The combination of high-dimensionality and disparity of time scales encountered in many problems in computational physics has motivated the development of coarse-grained (CG) models. In this paper, we advocate the paradigm of data-driven discovery for extracting governing equations by employing fine-scale simulation data. In particular, we cast the coarse-graining process under a probabilistic state-space model where the transition law dictates the evolution of the CG state variables and the emission law the coarse-to-fine map. The directed probabilistic graphical model implied, suggests that given values for the fine-grained (FG) variables, probabilistic inference tools must be employed to identify the corresponding values for the CG states and to that end, we employ Stochastic Variational Inference. We advocate a sparse Bayesian learning perspective which avoids overfitting and reveals the most salient features in the CG evolution law. The formulation adopted enables the quantification of a crucial, and often neglected, component in the CG process, i.e. the predictive uncertainty due to information loss. Furthermore, it is capable of reconstructing the evolution of the full, fine-scale system. We demonstrate the efficacy of the proposed framework in high-dimensional systems of random walkers.

Reconstructed Discontinuous Galerkin (rDG) methods are presented for solving diffusion equations based on a first-order hyperbolic system (FOHS) formulation. The idea is to combine the advantages of the FOHS formulation and the rDG methods in an effort to develop a more reliable, accurate, efficient, and robust method for solving the diffusion equations. The developed hyperbolic rDG methods can be made to have higher-order accuracy than conventional DG methods with fewer degrees of freedom. A number of test cases for different diffusion equations are presented to assess accuracy and performance of the newly developed hyperbolic rDG methods in comparison with the standard BR2 DG method. Numerical experiments demonstrate that the hyperbolic rDG methods are able to achieve the designed optimal order of accuracy for both solutions and their derivatives on regular, irregular, and heterogeneous girds, and outperform the BR2 method in terms of the magnitude of the error, the order of accuracy, the size of time steps, and the CPU times required to achieve steady state solutions, indicating that the developed hyperbolic rDG methods provide an attractive and probably an even superior alternative for solving the diffusion equations.

This work is concerned with the inverse source problem of locating multiple multipolar sources from boundary measurements for the Helmholtz equation. We develop simple and effective sampling schemes for location acquisition of the sources with a single wavenumber. Our algorithms are based on some novel indicator functions whose indicating behaviors could be used to locate multiple multipolar sources. The inversion schemes are totally "direct" in the sense that only simple integral calculations are involved in evaluating the indicator functions. Rigorous mathematical justifications are provided and extensive numerical examples are presented to demonstrate the effectiveness, robustness and efficiency of the proposed methods.

In [Fu et al., JCP 305(2016): 333-359], a family of high-order targeted ENO (TENO) schemes is proposed. The weighting strategy of TENO either applies a candidate stencil with its optimal weight, or removes its contribution completely when it is crossed by discontinuities. This ENO-like stencil selection procedure significantly diminishes the numerical dissipation induced by the nonlinear adaptations of classical WENO schemes. In this paper, the fifth-order TENO scheme is extended to simulate reactive flows in combination with an uncoupled method [1, 2], which splits the reaction source term of detailed chemistry from the flow equations. A set of benchmark cases including the two-dimensional self-sustained detonation is simulated to validate and compare the performance of the fifth-order WENO and TENO schemes. Numerical experiments demonstrate that TENO scheme is robust for simulating chemical reacting flows with using the uncoupled method. In particular, TENO scheme shows better performance in capturing both the shockwaves and the small-scale flow structures, e.g. shear layers and vortices.

In order to overcome the strong limitation of the computational cost for classic explicit schemes in quasi-geostrophic limit typically involved in oceanographic flows, in this study, a highly efficient numerical method for rotating oceanographic flows modeled by Saint-Venant system with Coriolis forces is developed. To efficiently deal with the rotating flows in the low Rossby and Froude number regime, the core idea is splitting fast varying flux terms into stiff and non-stiff parts, and implicitly approximating the fast dynamic waves using central difference method with an iteration algorithm and explicitly approximating slow dynamic waves using a finite-volume hyperbolic solver with minmod limiter. The proposed approach has a second order convergence rate in the quasi-geostrophic limit. The temporal evolution is approximated using a high-order implicit-explicit Runge-Kutta method. The proposed semi-implicit scheme is proved to be uniformly asymptotically consistent in the quasi-geostrophic limit when Froude and Rossby numbers→0. The proposed numerical methods are finally verified by numerical experiments of rotating shallow flows. The tests show that the proposed numerical scheme is stable and accurate with any grid size with low Rossby and Froude numbers, which leads to significant reduction of the computational cost comparing with classic explicit schemes.

Due to its computational efficiency, high-order finite difference (FD) method is attractive, but the difficulty of treating boundary hampers the practical application in complex flow simulation. In this work, we propose a novel high-order FD scheme based on discontinuous Galerkin (DG) boundary treatment (FDbDG) where a DG method based on variational principle is applied to provide the flow properties in the vicinity of the boundary with desirable derivative information in time. In order to carefully combine the finite element and finite difference, Hermite weighted essentially non-oscillatory (HWENO) interpolation is adopted to build the HWENO flux for interior FD scheme and HWENO reconstruction is used to construct the degrees of freedom in the DG flux for boundary variational method. Several typical test cases are selected to evaluate the treatment for FD boundary. Numerical results show the proposed FDbDG method can reach arbitrary order of accuracy including boundary region with non-essentially oscillations.

