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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.

Ginzburg--Landau (GL) functionals on graphs, which are relaxations of graph-cut functionals on graphs, have yielded a variety of insights in image segmentation and graph clustering. In this paper, we study large-graph limits of GL functionals by taking a functional-analytic view of graphs as nonlocal kernels. For a graph $W_n$ with $n$ nodes, the corresponding graph GL functional $\GL^{W_n}_\ep$ is an energy for functions on $W_n$. We minimize GL functionals on sequences of growing graphs that converge to functions called graphons. For such sequences of graphs, we show that the graph GL functional $\Gamma$-converges to a continuous and nonlocal functional that we call the \emph{graphon GL functional}. We also investigate the sharp-interface limits of the graph GL and graphon GL functionals, and we relate these limits to a nonlocal total variation. We express the limiting GL functional in terms of Young measures and thereby obtain a probabilistic interpretation of the variational problem in the large-graph limit. Finally, to develop intuition about the graphon GL functional, we compute the GL minimizer for several example families of graphons., Comment: 37 pages

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.

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.

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.

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., Comment: 51 pages; 14 figures; submitted for publication; minor typos corrected

The ongoing work to upgrade ALS to ALS-U demands strict RF requirements such as low jitter and low spurs frequency reference to meet its accelerator and science goals. A low phase noise dual frequency Master Oscillator (MO), where the two frequencies are related by a fractional ratio of 608/609 and flexible divide by four frequency outputs has been consolidated into a single chassis. Optical fiber clock distribution system has been selected over the old coax system used in ALS to distribute these signals to various clients across the facility, providing high electrical isolation between outputs and therefore lower phase errors. A Xilinx FPGA ties the MO chassis together by providing a RS-485 interface to monitor and control the system. The new system aims to deliver phase-continuous frequencies with a phase noise (integrated RMS jitter) from 1 Hz to 1 MHz of less than 200 femtosecond per output. This paper will discuss the design, implementation, performance and installation of the new MO generation and distribution system., Comment: Poster presented at LLRF Workshop 2023 (LLRF2023, arXiv: 2310.03199)

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 Dirichlet boundary condition is imposed through a specifically designed penalty formulation. We prove that the nonlocal Dirichlet energies with the penalty terms converge to local Dirichlet energies with Dirichlet boundary conditions in the sense of $\varGamma$-convergence.

We study nonlocal integral equations on bounded domains with finite-range nonlocal interactions that are localized at the boundary. We establish a Green's identity for the nonlocal operator that recovers the classical boundary integral, which, along with the variational analysis established previously, leads to the well-posedness of these nonlocal problems with various types of classical local boundary conditions. We continue our analysis via boundary-localized convolutions, using them to analyze the Euler-Lagrange equations, which permits us to establish global regularity properties and classical Sobolev convergence to their classical local counterparts.

We present a systematic study on a class of nonlocal integral functionals for functions defined on a bounded domain and the naturally induced function spaces. The function spaces are equipped with a seminorm depending on finite differences weighted by a position-dependent function, which leads to heterogeneous localization on the domain boundary. We show the existence of minimizers for nonlocal variational problems with classically-defined, local boundary constraints, together with the variational convergence of these functionals to classical counterparts in the localization limit. This program necessitates a thorough study of the nonlocal space; we demonstrate properties such as a Meyers-Serrin theorem, trace inequalities, and compact embeddings, which are facilitated by new studies of boundary-localized convolution operators.

We present discontinuous Galerkin (DG) methods for solving a first-order semi-linear hyperbolic system, which was originally proposed as a continuum model for a one-dimensional dimer lattice of topological resonators. We examine the energy-conserving or energy-dissipating property in relation to the choices of simple, mesh-independent numerical fluxes. We demonstrate that, with certain numerical flux choices, our DG method achieves optimal convergence in the $L^2$ norm. We provide numerical experiments that validate and illustrate the effectiveness of our proposed numerical methods.

