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We present a class of high-order Eulerian–Lagrangian Runge–Kutta finite volume methods that can numerically solve Burgers’ equation with shock formations, which could be extended to general scalar conservation laws. Eulerian–Lagrangian (EL) and semi-Lagrangian (SL) methods have recently seen increased development and have become a staple for allowing large time-stepping sizes. Yet, maintaining relatively large time-stepping sizes post shock formation remains quite challenging. Our proposed scheme integrates the partial differential equation on a space-time region partitioned by linear approximations to the characteristics determined by the Rankine–Hugoniot jump condition. We trace the characteristics forward in time and present a merging procedure for the mesh cells to handle intersecting characteristics due to shocks. Following this partitioning, we write the equation in a time-differential form and evolve with Runge–Kutta methods in a method-of-lines fashion. High-resolution methods such as ENO and WENO-AO schemes are used for spatial reconstruction. Extension to higher dimensions is done via dimensional splitting. Numerical experiments demonstrate our scheme’s high-order accuracy and ability to sharply capture post-shock solutions with large time-stepping sizes. [ABSTRACT FROM AUTHOR]

Surrogate modeling is of great practical significance for parametric differential equation systems. In contrast to classical numerical methods, using physics-informed deep learning-based methods to construct simulators for such systems is a promising direction due to its potential to handle high dimensionality, which requires minimizing a loss over a training set of random samples. However, the random samples introduce statistical errors, which may become the dominant errors for the approximation of low-regularity and high-dimensional problems. In this work, we present a deep adaptive sampling method for surrogate modeling of low-regularity parametric differential equations and illustrate the necessity of adaptive sampling for constructing surrogate models. In the parametric setting, the residual loss function can be regarded as an unnormalized probability density function (PDF) of the spatial and parametric variables. In contrast to the non-parametric setting, factorized joint density models can be employed to alleviate the difficulties induced by the parametric space. The PDF is approximated by a deep generative model, from which new samples are generated and added to the training set. Since the new samples match the residual-induced distribution, the refined training set can further reduce the statistical error in the current approximate solution through variance reduction. We demonstrate the effectiveness of the proposed method with a series of numerical experiments, including the physics-informed operator learning problem, the parametric optimal control problem with geometrical parametrization, and the parametric lid-driven 2D cavity flow problem with a continuous range of Reynolds numbers from 100 to 3200. [ABSTRACT FROM AUTHOR]

In contrast to the classical Allen–Cahn equation, the conservative Allen–Cahn equation with a nonlocal Lagrange multiplier not only satisfies the maximum bound principle (MBP) and energy dissipation law but also ensures mass conservation. Many existing schemes often fail to preserve all these properties at the discrete level or require high regularity in time on the exact solution for convergence analysis. In this paper, we construct a new class of low regularity integrators (LRIs) for time discretization of the conservative Allen–Cahn equation by repeatedly using Duhamel’s formula. The proposed first- and second-order LRI schemes are shown to conserve mass unconditionally and satisfy the MBP under some time step size constraints. Temporal error estimates for these schemes are derived under a low regularity assumption that the exact solution is only Lipschitz continuous in time, followed by a rigorous proof for energy stability of the corresponding time-discrete solutions. Various numerical experiments and comparisons in two and three dimensions are presented to verify the theoretical results and illustrate the performance of the LRI schemes, especially when the interfacial parameter approaches zero. [ABSTRACT FROM AUTHOR]

We present an algorithm for fast generation of quasi-uniform and variable-spacing nodes on domains whose boundaries are represented as computer-aided design (CAD) models, more specifically non-uniform rational B-splines (NURBS). This new algorithm enables the solution of partial differential equations within the volumes enclosed by these CAD models using (collocation-based) meshless numerical discretizations. Our hierarchical algorithm first generates quasi-uniform node sets directly on the NURBS surfaces representing the domain boundary, then uses the NURBS representation in conjunction with the surface nodes to generate nodes within the volume enclosed by the NURBS surface. We provide evidence for the quality of these node sets by analyzing them in terms of local regularity and separation distances. Finally, we demonstrate that these node sets are well-suited (both in terms of accuracy and numerical stability) for meshless radial basis function generated finite differences discretizations of the Poisson, Navier-Cauchy, and heat equations. Our algorithm constitutes an important step in bridging the field of node generation for meshless discretizations with isogeometric analysis. [ABSTRACT FROM AUTHOR]

