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• A novel positivity-preserving algorithm for implicit NS solver. • A residual correction to compute the correction factor. • A flux correction to enforce the positivity of the solution conservatively. • Positivity-preserving combined with implicit iterations. • Numerical experiments to verify the positivity-preserving capability. This paper presents a general positivity-preserving algorithm for implicit high-order finite volume schemes that solve compressible Euler and Navier-Stokes equations to ensure the positivity of density and internal energy (or pressure). Previous positivity-preserving algorithms are mainly based on the slope limiting or flux limiting technique, which rely on the existence of low-order positivity-preserving schemes. This dependency poses serious restrictions on extending these algorithms to temporally implicit schemes since it is difficult to know if a low-order implicit scheme is positivity-preserving. In the present paper, a new positivity-preserving algorithm is proposed in terms of the flux correction technique. And the factors of the flux correction are determined by a residual correction procedure. For a finite volume scheme that is capable of achieving a converged solution, we show that the correction factors are in the order of unity with additional high-order terms corresponding to the spatial and temporal rates of convergence. Therefore, the proposed positivity-preserving algorithm is accuracy-reserving and asymptotically consistent. The notable advantage of this method is that it does not rely on the existence of low-order positivity-preserving baseline schemes. Therefore, it can be applied to the implicit schemes solving Euler and especially Navier-Stokes equations. In the present paper, the proposed technique is applied to an implicit dual time-stepping finite volume scheme with temporal second-order and spatial high-order accuracy. The present positivity-preserving algorithm is implemented in an iterative manner to ensure that the dual time-stepping iteration will converge to the positivity-preserving solution. Another similar correction technique is also proposed to ensure that the solution remains positivity-preserving at each sub-iteration. Numerical results demonstrate that the proposed algorithm preserves positive density and internal energy in all test cases and significantly improves the robustness of the numerical schemes. [ABSTRACT FROM AUTHOR]

It is increasingly realized that taking stochastic effects into account is important in order to study biological cells. However, the corresponding mathematical formulation, the chemical master equation (CME), suffers from the curse of dimensionality and thus solving it directly is not feasible for most realistic problems. In this paper we propose a dynamical low-rank algorithm for the CME that reduces the dimensionality of the problem by dividing the reaction network into partitions. Only reactions that cross partitions are subject to an approximation error (everything else is computed exactly). This approach, compared to the commonly used stochastic simulation algorithm (SSA, a Monte Carlo method), has the advantage that it is completely noise-free. This is particularly important if one is interested in resolving the tails of the probability distribution. We show that in some cases (e.g. for the lambda phage) the proposed method can drastically reduce memory consumption and run time and provide better accuracy than SSA. • We propose a low-rank algorithm that does not rely on single orbital basis functions. • The proposed approach is better suited for biological/chemical reaction networks. • We demonstrate better accuracy and improved efficiency compared to SSA for some problems. • The proposed complexity reduction algorithm is noise free (in contrast to SSA). • We demonstrate that our algorithm accurately resolves the tails of the distribution (in contrast to SSA). [ABSTRACT FROM AUTHOR]

Steepest descent methods combining complex contour deformation with numerical quadrature provide an efficient and accurate approach for the evaluation of highly oscillatory integrals. However, unless the phase function governing the oscillation is particularly simple, their application requires a significant amount of a priori analysis and expert user input, to determine the appropriate contour deformation, and to deal with the non-uniformity in the accuracy of standard quadrature techniques associated with the coalescence of stationary points (saddle points) with each other, or with the endpoints of the original integration contour. In this paper we present a novel algorithm for the numerical evaluation of oscillatory integrals with general polynomial phase functions, which automates the contour deformation process and avoids the difficulties typically encountered with coalescing stationary points and endpoints. The inputs to the algorithm are simply the phase and amplitude functions, the endpoints and orientation of the original integration contour, and a small number of numerical parameters. By a series of numerical experiments we demonstrate that the algorithm is accurate and efficient over a large range of frequencies, even for examples with a large number of coalescing stationary points and with endpoints at infinity. As a particular application, we use our algorithm to evaluate cuspoid canonical integrals from scattering theory. A Matlab implementation of the algorithm is made available and is called PathFinder. • An algorithm for the evaluation of oscillatory integrals with polynomial phase. • The algorithm automates the numerical steepest descent method. • Our implementation is robust and efficient across all frequencies. [ABSTRACT FROM AUTHOR]

