience gained from recent LLRF projects at SLAC and efficiently reusing the core functionality of the hardware and code base developed for previous projects, in collaboration with LBNL, FNAL and JLAB. A prototype has been deployed and tested at station 26-3, demonstrating RF generation/control, interlocks, triggers, and waveform capture. Here, we describe the hardware, firmware and software infrastructure, highlight key features, and present initial test results.

physics.acc-ph↗

arXiv2025

On parameterized nonlocal-fractional transmission problems and associated function spaces

In this paper, we consider a family of seamlessly coupled nonlocal models associated with transmission conditions across an interface. The models are derived from the variation of a parameterized family of energies consisting of a fractional type Dirichlet energy on one subdomain and a nonlocal Dirichlet energy involving a finite range of interactions on another subdomain. We present the rigorous mathematical formulation and its well-posedness. We also investigate the behavior of the model in various limiting regimes.

math.AP↗

arXiv2025

A discontinuous Galerkin method for one-dimensional nonlocal wave problems

This paper presents a fully discrete numerical scheme for one-dimensional nonlocal wave equations and provides a rigorous theoretical analysis. To facilitate the spatial discretization, we introduce an auxiliary variable analogous to the gradient field in local discontinuous Galerkin (DG) methods for classical partial differential equations (PDEs) and reformulate the equation into a system of equations. The proposed scheme then uses a DG method for spatial discretization and the Crank-Nicolson method for time integration. We prove optimal L2 error convergence for both the solution and the auxiliary variable under a special class of radial kernels at the semi-discrete level. In addition, for general kernels, we demonstrate the asymptotic compatibility of the scheme, ensuring that it recovers the classical DG approximation of the local wave equation in the zero-horizon limit. Furthermore, we prove that the fully discrete scheme preserves the energy of the nonlocal wave equation. A series of numerical experiments are presented to validate the theoretical findings.

math.NA↗

arXiv2025

Nonlinear Optimal Recovery in Hilbert Spaces

This paper investigates solution strategies for nonlinear problems in Hilbert spaces, such as nonlinear partial differential equations (PDEs) in Sobolev spaces, when only finite measurements are available. We formulate this as a nonlinear optimal recovery problem, establishing its well-posedness and proving its convergence to the true solution as the number of measurements increases. However, the resulting formulation might not have a finite-dimensional solution in general. We thus present a sufficient condition for the finite dimensionality of the solution, applicable to problems with well-defined point evaluation measurements. To address the broader setting, we introduce a relaxed nonlinear optimal recovery and provide a detailed convergence analysis. An illustrative example is given to demonstrate that our formulations and theoretical findings offer a comprehensive framework for solving nonlinear problems in infinite-dimensional spaces with limited data.

math.NA↗

arXiv2025

DualMS: Implicit Dual-Channel Minimal Surface Optimization for Heat Exchanger Design

Heat exchangers are critical components in a wide range of engineering applications, from energy systems to chemical processing, where efficient thermal management is essential. The design objectives for heat exchangers include maximizing the heat exchange rate while minimizing the pressure drop, requiring both a large interface area and a smooth internal structure. State-of-the-art designs, such as triply periodic minimal surfaces (TPMS), have proven effective in optimizing heat exchange efficiency. However, TPMS designs are constrained by predefined mathematical equations, limiting their adaptability to freeform boundary shapes. Additionally, TPMS structures do not inherently control flow directions, which can lead to flow stagnation and undesirable pressure drops. This paper presents DualMS, a novel computational framework for optimizing dual-channel minimal surfaces specifically for heat exchanger designs in freeform shapes. To the best of our knowledge, this is the first attempt to directly optimize minimal surfaces for two-fluid heat exchangers, rather than relying on TPMS. Our approach formulates the heat exchange maximization problem as a constrained connected maximum cut problem on a graph, with flow constraints guiding the optimization process. To address undesirable pressure drops, we model the minimal surface as a classification boundary separating the two fluids, incorporating an additional regularization term for area minimization. We employ a neural network that maps spatial points to binary flow types, enabling it to classify flow skeletons and automatically determine the surface boundary. DualMS demonstrates greater flexibility in surface topology compared to TPMS and achieves superior thermal performance, with lower pressure drops while maintaining a similar heat exchange rate under the same material cost.

math.OC↗

arXiv2025

Nonlocal vector calculus on the sphere

We introduce a nonlocal vector calculus on the unit two-sphere using weakly singular integral operators. Within this framework, the operators are diagonalizable in terms of scalar and vector spherical harmonics, a property that facilitates the proof of a nonlocal Stokes theorem. This constitutes the first instance of such a theorem on a curved surface. Furthermore, our analysis demonstrates the strong convergence of these nonlocal operators to the classical differential operators of vector calculus as the interaction range tends to zero.

