The well-known Mori-Zwanzig theory tells us that model reduction leads to memory effect. For a long time, modeling the memory effect accurately and efficiently has been an important but nearly impossible task in developing a good reduced model. In this work, we explore a natural analogy between recurrent neural networks and the Mori-Zwanzig formalism to establish a systematic approach for developing reduced models with memory. Two training models — a direct training model and a dynamically coupled training model — are proposed and compared. We apply these methods to the Kuramoto-Sivashinsky equation and the Navier-Stokes equation. Numerical experiments show that the proposed method can produce reduced model with good performance on both short-term prediction and long-term statistical properties.

Quality improvement of interferometric data collected by gravitational-wave detectors such as Advanced LIGO and Virgo is mission critical for the success of gravitational-wave astrophysics. Gravitational-wave detectors are sensitive to a variety of disturbances of non-astrophysical origin with characteristic frequencies in the instrument band of sensitivity. Removing non-astrophysical artifacts that corrupt the data stream is crucial for increasing the number and statistical significance of gravitational-wave detections and enabling refined astrophysical interpretations of the data. Machine learning has proved to be a powerful tool for analysis of massive quantities of complex data in astronomy and related fields of study. We present two machine learning methods, based on random forest and genetic programming algorithms, that can be used to determine the origin of non-astrophysical transients in the LIGO detectors. We use two classes of transients with known instrumental origin that were identified during the first observing run of Advanced LIGO to show that the algorithms can successfully identify the origin of non-astrophysical transients in real interferometric data and thus assist in the mitigation of instrumental and environmental disturbances in gravitational-wave searches. While the datasets described in this paper are specific to LIGO, and the exact procedures employed were unique to the same, the random forest and genetic programming code bases and means by which they were applied as a dual machine learning approach are completely portable to any number of instruments in which noise is believed to be generated through mechanical couplings, the source of which is not yet discovered.

The discontinuous Galerkin (DG) method has attained increasing popularity for solving the incompressible Navier-Stokes (INS) equations in recent years. In this work, we present a novel DG discretization for solving the two-dimensional INS equations in which the inviscid term of the INS equations is split into two parts, the Stokes operator and the nonlinear convective term, and treated separately. The Stokes operator is discretized using the artificial compressibility flux which is provided by the (exact) solution of a Riemann problem associated with a local artificial compressibility perturbation of the Stokes system, while the nonlinear term is discretized in divergency form by using the local Lax-Friedrichs fluxes; thus, local conservativity is inherent. Unlike the existing artificial compressibility flux for the DG discretization of the INS equations which needs to solve a Riemann problem for a nonlinear system by numerical iteration, the separate treatment of the nonlinear term from the Stokes operator makes the Riemann problem become linear and can be solved explicitly and straightforwardly, therefore, no iterative procedure is further required. A number of test cases with a wide range of Reynolds number are presented to assess the performance of the proposed method, which demonstrates its potential to be an alternative approach for high order numerical simulations of incompressible flows.

Atomic-level structural characterization of flexible proteins, such as intrinsically disordered proteins and multi-domain proteins connected by flexible linkers, is challenging as they possess distinct conformations in physiological conditions. Significant efforts have been made to develop integrated approaches by combining small angle neutron/X-ray scattering experiments with molecular simulations to reveal the distinct atomic structures and the corresponding populations for these flexible proteins. One widely used method, the basis-set supported ensemble method, classifies the simulation-generated protein conformations into a set of structural basis and then derives the corresponding populations by fitting to the experimental data. This method makes an implicit assumption that protein conformations of similar structures have similar small angle scattering profiles.The present work demonstrates that, for various protein systems ranging from compact globular proteins and flexible multi-domain proteins through to intrinsically disordered proteins, this method provides inaccurate assessment of the structural ensemble of the protein molecules due to the breakdown of the assumption made. To alleviate this problem, a two-fold-clustering method is developed to cluster the simulation-generated protein structures using information on both 3D structure and scattering profiles. As benchmarked by both simulation and experimental results, this new method yields much more accurate populations of structural basis of protein molecules.