We investigate the Helmholtz equation with suitable boundary conditions and uncertainties in the wavenumber. Thus the wavenumber is modeled as a random variable or a random field. We discretize the Helmholtz equation using finite differences in space, which leads to a linear system of algebraic equations including random variables. A stochastic Galerkin method yields a deterministic linear system of algebraic equations. This linear system is high-dimensional, sparse and complex symmetric but, in general, not hermitian. We therefore solve this system iteratively with GMRES and propose two preconditioners: a complex shifted Laplace preconditioner and a mean value preconditioner. Both preconditioners reduce the number of iteration steps as well as the computation time in our numerical experiments. [ABSTRACT FROM AUTHOR]

It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high-dimensional random inputs, which are typically caused by the large number of stochastic basis functions required. It becomes crucial to properly choose effective basis functions, such that the dimension of the stochastic approximation space can be reduced. In this work, we focus on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC), and explore the gPC expansion based on the analysis of variance (ANOVA) decomposition. A concise form of the gPC expansion is presented for each component function of the ANOVA expansion, and an adaptive ANOVA procedure is proposed to construct the overall stochastic Galerkin system. Numerical results demonstrate the efficiency of our proposed adaptive ANOVA stochastic Galerkin method for both diffusion and Helmholtz problems. [ABSTRACT FROM AUTHOR]

In this work, we propose adaptive deep learning approaches based on normalizing flows for solving fractional Fokker–Planck equations (FPEs). The solution of a FPE is a probability density function (PDF). Traditional mesh-based methods are ineffective because of a unbounded computation domain, a large number of dimensions and a nonlocal fractional operator. To this end, we represent the solution with an explicit PDF model induced by a flow-based deep generative model, which constructs a transport map from a simple distribution to the target distribution. We consider two methods to approximate the fractional Laplacian. One method is the Monte Carlo approximation. The other method is to construct an auxiliary model with Gaussian radial basis functions (GRBFs) to approximate the solution such that we may take advantage of the fact that the fractional Laplacian of a Gaussian is known analytically. Based on these two different ways for the approximation of the fractional Laplacian, we propose two models to approximate stationary FPEs and one model to approximate time-dependent FPEs. To further improve the accuracy, we refine the training set and the approximate solution alternately. A variety of numerical examples is presented to demonstrate the effectiveness of our adaptive deep density approaches. [ABSTRACT FROM AUTHOR]

Solving high-dimensional Bayesian inverse problems (BIPs) with the variational inference (VI) method is promising but still challenging. The main difficulties arise from two aspects. First, VI methods approximate the posterior distribution using a simple and analytic variational distribution, which makes it difficult to estimate complex spatially-varying parameters in practice. Second, VI methods typically rely on gradient-based optimization, which can be computationally expensive or intractable when applied to BIPs involving partial differential equations (PDEs). To address these challenges, we propose a novel approximation method for estimating the high-dimensional posterior distribution. This approach leverages a deep generative model to learn a prior model capable of generating spatially-varying parameters. This enables posterior approximation over the latent variable instead of the complex parameters, thus improving estimation accuracy. Moreover, to accelerate gradient computation, we employ a differentiable physics-constrained surrogate model to replace the adjoint method. The proposed method can be fully implemented in an automatic differentiation manner. Numerical examples demonstrate two types of log-permeability estimation for flow in heterogeneous media. The results show the validity, accuracy, and high efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The convective Allen–Cahn (CAC) equation has been widely used for simulating multiphase flows of incompressible fluids, which contains an extra convective term but still maintains the same maximum bound principle (MBP) as the classic Allen–Cahn equation. Based on the operator splitting approach, we propose a second-order semi-Lagrangian exponential time differencing method for solving the CAC equation, that preserves the discrete MBP unconditionally. In our scheme, the AC equation part is first spatially discretized via the central finite difference scheme, then it is efficiently solved by using the exponential time differencing method with FFT-based fast implementation. The transport equation part is computed by combining the semi-Lagrangian approach with a cut-off post-processing within the finite difference framework. MBP stability and convergence analysis of our fully discretized scheme are presented. In particular, we conduct an improved error estimation for the semi-Lagrangian method with variable velocity, so that the error of our scheme is not spoiled by the reciprocal of the time step size. Extensive numerical tests in two and three dimensions are also carried out to validate the theoretical results and demonstrate the performance of our scheme. [ABSTRACT FROM AUTHOR]

Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution. Developing efficient and accurate solution strategies that account for errors on the space, time and parameter domains simultaneously is highly challenging. Indeed, it is well known that standard polynomial-based approximations on the parameter domain can incur errors that grow in time. In this work, we focus on advection–diffusion problems with parameter-dependent wind fields. A novel adaptive solution strategy is proposed that allows users to combine stochastic collocation on the parameter domain with off-the-shelf adaptive timestepping algorithms with local error control. This is a non-intrusive strategy that builds a polynomial-based surrogate that is adapted sequentially in time. The algorithm is driven by a so-called hierarchical estimator for the parametric error and balances this against an estimate for the global timestepping error which is derived from a scaling argument. [ABSTRACT FROM AUTHOR]