math.AP↗

arXiv2025

A Nonlocal Biharmonic Model with $Γ$-Convergence to Local Model and an Efficient Numerical Method

Nonlocal models and their associated theories have been extensively investigated in recent years. Among these, nonlocal versions of the classical Laplace operator have attracted the most attention, while higher-order nonlocal operators have been studied far less. In this work, we focus on the nonlocal counterpart of the classical biharmonic operator together with the clamped boundary condition ($u$ and $\frac{\partial u}{\partial n}$ are given on the boundary). We develop the variational formulation of a nonlocal biharmonic model, establish the existence and uniqueness of its solution, and analyze its convergence as the nonlocal horizon approaches zero. In addition, we propose an efficient finite element method to solve the nonlocal model and the numerical results verify the analytical properties of the nonlocal model and its solution.

math.AP↗

arXiv2024

Weighted nonlocal operators and their applications in semi-supervised learning

Motivated by problems in machine learning, we study a class of variational problems characterized by nonlocal operators. These operators are characterized by power-type weights, which are singular at a portion of the boundary. We identify a range of exponents on these weights for which the variational Dirichlet problem is well-posed. This range is determined by the ambient dimension of the problem, the growth rate of the nonlocal functional, and the dimension of the boundary portion on which the Dirichlet data is prescribed. We show the variational convergence of solutions to solutions of local weighted Sobolev functionals in the event of vanishing nonlocality.

math.AP↗

arXiv2024

$Γ$-convergence of Nonlocal Dirichlet Energies With Penalty Formulations of Dirichlet Boundary Data

We study nonlocal Dirichlet energies associated with a class of nonlocal diffusion models on a bounded domain subject to the conventional local Dirichlet boundary condition. The goal of this paper is to give a general framework to correctly impose Dirichlet boundary condition in nonlocal diffusion model. To achieve this, we formulate the Dirichlet boundary condition as a penalty term and use theory of $\varGamma$-convergence to study the correct form of the penalty term. Based on the analysis of $\varGamma$-convergence, we prove that the Dirichlet boundary condition can be correctly imposed in nonlocal diffusion model in the sense of $\varGamma$-convergence as long as the penalty term satisfies a few mild conditions. This work provides a theoretical foundation for approximate Dirichlet boundary condition in nonlocal diffusion model.

math.AP↗

arXiv2024

Error estimates of finite element methods for nonlocal problems using exact or approximated interaction neighborhoods

We study the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. The limit corresponds to the case when the horizon parameter, the radius of the spherical nonlocal interaction neighborhood of the nonlocal model, and the mesh size simultaneously approach zero. Two important cases are discussed: one involving the original nonlocal models and the other for nonlocal models with polygonal approximations of the nonlocal interaction neighborhood. Results of numerical experiments are also reported to substantiate the theoretical studies.

math.NA↗

arXiv2024

Ohta-Kawasaki energy for amphiphiles: asymptotics and phase-field simulations

We study the minimizers of a degenerate case of the Ohta-Kawasaki energy, defined as the sum of the perimeter and a Coulombic nonlocal term. We start by investigating radially symmetric candidates which give us insights into the asymptotic behaviors of energy minimizers in the large mass limit. In order to numerically study the problems that are analytically challenging, we propose a phase-field reformulation which is shown to Gamma-converge to the original sharp interface model. Our phase-field simulations and asymptotic results suggest that the energy minimizers exhibit behaviors similar to the self-assembly of amphiphiles, including the formation of lipid bilayer membranes.

math.OC↗

arXiv2024

Numerical analysis of a class of penalty discontinuous Galerkin methods for nonlocal diffusion problems

In this paper, we consider a class of discontinuous Galerkin (DG) methods for one-dimensional nonlocal diffusion (ND) problems. The nonlocal models, which are integral equations, are widely used in describing many physical phenomena with long-range interactions. The ND problem is the nonlocal analog of the classic diffusion problem, and as the interaction radius (horizon) vanishes, then the nonlocality disappears and the ND problem converges to the classic diffusion problem. Under certain conditions, the exact solution to the ND problem may exhibit discontinuities, setting it apart from the classic diffusion problem. Since the DG method shows its great advantages in resolving problems with discontinuities in computational fluid dynamics over the past several decades, it is natural to adopt the DG method to compute the ND problems. Based on [Du-Ju-Lu-Tian-CAMC2020], we develop the DG methods with different penalty terms, ensuring that the proposed DG methods have local counterparts as the horizon vanishes. This indicates the proposed methods will converge to the existing DG schemes as the horizon vanishes, which is crucial for achieving asymptotic compatibility. Rigorous proofs are provided to demonstrate the stability, error estimates, and asymptotic compatibility of the proposed DG schemes. To observe the effect of the nonlocal diffusion, we also consider the time-dependent convection-diffusion problems with nonlocal diffusion. We conduct several numerical experiments, including accuracy tests and Burgers' equation with nonlocal diffusion, and various horizons are taken to show the good performance of the proposed algorithm and validate the theoretical findings.