We propose an easy-to-implement iterative method for resolving the implicit (or semi-implicit) schemes arising in solving reaction-diffusion (RD) type equations. We formulate the nonlinear time implicit scheme as a min-max saddle point problem and then apply the primal-dual hybrid gradient (PDHG) method. Suitable precondition matrices are applied to the PDHG method to accelerate the convergence of algorithms under different circumstances. Furthermore, our method is applicable to various discrete numerical schemes with high flexibility. From various numerical examples tested in this paper, the proposed method converges properly and can efficiently produce numerical solutions with sufficient accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modification of an acoustic-transport operator splitting Lagrange-projection method for simulating compressible flows with gravity. The original method involves two steps that respectively account for acoustic and transport effects. Our work proposes a simple modification of the transport step, and the resulting modified scheme turns out to be a flux-splitting method. This new numerical method is less computationally expensive in the low-Mach regime, more memory efficient, and easier to implement than the original one. We prove stability properties for this new scheme by showing that under classical CFL conditions, the method is positivity preserving for mass, energy and entropy satisfying. The flexible flux-splitting structure of the method enables straightforward extensions of the method to multi-dimensional problems (with respect to space) and high-order discretizations that are presented in this work. We also propose an interpretation of the flux-splitting solver as a relaxation approximation. Both the stability and the accuracy of the new method are tested against one-dimensional and two-dimensional numerical experiments that involve highly compressible flows and low-Mach regimes. • The all-regime acoustic/transport splitting solver proposed in [16] can be modified into a flux-splitting, one-step version. • The resulting method is less computationally expensive, more memory efficient, and easier to implement than the original one. • Stability properties are derived by interpreting the scheme as a convex combination of the advection/pressure steps. • This recasting allows an easy high-order extension as the new method is a standard flux-based solver. • Following the lines of [32] , we equip our scheme with the well-balanced property for gravity source terms. [ABSTRACT FROM AUTHOR]

For band-structure computation, the Bloch mode synthesis (BMS) or the generalized Bloch mode synthesis (GBMS) method is a series of model-reduction computational algorithms based on the Craig-Bampton method. A hybrid Bloch mode synthesis (HBMS) method with a hybrid Floquet-Bloch periodic boundary condition is proposed in this paper. A force-based Floquet-Bloch periodic boundary condition is derived and combined with the default displacement-based Floquet-Bloch periodic boundary condition to form the hybrid Floquet-Bloch periodic boundary condition. The hybrid BMS method has been developed by combining the hybrid Floquet-Bloch periodic boundary conditions with the free-interface component mode synthesis method. The HBMS method considers the influence of higher-order modes on periodic boundary structures so that the hybrid Floquet-Bloch periodic boundary conditions can be applied to the reduced-order modal space without introducing errors, which improves the computational efficiency. According to different truncation errors, the HBMS method has been developed into three algorithms, and they each have a different precision-efficiency ratio. Numerical experiments demonstrate that the developed method outperforms in accuracy and computational efficiency compared to the default BMS/GBMS method. • The force and displacement balance on the Bloch periodic boundary are considered. • Higher-order modes are considered by introducing force Bloch boundary conditions. • The Bloch boundary conditions are directly imposed on the reduced order model. • A new hybrid Bloch mode synthesis method for phononic was developed. • The proposed method is developed into three algorithms for practicality. [ABSTRACT FROM AUTHOR]

In this paper, we present an efficient adaptive multigrid strategy for the geometry optimization of large-scale three dimensional molecular mechanics. The resulting method can achieve significantly reduced complexity by exploiting the intrinsic low-rank property of the material configurations and by combining the state-of-the-art adaptive techniques with the hierarchical structure of multigrid algorithms. To be more precise, we develop a oneway multigrid method with adaptive atomistic/continuum (a/c) coupling, e.g., blended ghost force correction (BGFC) [1] approximations with gradient-based a posteriori error estimators on the coarse levels. We utilize state-of-the-art 3D mesh generation techniques to effectively implement the method. For 3D crystalline defects, such as vacancies, micro-cracks and dislocations, compared with brute-force optimization, complexity with superior rates can be observed numerically, and the strategy has a five-fold acceleration in terms of CPU time for systems with 108 atoms. • An efficient adaptive multigrid strategy for large-scale geometry optimization is proposed. • The computational cost for 3D geometry optimization can be significantly reduced. • The low-rank property of materials and the multigrid algorithms with adaptivity is combined. • Effective 3D mesh generation and adaptation for a/c coupling methods is implemented. • 3D simulation for various types of complex defect configurations is provided. [ABSTRACT FROM AUTHOR]