The liquid drop model, originally used to model atomic nuclei, describes the competition between surface tension and Coulomb force. To help understand how a ball loses stability and becomes prone to fission, we calculate the minimum energy path of the fission process, and study the bifurcation branch conjectured by Bohr and Wheeler. We then present the two-dimensional analogue for comparison. Our study is conducted with the help of numerical simulations via a phase-field approach., Comment: 61 pages; 53 figures; minor revisions made; several references added

We analyze the convergence of a nonlocal gradient descent method for minimizing a class of high-dimensional non-convex functions, where a directional Gaussian smoothing (DGS) is proposed to define the nonlocal gradient (also referred to as the DGS gradient). The method was first proposed in [42], in which multiple numerical experiments showed that replacing the traditional local gradient with the DGS gradient can help the optimizers escape local minima more easily and significantly improve their performance. However, a rigorous theory for the efficiency of the method on nonconvex landscape is lacking. In this work, we investigate the scenario where the objective function is composed of a convex function, perturbed by a oscillating noise. We provide a convergence theory under which the iterates exponentially converge to a tightened neighborhood of the solution, whose size is characterized by the noise wavelength. We also establish a correlation between the optimal values of the Gaussian smoothing radius and the noise wavelength, thus justify the advantage of using moderate or large smoothing radius with the method. Furthermore, if the noise level decays to zero when approaching global minimum, we prove that DGS-based optimization converges to the exact global minimum with linear rates, similarly to standard gradient-based method in optimizing convex functions. Several numerical experiments are provided to confirm our theory and illustrate the superiority of the approach over those based on the local gradient.

We describe and analyze nonlocal integro-differential equations with classical local boundary conditions. The interaction kernel of the nonlocal operator has horizon parameter dependent on position in the domain, and vanishes as the boundary of the domain is approached. This heterogeneous localization allows for boundary values to be captured in the trace sense. We state and prove a nonlocal Green's identity for these nonlocal operators that involve local boundary terms. We use this identity to state and establish the well-posedness of variational formulations of the nonlocal problems with several types of classical boundary conditions. We show the consistency of these nonlocal boundary-value problems with their classical local counterparts in the vanishing horizon limit via the convergence of solutions. The Poisson data for the local boundary-value problem is permitted to be quite irregular, belonging to the dual of the classical Sobolev space. Heterogeneously mollifying this Poisson data for the local problem on the same length scale as the horizon and using the regularity of the interaction kernel, we show that the solutions to the nonlocal boundary-value problem with the mollified Poisson data actually belong to the classical Sobolev space, and converge weakly to the unique variational solution of the classical Poisson problem with original Poisson data.

This paper considers numerical discretization of a nonlocal conservation law modeling vehicular traffic flows involving nonlocal inter-vehicle interactions. The nonlocal model involves an integral over the range measured by a horizon parameter and it recovers the local Lighthill-Richards-Whitham model as the nonlocal horizon parameter goes to zero. Good numerical schemes for simulating these parameterized nonlocal traffic flow models should be robust with respect to the change of the model parameters but this has not been systematically investigated in the literature. We fill this gap through a careful study of a class of finite volume numerical schemes with suitable discretizations of the nonlocal integral, which include several schemes proposed in the literature and their variants. Our main contributions are to demonstrate the asymptotically compatibility of the schemes, which includes both the uniform convergence of the numerical solutions to the unique solution of nonlocal continuum model for a given positive horizon parameter and the convergence to the unique entropy solution of the local model as the mesh size and the nonlocal horizon parameter go to zero simultaneously. It is shown that with the asymptotically compatibility, the schemes can provide robust numerical computation under the changes of the nonlocal horizon parameter., Comment: 24 pages, 5 figures

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.