Simulation of neutron transport process plays an important role in nuclear reactor computation and the numerical technique becomes the focus of nuclear reactor engineering. This paper provides a neutron finite volume lattice Boltzmann method (NFV-LBM) for solving the neutron discrete velocity Boltzmann equation (NDVBE), in which the NDVBE is deduced from the neutron transport equation (NTE) and the NFV-LBM is obtained by integrating the NDVBE. The macroscopic conservation equations recovered from the NDVBE via multi-scale expansion shows that the NDVBE has higher-order accuracy than diffusion theory, and the numerical solutions of neutron transport problems reveal the flexibility and applicability of NFV-LBM. This paper may provide some alternative perspectives for solving the NTE and some new ideas for researching the relationship between the NTE and other approximations.

This paper presents an implicit discrete unified gas-kinetic scheme (DUGKS) for steady state flow simulation in all flow regimes. The DUGKS is a multi-scale finite volume method (FVM), which is able to recover accurately the Navier-Stokes solutions in the continuum regime and the free molecular transport in collisionless regime. In the transition regime, the DUGKS can present reliable solution as well due to the close coupling of particle transport and collision in the flux evaluation at a cell interface. In this paper, an implicit DUGKS is constructed with predicted iterative steps for the updating of macroscopic flow variables, then the updating of microscopic gas distribution function in a discrete velocity space. The lower-upper symmetric Gauss-Seidel (LU-SGS) factorization is applied to solve the implicit equations. The fast convergence of implicit discrete unified gas-kinetic scheme (IDUGKS) can be achieved through the adoption of a numerical time step with a large CFL number. Some numerical test cases, including the Couette flow, the lid-driven cavity flows under different Knudsen numbers and the hypersonic flow in transition flow regime around a circular cylinder, have been performed to validate the proposed IDUGKS. The computational efficiency of the IDUGKS for steady state flow computations in all flow regimes can be improved by one or two orders of magnitude in comparison with the explicit DUGKS.

In this paper we investigate the plane wave discontinuous Galerkin method for three-dimensional anisotropic time-harmonic Maxwell's equations with diagonal matrix coefficients. By introducing suitable transformations, we define new plane wave basis functions and derive error estimates of the approximate solutions generated by the proposed discretization method for the considered homogeneous equations. In the error estimates, some dependence of the error bounds on the condition number of the coefficient matrix is explicitly given. Combined with local spectral element method, we further prove a convergence result for the nonhomogeneous case. Numerical results verify the validity of the theoretical results, and indicate that the resulting approximate solutions generated by the PWDG possess high accuracies.

Numerical integration of stiff stochastic differential equations based on stochastic computational singular perturbation (SCSP) was recently developed in [62]. In this paper, a modified stochastic computational singular perturbation (MSCSP) method is considered. Similar to what was proposed in [26] for deterministic chemical reaction systems, the current study applies the sensitivity derivatives of the forcing terms with respect to the state variables to measure the reaction scales, which leads to a quasi-steady state equation for the fast species. This yields explicit large-step integrators for stochastic fast-slow stiff differential equations systems, which removes the expensive eigen-calculations of the standard SCSP integrators. The efficiency of the MSCSP integrators is demonstrated with the benchmark stochastic Davis-Skodje model and a nonlinear catalysis model under certain stochastic disturbances.

An implementation of the Monte Carlo (MC) phase space generators coupled with adaptive MC integration/simulation program FOAM is presented. The first program is a modification of the classic phase space generator GENBOD interfaced with the adaptive sampling integrator/generator FOAM. On top of this tool the algorithm suitable for generation of the phase space for an reaction with two leading particles is presented (double-peripheral process with central production of particles). At the same time, it serves as an instructive example of construction of a self-adaptive phase space generator/integrator with a modular structure for specialized particle physics calculations.

High intensity focused ultrasound is a non-invasive method for treatment of diseased tissue that uses a beam of ultrasound to generate heat within a small volume. A common challenge in application of this technique is that heterogeneity of the biological medium can defocus the ultrasound beam. Here we reduce the problem of refocusing the beam to the inverse problem of estimating the acoustic aberration due to the biological tissue from acoustic radiative force imaging data. We solve this inverse problem using a Bayesian framework with a hierarchical prior and solve the inverse problem using a Metropolis-within-Gibbs algorithm. The framework is tested using both synthetic and experimental datasets. We demonstrate that our approach has the ability to estimate the aberrations using small datasets, as little as 32 sonication tests, which can lead to significant speedup in the treatment process. Furthermore, our approach is compatible with a wide range of sonication tests and can be applied to other energy-based measurement techniques.

Random rough textures can increase the absorbing efficiency of solar cells by trapping the optical light and increasing the optical path of photons. In this paper, we are concerned with optimal design of random rough surfaces in thin-film solar cells. We formulate the design problem as a random PDE constrained optimization problem and employ gradient-based methods for solving the problem numerically. To evaluate the gradient of the objective function, the Monte-Carlo method is used for sampling the probability space and the adjoint state method is employed to calculate the gradient at each sample. Numerical examples are shown to test the efficiency of the proposed algorithm. It is demonstrated that optimally obtained random textures yield an enormous absorption enhancement and a higher photon absorptance than that of existing random textures.