This article presents a new method to identify the main patterns describing the flow motion in complex flows. The algorithm is an extension of the higher-order dynamic mode decomposition (HODMD), which compresses the snapshots from the analysed database and progressively updates new compressed snapshots on-the-fly , so it is denoted as HODMD on-the-fly (HODMD-of). This algorithm can be applied in parallel to the numerical simulations (or experiments), and it exhibits two main advantages over offline algorithms: (i) it automatically selects on-the-fly the number of necessary snapshots from the database to identify the relevant dynamics; and (ii) it can be used from the beginning of a numerical simulation (or experiment), since it uses a sliding-window to automatically select, also on-the-fly, the suitable interval to perform the data analysis, i.e. it automatically identifies and discards the transient dynamics. The HODMD-of algorithm is suitable to build reduced order models, which have a much lower computational cost than the original simulation. The performance of the method has been tested in three different cases: the axi-symmetric synthetic jet, the three-dimensional wake of a circular cylinder and the turbulent wake behind a wall-mounted square cylinder. The obtained speed-up factors are around 7 with respect to HODMD; this value depends on the simulation and the configuration of the hyperparameters. HODMD-of also provides a significant reduction of the memory requirements, between 40 − 80 % amongst the two- and three-dimensional cases studied in this paper. • Novel algorithm based on HODMD that combines rank reduction strategies with the on-the-fly methodology. • The algorithm automatically selects the optimal time interval to extract the main flow patterns. • Significant reduction of the memory requirements and speed-up related with standard HODMD. • Suitable to build reduced order models. [ABSTRACT FROM AUTHOR]

• Designed coupled implicit-explicit temporal and spatial splitting method. • Formulated conditions for stability and designed appropriate spaces. • Multiscale spaces with CEM-GMsFEM are used in constructing the splitting method. • Numerical results confirm our theoretical findings. Many multiscale problems have a high contrast, which is expressed as a very large ratio between the media properties. The contrast is known to introduce many challenges in the design of multiscale methods and domain decomposition approaches. These issues to some extent are analyzed in the design of spatial multiscale and domain decomposition approaches. However, some of these issues remain open for time dependent problems as the contrast affects the time scales, particularly, for explicit methods. For example, in parabolic equations, the time step is d t = H 2 / κ m a x , where κ m a x is the largest diffusivity. In this paper, we address this issue in the context of parabolic equation by designing a splitting algorithm. The proposed splitting algorithm treats dominant multiscale modes in the implicit fashion, while the rest in the explicit fashion. The contrast-independent stability of these algorithms requires a special multiscale space design, which is the main purpose of the paper. We show that with an appropriate choice of multiscale spaces we can achieve an unconditional stability with respect to the contrast. This could provide computational savings as the time step in explicit methods is adversely affected by the contrast. We discuss some theoretical aspects of the proposed algorithms. Numerical results are presented. [ABSTRACT FROM AUTHOR]

In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasi-linear parabolic partial differential equations (PDEs), which is related to the FBSDEs from the Pardoux-Peng theory. The algorithms rely on a learning process by minimizing the path-wise difference between two discrete stochastic processes, which are defined by the time discretization of the FBSDEs and the DNN representation of the PDE solution, respectively. The proposed algorithms are shown to generate DNN solution for a 100-dimensional Black–Scholes–Barenblatt equation, which is accurate in a finite region in the solution space, and has a convergence rate close to that of the Euler–Maruyama scheme used for discretizing the FBSDEs. • FBSDE deep neural network (DNN) with loss comparing two stochastic processes for the PDEs from Pardoux-Peng theory. • Strong 1/2 Convergence as the Euler-Maruyama method in solving a 100-dimensional Black-Scholes-Barenblatt equation. • Use of Richardson extrapolation for accuracy enhancement. • Multiscale DNN captures temporal high frequency content of the PDEs more accurately. [ABSTRACT FROM AUTHOR]

• Lack of standardized format for Adaptive Mesh Refinement post-processing tools. • LightAMR is a new compact way to fully describe Adaptive Mesh Refinement meshes. • Tree pruning algorithm for data redundancy can be built over the lightAMR format. • LightAMR mesh description is compressible by a factor ∼1000 on RAMSES uses cases. • Lossless compression of lightAMR float scalar fields achieves a ratio of 1.2-1.5. The evolution of parallel I/O library as well as new concepts such as 'in transit' and 'in situ' visualization and analysis have been identified as key technologies to circumvent I/O bottleneck in pre-exascale applications. Nevertheless, data structure and data format can also be improved for both reducing I/O volume and improving data interoperability between data producer and data consumer. In this paper, we propose a very lightweight and purpose-specific post-processing data model for AMR meshes, called lightAMR. Based on this data model, we introduce a tree pruning algorithm that removes data redundancy from a fully threaded AMR octree. In addition, we present two lossless compression algorithms, one for the AMR grid structure description and one for AMR double/single precision physical quantity scalar fields. Then we present performance benchmarks on RAMSES simulation datasets of this new lightAMR data model and the pruning and compression algorithms. We show that our pruning algorithm can reduce the total number of cells from RAMSES AMR datasets by 10-40% without loss of information. Finally, we show that the RAMSES AMR grid structure can be compacted by ∼3 orders of magnitude and the float scalar fields can be compressed by a factor ∼1.2 for double precision and ∼ 1.3 − 1.5 in single precision with a compression speed of ∼1 GB/s. [ABSTRACT FROM AUTHOR]