We propose and study a nonlocal conservation law modelling traffic flow in the existence of inter-vehicle communication. It is assumed that the nonlocal information travels at a finite speed and the model involves a space-time nonlocal integral of weighted traffic density. The well-posedness of the model is established under suitable conditions on the model parameters and by a suitably-defined initial condition. In a special case where the weight kernel in the nonlocal integral is an exponential function, the nonlocal model can be reformulated as a $2\times2$ hyperbolic system with relaxation. With the help of this relaxation representation, we show that the Lighthill-Whitham-Richards model is recovered in the equilibrium approximation limit., Comment: 32 pages

Currently ALS is undergoing an upgrade to ALSU to produce 100 times brighter soft X-ray light. The LLRF system for Accumulator Ring (AR) is composed of two identical LLRF stations, for driving RF amplifiers. The closed loop RF amplitude and phase stability is measured as $< 0.1\%$ and $< 0.1^\circ$ respectively, using the non-IQ digital down conversion together with analog up/down conversion, under a system-on-chip architecture. Realtime interlock system is implemented with $< 2 \mu$s latency, for machine protection against arc flash and unexpected RF power. Control interfaces are developed to enable PLC-FPGA-EPICS communication to support operation, timing, cavity tuning, and interlock systems. The LLRF system handles alignment of buckets to swap beams between AR and Storage Ring by synchronous phase loop ramping between the two cavities. The system also includes an optimization routine to characterize the loop dynamics and determine optimal operating point using a built-in network analyzer feature. A cavity emulator of 31 kHz bandwidth is integrated with the LLRF system to validate the performance of the overall system being developed., Comment: Talk presented at LLRF Workshop 2022 (LLRF2022, arXiv:2208.13680)

Large scale laser facilities are needed to advance the energy frontier in high energy physics and accelerator physics. Laser plasma accelerators are core to advanced accelerator concepts aimed at reaching TeV electron electron colliders. In these facilities, intense laser pulses drive plasmas and are used to accelerate electrons to high energies in remarkably short distances. A laser plasma accelerator could in principle reach high energies with an accelerating length that is 1000 times shorter than in conventional RF based accelerators. Notionally, laser driven particle beam energies could scale beyond state of the art conventional accelerators. LPAs have produced multi GeV electron beams in about 20 cm with relative energy spread of about 2 percent, supported by highly developed laser technology. This validates key elements of the US DOE strategy for such accelerators to enable future colliders but extending best results to date to a TeV collider will require lasers with higher average power. While the per pulse energies envisioned for laser driven colliders are achievable with current lasers, low laser repetition rates limit potential collider luminosity. Applications will require rates of kHz to tens of kHz at Joules of energy and high efficiency, and a collider would require about 100 such stages, a leap from current Hz class LPAs. This represents a challenging 1000 fold increase in laser repetition rates beyond current state of the art. This whitepaper describes current research and outlook for candidate laser systems as well as the accompanying broadband and high damage threshold optics needed for driving future advanced accelerators., Comment: contribution to Snowmass 2021

Particle accelerators for high energy physics will generate TeV-scale particle beams in large, multi-Km size machines colliding high brightness beams at the interaction point [1-4]. The high luminosity in such machines is achieved by producing very small asymmetric beam size at the interaction point, with short durations to minimize beam-beam effects. Tuning energy, timing and position of the beam for optimal performance will require high-precision controls of amplitude and phase of high-frequency electromagnetic fields and real-time processing of complex algorithms. The stability of the colliding beams has a large impact on the collider's effective luminosity. Therefore, the technology readiness level of diagnostic and control systems will be a major consideration in the collider design. The technical requirements of such systems depend on the specifics of beam parameters, such as transverse and longitudinal dimensions, charge/pulse and beam pulse format, which are driven by the accelerating technology of choice. While feedback systems with single bunch position monitor resolution below 50 nm and latency <300 ns have been demonstrated in beam test facilities, many advanced collider concepts make use of higher repetition rates, brighter beams and higher accelerating frequencies, and will require better performance, up to 1-2 order of magnitude, demanding aggressive R&D to be able to deliver and maintain the targeted luminosity., Comment: contribution to Snowmass 2021

In this paper, we introduce a nonlocal model for linear steady Stokes system with physical no-slip boundary condition. We use the idea of volume constraint to enforce the no-slip boundary condition and prove that the nonlocal model is well-posed. We also show that and the solution of the nonlocal system converges to the solution of the original Stokes system as the nonlocality vanishes., Comment: accepted for publication in Numerical Mathematics: Theory, Methods and Applications