Molecular beam epitaxy (MBE) is an important and challenging research topic in material science. In this paper, we propose a new fully discrete scheme for the well-celebrated continuum MBE model with slope selection. First of all, we use a multi-step strategy to discretize the MBE model in time. The obtained semi-discrete scheme is proved to possess properties of total mass conservation, unconditionally energy stability and uniquely solvability. The rigorous error estimate is then conducted to show its second-order convergence. The semi-discrete scheme is further discretized in space using the Fourier pseudo-spectral method. The fully discrete scheme is also shown to preserve mass-conservation and energy-dissipation properties. Afterward, several numerical examples are presented to validate the accuracy and efficiency of our proposed scheme. In particular, the scaling law for the roughness growing and effective energy decaying are captured during long-time coarsening dynamic simulations. The idea proposed in this paper could be readily utilized to design accurate and stable numerical approximations for many other energy-based phase field models.

Superconvergence for the lowest-order edge finite elements on strongly regular triangulation is studied. By the averaging technique, superconvergence of order O(h²) is established at the midpoint of the interior edge for both the finite element solution and the curl of the finite element solution. Numerical results justifying our theoretical analysis are presented.

In this paper, we study the equation system governing the movement of an immersed interface in incompressible fluid flows, and propose an efficient method for its numerical solution. The particularity of the current model is the inextensibility constraint imposed on the interface. We are interested in constructing a suitable variational formulation associated to this problem and the well-posedness of the weak problem. The significance of this variational formulation is that both the inextensibility of the interface and fluid incompressibility are strictly satisfied, and the well-posedness of the associated weak problem is rigorously proved. To the best of the authors' knowledge, no other models can be claimed to posses these properties. In fact our new formulation renders the inextensibility and the incompressibility constraints into a unique saddle point problem. Then, based on the proposed variational framework, we design an efficient spectral method for numerical approximations of the weak solution. The main contribution of this work are threefold: 1) a variational framework for the weak solutions of the immersed interface/incompressible equations and rigorous proof of the well-posedness of the weak problem; 2) a spectral method for solving the weak problem, together with a detailed stability analysis for the numerical solutions; 3) efficient implementation technique for the proposed method and some numerical experiments carried out to confirm the theoretical claims.

Numerical schemes for systems of transport equations are commonly constrained by a stability condition of Courant-Friedrichs-Lewy (CFL) type. We consider a system modeling the steady transport of photons and electrons in the field of radiotherapy. Naive discretizations of such a system are commonly constrained by a very restrictive CFL condition. This issue is circumvented by constructing an implicit scheme based on a relaxation approach.
We use an entropy-based moment model, namely the $M_1$ model. Such a system of equations possesses the non-linear flux terms of a hyperbolic system but no time derivative. The flux terms are well-defined only under a condition on the unknowns, called realizability, which corresponds to the positivity of an underlying kinetic distribution function.
The present numerical approach is applicable to non-linear systems which possess no hyperbolic operator, and it preserves the realizability property. However, the discrete equations are non-linear, and we propose a numerical method to solve such non-linear systems.
Our approach is tested on academic and practical cases in 1D, 2D, and 3D and it is shown to require significantly less computational power than reference methods.

In this paper, we mainly propose two kinds of high-accuracy schemes for the coupled nonlinear Schrödinger (CNLS) equations, based on the Fourier pseudospectral method (FPM), the high-order compact method (HOCM) and the Hamiltonian boundary value methods (HBVMs). With periodic boundary conditions, the proposed schemes admit the global energy conservation law and converge with even-order accuracy in time. Numerical results are presented to demonstrate the accuracy, energy-preserving and long-time numerical behaviors. Compared with symplectic Runge-Kutta methods (SRKMs), the proposed schemes are assuredly more effective to preserve energy, which is consistent with our theoretical analysis.