• The sound propagation problems of classical/generalized line sources are solved. • The proposed algorithm and program are unconditionally stable. • The proposed algorithm is naturally parallel. • The normal mode solver based on the spectral method can accurately simulate seabeds. A high-precision numerical sound field is the basis of underwater target detection, positioning and communication. A line source in a plane is a common type of sound source in computational ocean acoustics. The exciting waveguide in a range-dependent ocean environment is often structurally complicated; however, traditional algorithms often assume that the waveguide has a simple seabed boundary and that the line source is located at a horizontal range of 0 m, although this ideal situation is rarely encountered in the actual ocean. In this paper, a novel algorithm is designed that can solve for the sound field excited by a line source at any position in a range-dependent ocean environment. The proposed algorithm uses the classic stepwise approximation approach to address the range dependence of the environment and uses the Chebyshev–Tau spectral method to solve for the horizontal wavenumbers and modes of approximately range-independent segments. Once the modal information of these flat segments has been obtained, a global matrix is constructed to solve for the coupling coefficients of all segments, and finally, the complete sound field is synthesized. Numerical experiments using a robust numerical program developed based on this algorithm verify the correctness and usability of our novel algorithm and software. Furthermore, a detailed analysis and test of the computational cost of this algorithm show that it is efficient. [ABSTRACT FROM AUTHOR]

• Design temporal splitting together with multiscale methods to reduce the computational cost. • Study temporal splitting and spatial multiscale decomposition to extract implicit and explicit modes. • Use of GMsFEM within temporal splitting methods appropriately. In this paper, we study temporal splitting algorithms for multiscale problems. The exact fine-grid spatial problems typically require some reduction in degrees of freedom. Multiscale algorithms are designed to represent the fine-scale details on a coarse grid and, thus, reduce the problems' size. When solving time-dependent problems, one can take advantage of the multiscale decomposition of the solution and perform temporal splitting by solving smaller-dimensional problems, which is studied in the paper. In the proposed approach, we consider the temporal splitting based on various low dimensional spatial approximations. Because a multiscale spatial splitting gives a "good" decomposition of the solution space, one can achieve an efficient implicit-explicit temporal discretization. We present a recently developed theoretical result in [15] and adopt in this paper for multiscale problems. Numerical results are presented to demonstrate the efficiency of the proposed splitting algorithm. [ABSTRACT FROM AUTHOR]

• The Darwin particle-in-cell method is a non-separable canonical Hamiltonian system. • A new, explicit time-stepping is applied, based on Tao's recent paper. • Semi-implicit method avoids the plasma frequency time-step constraint. • Results are obtained to demonstrate accuracy and long-term stability. • Energy conservation, Weibel growth and saturation, Alfvén-branch waves accurate. A new approach to Darwin particle-in-cell plasma simulation is described. Using a finite-element approach, the vector potential is assured to be exactly solenoidal. This allows writing the system of multi-species particles as an action-at-a-distance Hamiltonian system. Applying recently-developed explicit symplectic methods to this gives a time-explicit algorithm with all desired properties. Elliptic systems for the electrostatic and magneto-static portions of the problem are inverted efficiently by an algebraic multi-grid algorithm. The algorithm is implemented in a two-dimensional Cartesian-geometry code and tested by application to Weibel instability and to Alfvén wave dynamics. Results show the effectiveness of this approach. Extensions to arbitrary meshes and to a partially time-implicit scheme are briefly discussed. [ABSTRACT FROM AUTHOR]

By utilizing the perfectly matched layer (PML) and source transfer techniques, the diagonal sweeping domain decomposition method (DDM) was recently developed for solving the high-frequency Helmholtz equation in R n , which uses 2 n sweeps along respective diagonal directions with checkerboard domain decomposition. Although this diagonal sweeping DDM is essentially multiplicative, it is highly suitable for parallel computing of the Helmholtz problem with multiple right-hand sides when combined with the pipeline processing since the number of sequential steps in each sweep is much smaller than the number of subdomains. In this paper, we propose and analyze a trace transfer-based diagonal sweeping DDM. A major advantage of changing from source transfer to trace transfer for information passing between neighbor subdomains is that the resulting diagonal sweeps become easier to analyze and implement and more efficient, since the transferred traces have only 2 n cardinal directions between neighbor subdomains while the transferred sources come from a total of 3 n − 1 cardinal and corner directions. We rigorously prove that the proposed diagonal sweeping DDM not only gives the exact solution of the global PML problem in the constant medium case but also does it with at most one extra round of diagonal sweeps in the two-layered media case, which lays down the theoretical foundation of the method. Performance and parallel scalability of the proposed DDM as direct solver or preconditioner are also numerically demonstrated through extensive experiments in two and three dimensions. [ABSTRACT FROM AUTHOR]