math.NA↗

arXiv2024

Computational and analytical studies of a new nonlocal phase-field crystal model in two dimensions

A nonlocal phase-field crystal (NPFC) model is presented as a nonlocal counterpart of the local phase-field crystal (LPFC) model and a special case of the structural PFC (XPFC) derived from classical field theory for crystal growth and phase transition. The NPFC incorporates a finite range of spatial nonlocal interactions that can account for both repulsive and attractive effects. The specific form is data-driven and determined by a fitting to the materials structure factor, which can be much more accurate than the LPFC and previously proposed fractional variant. In particular, it is able to match the experimental data of the structure factor up to the second peak, an achievement not possible with other PFC variants studied in the literature. Both LPFC and fractional PFC (FPFC) are also shown to be distinct scaling limits of the NPFC, which reflects the generality. The advantage of NPFC in retaining material properties suggests that it may be more suitable for characterizing liquid-solid transition systems. Moreover, we study numerical discretizations using Fourier spectral methods, which are shown to be convergent and asymptotically compatible, making them robust numerical discretizations across different parameter ranges. Numerical experiments are given in the two-dimensional case to demonstrate the effectiveness of the NPFC in simulating crystal structures and grain boundaries.

math.NA↗

arXiv2024

Nonlocal Stokes equation with relaxation on the divergence free equation

In this paper, we consider a new nonlocal approximation to the linear Stokes system with periodic boundary conditions in two and three dimensional spaces . A relaxation term is added to the equation of nonlocal divergence free equation, which is reminiscent to the relaxation of local Stokes equation with small artificial compressibility. Our analysis shows that the well-posedness of the nonlocal system can be established under some mild assumptions on the kernel of nonlocal interactions. Furthermore, the new nonlocal system converges to the conventional, local Stokes system in second order as the horizon parameter of the nonlocal interaction goes to zero. The study provides more theoretical understanding to some numerical methods, such as smoothed particle hydrodynamics, for simulating incompressible viscous flows.

math.AP↗

arXiv2024

Asymptotically compatible schemes for nonlinear variational models via Gamma-convergence and applications to nonlocal problems

We present a study on asymptotically compatible Galerkin discretizations for a class of parametrized nonlinear variational problems. The abstract analytical framework is based on variational convergence, or Gamma-convergence. We demonstrate the broad applicability of the theoretical framework by developing asymptotically compatible finite element discretizations of some representative nonlinear nonlocal variational problems on a bounded domain. These include nonlocal nonlinear problems with classically-defined, local boundary constraints through heterogeneous localization at the boundary, as well as nonlocal problems posed on parameter-dependent domains.

math.NA↗

arXiv2023

Revealing Excited States of Rotational Bose-Einstein Condensates

Rotational Bose-Einstein condensates can exhibit quantized vortices as topological excitations. In this study, the ground and excited states of the rotational Bose-Einstein condensates are systematically studied by calculating the stationary points of the Gross-Pitaevskii energy functional. Various excited states and their connections at different rotational frequencies are revealed in solution landscapes constructed with the constrained high-index saddle dynamics method. Four excitation mechanisms are identified: vortex addition, rearrangement, merging, and splitting. We demonstrate changes in the ground state with increasing rotational frequencies and decipher the evolution of the stability of ground states.

cond-mat.quant-gas↗

arXiv2023

Stretched Non-negative Matrix Factorization

An algorithm is described and tested that carries out a non negative matrix factorization (NMF) ignoring any stretching of the signal along the axis of the independent variable. This extended NMF model is called StretchedNMF. Variability in a set of signals due to this stretching is then ignored in the decomposition. This can be used, for example, to study sets of powder diffraction data collected at different temperatures where the materials are undergoing thermal expansion. It gives a more meaningful decomposition in this case where the component signals resemble signals from chemical components in the sample. The StretchedNMF model introduces a new variable, the stretching factor, to describe any expansion of the signal. To solve StretchedNMF, we discretize it and employ Block Coordinate Descent framework algorithms. The initial experimental results indicate that StretchedNMF model outperforms the conventional NMF for sets of data with such an expansion. A further enhancement to StretchedNMF for the case of powder diffraction data from crystalline materials called Sparse-StretchedNMF, which makes use of the sparsity of the powder diffraction signals, allows correct extractions even for very small stretches where StretchedNMF struggles. As well as demonstrating the model performance on simulated PXRD patterns and atomic pair distribution functions (PDFs), it also proved successful when applied to real data taken from an in situ chemical reaction experiment.

cond-mat.mtrl-sci↗