This work develops a second-order two-scale (SOTS) model based on homogenization method to predict thermo-mechanical coupling performance of porous materials with quasi-periodic structures. For the kinds of porous materials, the corresponding material coefficients are dependent on the macroscopic variable and the radiation effect at microscale is considered in this paper. The quasi-periodic properties of the thermo-mechanical coupling models which consider mutual interaction between temperature and displacement fields are proposed at first. Then, the two-scale formulas for the thermo-mechanical coupling problems with radiation boundary conditions are derived successively, and the finite element algorithms based on the SOTS model are brought forward in detail. Finally, by some typical examples, the effectiveness and validity of the proposed algorithms are confirmed. The computational results demonstrate that the SOTS method is efficient and valid to predict the thermo-mechanical coupling properties, and can acquire the microscale information of the porous materials accurately.

Turbidity current over a Gaussian-bump is investigated numerically using the upwinding combined compact difference (UCCD) scheme and the immersed boundary (IB) method in Cartesian grids. In the prediction of lock-exchange gravity-driven flow motion, the initial discontinuous concentration field is smoothed to avoid numerical oscillation by solving the Hamilton-Jacobi equation. The UCCD spatial scheme with sixth-order accuracy which introduces less dispersion errors is then used to discretize advection and diffusion terms in the calculation of concentration transport equation. Direct forcing IB method is employed to treat solid object bumps in the fluid flow. The incompressible Navier-Stokes solutions are obtained through the projection method. Analysis of the smoothing procedure, grid sensitivity and the effect of the Schmidt numbers is performed for the turbidity current problem to validate the proposed numerical algorithm, which is shown to be capable of accurately demonstrating their results. Finally, several problems of turbidity current over a three-dimensional seafloor are investigated. The front locations of the currents interacting with the bump are predicted under different Reynolds numbers. Also, the current properties, namely the suspended particle mass, sedimentation rate and energy budget, are compared with the available numerical results.

In this work, we propose a generalized multiscale inversion algorithm for heterogeneous problems that aims at solving an inverse problem on a computational coarse grid. Previous inversion techniques for multiscale problems seek a coarse-grid medium properties, e.g., permeability and conductivity, and by doing so, they assume that there exists a homogenized representation of the underlying fine-scale permeability field on a coarse grid. Generally such assumptions do not hold for highly heterogeneous fields, e.g., fracture media or channelized fields, where the width of channels are very small compared to the scale of coarse grids. In these cases, grid refinement can lead to many degrees of freedom, and thus numerically unattractive to apply. The proposed algorithm is based on the Generalized Multiscale Finite Element Method (GMsFEM), which uses local spectral problems to identify non-localized features, i.e., channels (high-conductivity inclusions that connect the boundaries of the coarse-grid block). The inclusion of these features in the coarse space enables one to achieve a good accuracy. The approach is valid under the assumption that the solution can be well represented in a reduced-dimensional space spanned by multiscale basis functions. In practice, these basis functions are non-observable as we do not identify the fine-scale features of the permeability field. Our inversion algorithm finds the discretization parameters of the resulting system on the coarse grid. By doing so, we identify the appropriate coarse-grid parameters representing the permeability field instead of fine-grid permeability field. We illustrate the potential of the approach by numerical results for fractured media.

We present DAFMPB (DASHMM-accelerated Adaptive Fast Multipole Poisson-Boltzmann solver) for rapid evaluation of the electrostatic potentials and forces, and total solvation-free energy in biomolecular systems modeled by the linearized Poisson-Boltzmann (LPB) equation. DAFMPB first reformulates the LPB into a boundary integral equation and then discretizes it using the node-patch scheme [33]. It solves the resulting linear system using GMRES, where it adopts the DASHMM library [14] to accelerate the matrix-vector multiplication in each iteration. DASHMM is built on top of a global address space allowing the user of DAFMPB to operate on both shared and distributed memory computers with modification of their code. This paper is a brief summary of the program, including the algorithm, implementation, installation and usage.