• Development of numerical methods to compute the ablation of metallic thermal protection system. • Extension of the operator splitting strategy to the five-equation model with dissipative effects. • Several approaches are detailed in order to prevent the numerical diffusion of the material interface in the transport step. • An ALE formulation of numerical schemes for both fluid and solid domains is given to capture accurately the melting front. In this paper, numerical methods are developed and detailed in order to be able to compute the ablation of metallic thermal protection system. In this complex multi-physics problem, the thermal state inside a solid domain and a two-phase viscous flow have to be computed. Since the two fluid phases are non-miscible, an extension of the five-equation model to dissipative effects is considered. An operator splitting strategy is used to separate the different phenomena according to their own propagation speed. An implicit time integration is performed for the acoustic+dissipative step while the transport step is computed with an explicit scheme. The hyperbolic part of the acoustic+dissipative step is solved in a non-conservative form using a Godunov-type scheme based on a simple Riemann solver. A classical discretization is used for the dissipative terms, and also for the heat equation inside the solid domain. Several approaches are detailed in order to prevent the numerical diffusion of the material interface in the transport step. Finally, since moving grids are used to capture accurately the melting front, an ALE formulation of the numerical schemes for both fluid and solid domains is given in a multidimensional framework. A fluid-solid coupling algorithm is then proposed to compute such a complex multi-physics problem. Numerical simulations show the validity and the robustness of the implicit-explicit scheme used for the discretization of the five-equation system. The last test case, namely the melting of an aluminium solid block by a lid-driven cavity filled with air, shows that the numerical tools developed here are robust enough to compute complex configurations involving a two-phase flow with high density ratios and a solid part. [ABSTRACT FROM AUTHOR]

• The proposed verification scheme requires only errors on grid quantities. • Rigorously prove convergence of the Vlasov solution is implied by convergence of the electric field or charge distribution. • Confirm by analysis and experiment differing convergence rates, with respect to particle number, of various grid quantities. As simulations of kinetic plasmas continue to increase in scope and complexity, a rigorous and straightforward method for verifying particle-in-cell (PIC) implementations is necessary to ensure their correctness. In this paper, we present a deterministic method for the rigorous verification of multidimensional, multispecies, electrostatic particle-in-cell codes based on the method of manufactured solutions. Specifically, we prove that rigorous verification is possible through the exclusive examination of errors of grid quantities (i.e., moments and/or fields), allowing for a very light-weight and non-intrusive implementation in existing PIC codes. This is a marked improvement over earlier PIC verification approaches (only demonstrated with one species in 1D-1V), which rely on the comparison of cumulative distribution functions, and are invasive on the code base, introduce additional stochastic noise, are significantly more computationally expensive, and lack rigorous convergence properties. Interestingly, we show that different grid quantities feature different rates of convergence with the number of particles and mesh size, impacting the verification process. These theoretical results are confirmed numerically with a multi-species 2D-2V particle-in-cell code, including the ability of the method to detect order reduction due to an incorrect implementation. [ABSTRACT FROM AUTHOR]

• Machine learning framework for the reconstruction of unsteady fluid flows fields. • Variational approach for a robust reduction of the high dimensional field. • Investigations on three canonical cases. • Combination of linear or nonlinear reduction with regressive or direct reconstruction. This paper investigates the use of data-driven methods for the reconstruction of unsteady fluid flow fields. The proposed framework is based on the combination of machine learning tools: dimensionality reduction to extract dominant spatial directions from data, reconstruction algorithm to recover encoded data by limited measurements and cross-validation for hyperparameter optimization. For the encoding part, linear and nonlinear extraction of patterns are considered: proper orthogonal decomposition (POD), linear autoencoder (LAE) and variational autoencoder (VAE). For the reconstruction part, regressive reconstruction (neural network, linear, support vector, gradient boosting) and library-based reconstruction are compared, each method being cross-validated to ensure good generalization on testing data. The position of sensors is optimized using an enhanced clustering algorithm. The robustness of regressive reconstructions to noise measurements is also addressed, showing the benefits of variational approaches in the reduction phase. The strategy is tested for three increasing complexity flows: 2D vortex shedding (R e = 200), 2D spatial mixing layer and 3D vortex shedding (R e = 20000). The results suggest that proper machine learning approaches to fluid flow data can lead to effective reconstruction models that can be used for the rapid estimation of complex flows. [ABSTRACT FROM AUTHOR]