In this paper, we investigate residual neural network (ResNet) method to solve ordinary differential equations. We verify the accuracy order of ResNet ODE solver matches the accuracy order of the data. Forward Euler, Runge–Kutta2 and Runge–Kutta4 finite difference schemes are adapted generating three learning data sets, which are applied to train three ResNet ODE solvers independently. The well trained ResNet solvers obtain 2nd, 3rd and 5th orders of one step errors and behave just as its counterpart finite difference method for linear and nonlinear ODEs with regular solutions. In particular, we carry out (1) architecture study in terms of number of hidden layers and neurons per layer to obtain optimal network structure; (2) target study to verify the ResNet solver is as accurate as its finite difference method counterpart; (3) solution trajectory simulations. A sequence of numerical examples are presented to demonstrate the accuracy and capability of ResNet solver. [ABSTRACT FROM AUTHOR]

The study of uncertainty propagation poses a great challenge to design high fidelity numerical methods. Based on the stochastic Galerkin formulation, this paper addresses the idea and implementation of the first flux reconstruction scheme for hyperbolic conservation laws with random inputs. High-order numerical approximation is adopted simultaneously in physical and random space, i.e., the modal representation of solutions is based on an orthogonal polynomial basis and the nodal representation is based on solution collocation points. Therefore, the numerical behaviors of the scheme in the (physical-random) phase space can be designed and understood uniformly. A family of filters is developed in multi-dimensional cases to mitigate the Gibbs phenomenon arising from discontinuities in both physical and random space. The filter function is switched on and off by the dynamic detection of discontinuous solutions, and a slope limiter is employed to preserve the positivity of physically realizable solutions. As a result, the proposed method is able to capture the stochastic flow evolution where resolved and unresolved regions coexist. Numerical experiments including a wave propagation, a Burgers’ shock, a one-dimensional Riemann problem, and a two-dimensional shock-vortex interaction problem are presented to validate the current scheme. The order of convergence of the high-order scheme is identified. The capability of the scheme for simulating smooth and discontinuous stochastic flow dynamics is demonstrated. The open-source codes to reproduce the numerical results are available under the MIT license (Xiao et al. in FRSG: stochastic Galerkin method with flux reconstruction. , (2021). ). [ABSTRACT FROM AUTHOR]

The generalized polynomial chaos (gPC) method is one of the most popular method for uncertainty quantification. Being essentially a spectral approach, the gPC method exhibits the spectral convergence rate which heavily depends on the regularity of the solution in the random space. Many regularity studies have been made for stochastic elliptic and parabolic equations while regularities studies of stochastic hyperbolic equations has long been infeasible due to its intrinsic difficulties. In this paper, we investigate the impact of uncertainty on the time-dependent radiative transfer equation (RTE) with nonhomogeneous boundary conditions, which sits somewhere between hyperbolic and parabolic equations. We theoretically prove the a-priori bound of the solution, its continuity with respect to the scattering coefficient, and its regularity in the random space. These studies can serve as a building block in understanding the influence of uncertainties in the passage from hyperbolic to parabolic equations. Moreover, we vigorously justify the validity of the gPC expansion ansatz based on the regularity study. Then the stochastic Galerkin method of the gPC approach is employed to discretize the random variable. We further conduct a delicate analysis to show the exponential decay rate of the gPC coefficients and establish the error estimates of the stochastic Galerkin approximation for both one-dimensional and multi-dimensional random space cases. Numerical tests are presented to verify our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, a spectral Petrov–Galerkin method is developed to solve an optimal control problem governed by a two-sided space-fractional diffusion-reaction equation with additive fractional noise. In order to compensate weak singularities of the solution near boundaries, regularities of both the fractional noise and the optimal control problem are analyzed in weighted Sobolev space. The spectral Petrov–Galerkin method is presented by employing truncated spectral expansion of the fractional Brownian motion (fBm) type noise, and error estimates are given based on the obtained regularity of the optimal control problem. Numerical experiments are carried out to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

Accurate simulations of the Stokes–Darcy system face many difficulties including the coupling of flows in two different subdomains via interface conditions, the incompressibility constraint for the free flow and uncertainties in model parameters. In this report, we propose and study efficient, decoupled, artificial compressibility (AC) ensemble schemes based on a recently developed SAV approach for fast computation of Stokes–Darcy flow ensembles. The proposed algorithms (1) do not require any time step condition and (2) decouple the computation of the velocity and pressure in the free flow region, and (3) result in a common coefficient matrix for all realizations after spatial discretization for which efficient iterative linear solvers such as block CG or block GMRES can be used to greatly reduce the computational cost. We prove the long time stability under two parameter conditions, without any timestep constraints. In particular, for one single simulation, they are unconditionally stable schemes. Several numerical tests are presented to demonstrate the efficiency of the algorithms and illustrate their applications in realistic flow problems. [ABSTRACT FROM AUTHOR]