For some spatially nonlocal diffusion models with a finite range of nonlocal interactions measured by a positive parameter $\delta$, we review their formulation defined on a bounded domain subject to various conditions that correspond to some inhomogeneous data. We consider their consistency to similar inhomogeneous boundary value problems of classical partial differential equation (PDE) models as the nonlocal interaction kernel gets localized in the local $\delta\to 0$ limit, and at the same time, for rescaled fractional type kernels, to corresponding inhomogeneous nonlocal boundary value problems of fractional equations in the global $\delta\to \infty$ limit. Such discussions help to delineate issues related to nonlocal problems defined on a bounded domain with inhomogeneous data., Comment: 31 pages, 5 figures

Machine learning models based on convolutional neural networks have been used for predicting space groups of crystal structures from their atomic pair distribution function (PDF). However, the PDFs used to train the model are calculated using a fixed set of parameters that reflect specific experimental conditions, and the accuracy of the model when given PDFs generated with different choices of these parameters is unknown. In this paper, we report that the results of the top-1 accuracy and top-6 accuracy are robust when applied to PDFs of different choices of experimental parameters $r_\text{max}$, $Q_\text{max}$, $Q_\text{damp}$ and atomic displacement parameters., Comment: 15 pages, 5 figures

In this paper we consider the functional \begin{equation*} E_{p,\la}(\Omega):=\int_\Omega \dist^p(x,\pd \Omega )\d x+\la \frac{\H^1(\pd \Omega)}{\H^2(\Omega)}. \end{equation*} Here $p\geq 1$, $\la>0$ are given parameters, the unknown $\Omega$ varies among compact, convex, Hausdorff two-dimensional sets of $\R^2$, $\pd \Omega$ denotes the boundary of $\Omega$, and $\dist(x,\pd \Omega):=\inf_{y\in\pd \Omega}|x-y|$. The integral term $\int_\Omega \dist^p(x,\pd \Omega )\d x$ quantifies the "easiness" for points in $\Omega$ to reach the boundary, while $\frac{\H^1(\pd \Omega)}{\H^2(\Omega)}$ is the perimeter-to-area ratio. The main aim is to prove existence and $C^{1,1}$-regularity of minimizers of $\E$.

We consider the minimization of an average distance functional defined on a two-dimensional domain $\Omega$ with an Euler elastica penalization associated with $\pd \Omega$, the boundary of $\Omega$. The average distance is given by \begin{equation*} \int_{\Omega} \dist^p(x,\pd \Omega )\d x \end{equation*} where $p\geq 1$ is a given parameter, and $\dist(x,\pd \Omega)$ is the Hausdorff distance between $\{x\}$ and $\pd \Omega$. The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve ${\pd \Omega}$, which is proportional to the integrated squared curvature defined on $\pd \Omega$, as given by \begin{equation*} \la \int_{\pd \Omega} \kappa_{\pd \Omega}^2\d\H_{\llcorner \pd \Omega}^1, \end{equation*} where $\kappa_{\pd \Omega}$ denotes the (signed) curvature of $\pd \Omega$ and $\la>0$ denotes a penalty constant. The domain $\Omega$ is allowed to vary among compact, convex sets of $\mathbb{R}^2$ with Hausdorff dimension equal to $2$\tcr{.} Under no a priori assumptions on the regularity of the boundary $\pd \Omega$, we prove the existence of minimizers of $E_{p,\la}$. Moreover, we establish the $C^{1,1}$-regularity of its minimizers. An original construction of a suitable family of competitors plays a decisive role in proving the regularity.

We study the propagation of singularities in solutions of linear convection equations with spatially heterogeneous nonlocal interactions. A spatially varying nonlocal horizon parameter is adopted in the model, which measures the range of nonlocal interactions. Via heterogeneous localization, this can lead to the seamless coupling of the local and nonlocal models. We are interested in understanding the impact on singularity propagation due to the heterogeneities of nonlocal horizon and the local and nonlocal transition. We first analytically derive equations to characterize the propagation of different types of singularities for various forms of nonlocal horizon parameters in the nonlocal regime. We then use asymptotically compatible schemes to discretize the equations and carry out numerical simulations to illustrate the propagation patterns in different scenarios.