Usually, the temperature dependence of specific heats is neglected or the specific heats are frozen in interface computations for compressible two-material flows. In this paper, we present a practical interface treatment to faithfully capture the effect of high temperature on interface evolutions. A general technique for solving the Riemann problem equipped with a wide variety of equations of state (EOS) is established. In a unified framework for computing the interfacial states, it provides a convenient way to deal with the thermally perfect gas (PG) that considers the effect of temperature on specific heats. The algorithm of the complete and exact solution to Riemann problem with thermally PG is also designed in detail. Based on this technique, the modified ghost fluid method with an approximate Riemann solver is further extended to handle the interface of two-material flows involving thermally PG with chemical reactions. Several typical problems are selected to validate and test the present algorithm for the interaction between the thermally PG and other gases or liquids. The results indicate that the present algorithm enables an effective implementation for simulating various two-material multi-species flows with different types of EOS. As temperature increases, the behavior of the interfacial flows under the assumption of thermally PG EOS gradually differs from that under the assumption of calorically PG EOS. • Unified framework for simulating two-material multi-species flows. • Exact and approximate Riemann solvers for thermally perfect gases. • Modified ghost fluid method for high-temperature effects with chemical reactions. • Comparison of thermally/calorically perfect gases. [ABSTRACT FROM AUTHOR]

• Introduce a factorization for the inverse of discrete Fourier integral operators. • Use the butterfly factorization and the peeling algorithm to efficiently invert discrete Fourier integral operators. • Introduce an algorithm to invert hierarchical matrices arising during the inversion of Fourier integral operators. • Apply the proposed approach to 1D and 2D Fourier integral operators to demonstrate its numerical efficiency. This paper introduces a factorization for the inverse of discrete Fourier integral operators of size N × N that can be applied in quasi-linear time. The factorization starts by approximating the operator with the butterfly factorization. Next, a hierarchical matrix representation is constructed for the hermitian matrix arising from composing the Fourier integral operator with its adjoint. This representation is inverted efficiently with a new algorithm based on the hierarchical interpolative factorization. By combining these two factorizations, an approximate inverse factorization for the Fourier integral operator is obtained as a product of O (log ⁡ N) sparse matrices with O (N) entries. The resulting approximate inverse factorization can be used as a direct solver or as a preconditioner. Numerical examples on 1D and 2D Fourier integral operators, including a generalized Radon transform, demonstrate the performance of this new approach. [ABSTRACT FROM AUTHOR]

• A linearly operator splitting algorithm is proposed for the nonlocal VCH equation. • The energy stabilities for both subproblems are proved. • The stability and convergence of the operator splitting algorithm are studied. • A semi-implicit SDC method is further used to improve time accuracy. Recently, the viscous Cahn-Hilliard (VCH) equation has been proposed as a phenomenological continuum model for phase separation in glass and polymer systems where intermolecular friction forces become important. Compared with the classical local VCH model, the nonlocal VCH model equipped with nonlocal diffusion operator can describe more practical phenomena for modeling phase transitions of microstructures in materials. This paper presents a high order fast explicit method based on operator splitting and spectral deferred correction (SDC) for solving the nonlocal VCH equation. We start with a second-order operator splitting spectral scheme, which is based on the Fourier spectral method and the strong stability preserving Runge-Kutta (SSP-RK) method. The scheme takes advantage of applying the fast Fourier transform (FFT) and avoiding nonlinear iteration. The stability and convergence analysis of the obtained numerical scheme are analyzed. To improve the temporal accuracy, the semi-implicit SDC method is then introduced. Various numerical simulations are performed to validate the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

WENO schemes are a popular class of shock-capturing schemes which adopt an adaptive-stencil approach to interpolation. WENO schemes rely on smoothness indicators to assess the relative smoothness of the solution within the sub-stencils. Computing these smoothness indicators is the most expensive operation in the WENO reconstruction procedure. In this paper, an efficient algorithm is proposed to compute these quantities without sacrificing the positivity property of the smoothness indicators. The proposed algorithm involves linear combinations of the undivided differences which can be computed efficiently in a recursive manner. This allows the computation of the smoothness indicators to be performed using significantly fewer floating-point operations compared to conventional implementations. Moreover, the proposed algorithm is simple to implement and involves fewer constants. [ABSTRACT FROM AUTHOR]

Originated from the complex network theory, the Lagrangian flow network (LFN) has motivated various tools to analyze fluid flows' transport and mixing behaviors. These tools include the so-called finite time escape rate (FTER), the transition matrix, and also the finite time entropy. This paper proposes novel Eulerian algorithms to compute these LFN quantities in two dimensions based on a partial differential equation (PDE) formulation for the underlying flow map. Compared to the typical Lagrangian computations based on ray tracing, our numerical approaches are computationally efficient. Moreover, for velocity fields obtained numerically by computational fluid dynamic (CFD) solvers, our approach does not require further interpolation of the velocity field but can use the discrete velocity data directly. Finally, we will apply our proposed algorithms to several test examples, including a real-life dataset, to demonstrate the performance of the method. [ABSTRACT FROM AUTHOR]

• Method (modBIP) presented to pre-train random coefficients of extreme learning machines (ELM). • modBIP can work with any activation function and is more accurate than the BIP method. • ELM/modBIP is highly accurate and accuracy is insensitive to initial random coefficients. In extreme learning machines (ELM) the hidden-layer coefficients are randomly set and fixed, while the output-layer coefficients of the neural network are computed by a least squares method. The randomly-assigned coefficients in ELM are known to influence its performance and accuracy significantly. In this paper we present a modified batch intrinsic plasticity (modBIP) method for pre-training the random coefficients in the ELM neural networks. The current method is devised based on the same principle as the batch intrinsic plasticity (BIP) method, namely, by enhancing the information transmission in every node of the neural network. It differs from BIP in two prominent aspects. First, modBIP does not involve the activation function in its algorithm, and it can be applied with any activation function in the neural network. In contrast, BIP employs the inverse of the activation function in its construction, and requires the activation function to be invertible (or monotonic). The modBIP method can work with the often-used non-monotonic activation functions (e.g. Gaussian, swish, Gaussian error linear unit, and radial-basis type functions), with which BIP breaks down. Second, modBIP generates target samples on random intervals with a minimum size, which leads to highly accurate computation results when combined with ELM. The combined ELM/modBIP method is markedly more accurate than ELM/BIP in numerical simulations. Ample numerical experiments are presented with shallow and deep neural networks for function approximation and boundary/initial value problems with partial differential equations. They demonstrate that the combined ELM/modBIP method produces highly accurate simulation results, and that its accuracy is insensitive to the random-coefficient initializations in the neural network. This is in sharp contrast with the ELM results without pre-training of the random coefficients. [ABSTRACT FROM AUTHOR]

• Interface-capturing is based on a second-order unstructured finite-volume method. • A central interpolation technique is applied for cell-faces approximations. • The numerical method is quite suitable to work within the signed distance function. • 2D and 3D benchmarks validate the capabilities and performance of the algorithm. • The interface accurately returns to its original shape for the single vortex problem. In this paper, we introduce a second-order unstructured finite-volume method developed to solve a conservative level-set equation in two- and three-dimensional geometries. The characteristic function for the flow is defined at the center of each cell while the face-normal velocities are calculated at the mid-points of the corresponding cell faces. An interpolation method independent of cell shape and based on a central scheme is applied for the approximations at the cell faces. The capabilities and performance of the proposed scheme are validated using several 2D and 3D tests. Results show that the present numerical method is quite suitable for working within the smooth framework resulting from the signed distance function. Moreover, this method yields very accurate results in interface-capturing problems such as the single vortex deformation. [ABSTRACT FROM AUTHOR]

• A new algorithm incorporates model reduction in the spatial and temporal domains. • We study parabolic problems with heterogeneous coefficients and rough initial data. • We derive convergence analysis that weakly depends on the heterogeneous coefficients. • The convergence is rigorously studied, which greatly improves the current result. • Extensive numerical tests are performed to show the fast convergence of our algorithm. We propose in this paper the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm to solve parabolic equations with heterogeneous coefficients efficiently. This algorithm combines the advantages of multiscale methods that can deal with heterogeneity in the spatial domain effectively, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. We derive the convergence rate of this algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step and the fine-scale time step. Extensive numerical tests are presented to demonstrate the performance of our algorithm, which verify our theoretical results perfectly. [ABSTRACT FROM AUTHOR]

• Hybrid flux algorithm to find fluxes across cell faces in same phase and different phase. • Resolves the profiles where both smooth and discontinuous solutions coexist. • Automatically adjust the weight between MUSCL and THINC schemes using a slope-weight parameter. This paper presents a single-pressure-field two-fluid model with finite-volume discretization to solve the equations of motion of compressible multiphase flows. To capture the discontinuities caused by shock waves and fluid interfaces, we propose a generalized discontinuity sharpening technique that combines the conventional monotonic upstream scheme for conservation law (MUSCL) and tangent of hyperbola interface capturing (THINC) schemes. In addition, a slope ratio-weighted parameter, ζ , is used to control the proportion of values reconstructed by MUSCL and THINC, and we show that the present method can retain sharp interfaces when the value of the parameter β in the THINC scheme is set ranging from 1.6 to 3.0. Fluxes across various interfaces are evaluated using a hybrid AUSMD-type flux algorithm, where the mass flux and pressure induced on the cell faces are calculated using an approximate Riemann solver. The accuracy and robustness of the proposed method are validated by solving a series of one- and two-dimensional single-phase flows. Furthermore, complex wave patterns arising from two-dimensional shock bubble/water-column interactions are examined, which indicate that compared with the existing schemes applied to two-fluid modeling, the proposed scheme significantly sharpens the interfaces and captures more details of the flow features. Finally, simulations of a three-dimensional example of the liquid jet crossflow are conducted. The proposed scheme shows more details of the fluid interface, including the interfacial instabilities on the windward side of the liquid jet and droplet formation due to the breakup phenomenon in the downstream of the crossflow, than the existing schemes. [ABSTRACT FROM AUTHOR]

• A neural network enhanced Boltzmann model is proposed. • The mechanical and neural models are unified into a differentiable architecture and the neural- ODE-type training strategy is constructed. • A general numerical scheme is designed to solve the universal Boltzmann equation. • Numerical experiments of homogeneous and inhomogeneous systems are provided to validate the current method. One of the biggest challenges for simulating the Boltzmann equation is the evaluation of fivefold collision integral. Given the recent successes of deep learning and the availability of efficient tools, it is an obvious idea to try to substitute the calculation of the collision operator by the evaluation of a neural network. However, it is unlcear whether this preserves key properties of the Boltzmann equation, such as conservation, invariances, the H-theorem, and fluid-dynamic limits. In this paper, we present an approach that guarantees the conservation properties and the correct fluid dynamic limit at leading order. The concept originates from a recently developed scientific machine learning strategy which has been named "universal differential equations". It proposes a hybridization that fuses the deep physical insights from classical Boltzmann modeling and the desirable computational efficiency from neural network surrogates. The construction of the method and the training strategy are demonstrated in detail. We conduct an asymptotic analysis and illustrate the multi-scale applicability of the method. The numerical algorithm for solving the neural network-enhanced Boltzmann equation is presented as well. Several numerical test cases are investigated. The results of numerical experiments show that the time-series modeling strategy enjoys the training efficiency on this supervised learning task. [ABSTRACT FROM AUTHOR]

• First dynamical low-rank algorithm that is mass, momentum, and energy conservative. • Can be combined with an explicit integrator that maintains conservation. • Conserves the underlying continuity equations in addition to the invariants. • Low-rank breaks the curse of dimensionality for high-dimensional kinetic equations. The primary challenge in solving kinetic equations, such as the Vlasov equation, is the high-dimensional phase space. In this context, dynamical low-rank approximations have emerged as a promising way to reduce the high computational cost imposed by such problems. However, a major disadvantage of this approach is that the physical structure of the underlying problem is not preserved. In this paper, we propose a dynamical low-rank algorithm that conserves mass, momentum, and energy as well as the corresponding continuity equations. We also show how this approach can be combined with a conservative time and space discretization. [ABSTRACT FROM AUTHOR]

Lorentz invariant structure-preserving algorithms possess reference-independent secular stability, which is vital for simulating relativistic multi-scale dynamical processes. The splitting method has been widely used to construct structure-preserving algorithms, but without exquisite considerations, it can easily break the Lorentz invariance of continuous systems. In this paper, we introduce a Lorentz invariant splitting technique to construct high order explicit Lorentz invariant volume-preserving algorithms (LIVPAs) for simulating charged particle dynamics. Using this method, long-term stable explicit LIVPAs with different orders are developed and their performances of Lorentz invariance and long-term stability are analyzed in detail. [ABSTRACT FROM AUTHOR]

• We present the SSLE approach for Bayesian model inversion based on SSE and SLE. • Posterior quantities are obtained analytically from the SSLE coefficients. • The original SSE algorithm is enhanced with an active learning enrichment scheme. • This method generalizes and drastically improves the efficiency of the SLE approach. • We showcase the method on three problems of different complexity and dimensionality. In this paper we propose a new sampling-free approach to solve Bayesian model inversion problems that is an extension of the previously proposed spectral likelihood expansions (SLE) method. Our approach, called stochastic spectral likelihood embedding (SSLE), uses the recently presented stochastic spectral embedding (SSE) method for local spectral expansion refinement to approximate the likelihood function at the core of Bayesian inversion problems. We show that, similar to SLE, this approach results in analytical expressions for key statistics of the Bayesian posterior distribution, such as evidence, posterior moments and posterior marginals , by direct post-processing of the expansion coefficients. Because SSLE and SSE rely on the direct approximation of the likelihood function, they are in a way independent of the computational/mathematical complexity of the forward model. We further enhance the efficiency of SSLE by introducing a likelihood specific adaptive sample enrichment scheme. To showcase the performance of the proposed SSLE, we solve three problems that exhibit different kinds of complexity in the likelihood function: multimodality, high posterior concentration and high nominal dimensionality. We demonstrate how SSLE significantly improves on SLE, and present it as a promising alternative to existing inversion frameworks. [ABSTRACT FROM AUTHOR]