A method for the multifidelity Monte Carlo (MFMC) estimation of statistical quantities is proposed which is applicable to computational budgets of any size. Based on a sequence of optimization problems each with a globally minimizing closed-form solution, this method extends the usability of a well known MFMC algorithm, recovering it when the computational budget is large enough. Theoretical results verify that the proposed approach is at least as optimal as its namesake and retains the benefits of multifidelity estimation with minimal assumptions on the budget or amount of available data, providing a notable reduction in variance over simple Monte Carlo estimation. [ABSTRACT FROM AUTHOR]

Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in the multilevel hierarchy is needed to obtain efficiency gains. In this work, we show that strong pairwise coupling indeed is also important when MLMC is applied to stochastic partial differential equations (SPDE) of reaction-diffusion type, as it can improve the rate of convergence and thus improve tractability. For the MLMC method with strong pairwise coupling that was developed and studied numerically on filtering problems in (Chernov in Num Math 147:71-125, 2021), we prove that the rate of computational efficiency is higher than for existing methods. We also provide numerical comparisons with alternative coupling ideas on linear and nonlinear SPDE to illustrate the importance of this feature. [ABSTRACT FROM AUTHOR]

In Zhu et al. (SIAM J Sci Comput 43: A3009–A3031, 2021), we proposed a new framework of troubled-cell indicator (TCI) using K-means clustering for the discontinuous Galerkin (DG) methods. However, there are two user-tunable parameters in the framework that depend on the polynomial degree of the solution space, the indication variable and even the test problem, which circumscribe the application of the framework. To overcome this drawback, we introduce two simple techniques in this paper: one is to modify the indication variables and the other is to apply a statistical normalization to the modified values. Coupled with four different indication variables, the modified framework is tested via the classical benchmark problems and produces close results under the same setting of the parameters. The discontinuities are overall well captured and the solutions are free of spurious oscillations. The numerical results demonstrate the effectiveness and flexibility of the modified framework and the success in unifying the parameters. Existing TCIs/limiters for the DG methods can be easily implemented into this framework. [ABSTRACT FROM AUTHOR]

In this paper, we study nonlinear filtering problems via solving their corresponding Zakai equations. Using the splitting-up technique, we approximate the Zakai equation with two equations consisting of a first-order stochastic partial differential equation and a deterministic second-order partial differential equation. For the splitting-up equations, we use a spectral Galerkin method for the spatial discretization and a finite difference scheme for the temporal discretization. The main results are an error estimate for the semi-discretized scheme with respect to the spatial variable, and an error estimate for the full discretized scheme. To improve the numerical performance, we apply an adaptive technique to accurately locate the support domain of the solution in each time iteration. Finally, we present numerical experiments to demonstrate our theoretical analysis. [ABSTRACT FROM AUTHOR]

We develop a randomized Newton’s method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton’s method randomly chooses equations from the overdetermined nonlinear system resulting from the neural network discretization and solves the nonlinear system adaptively. We theoretically prove that the randomized Newton’s method has a quadratic convergence locally. We also apply this new method to various numerical examples, from one to high-dimensional differential equations, to verify its feasibility and efficiency. Moreover, the randomized Newton’s method can allow the neural network to “learn” multiple solutions for nonlinear systems of differential equations, such as pattern formation problems, and provides an alternative way to study the solution structure of nonlinear differential equations overall. [ABSTRACT FROM AUTHOR]

Data-driven discovery of partial differential equations (PDEs) from observed data in machine learning has been developed by embedding the discovery problem. Recently, the discovery of traditional ODEs dynamics using linear multistep methods in deep learning have been discussed in [Racheal and Du, SIAM J. Numer. Anal. 59 (2021) 429-455; Du et al. ]. We extend this framework to the data-driven discovery of the time-fractional PDEs, which can effectively characterize the ubiquitous power-law phenomena. In this paper, identifying source function of subdiffusion with noisy data using L 1 approximation in deep neural network is presented. In particular, two types of networks for improving the generalization of the subdiffusion problem are designed with noisy data. The numerical experiments are given to illustrate the availability with high noise levels using deep learning. To the best of our knowledge, this is the first topic on the discovery of subdiffusion in deep learning with noisy data. [ABSTRACT FROM AUTHOR]

In this paper, we investigate hyperbolic systems with random inputs based on generalized polynomial chaos (gPC) approximations, which is one of the most popular methods for uncertainty quantification (UQ) and can be implemented with either the stochastic Galerkin (SG) method or the stochastic collocation (SC) method. One of the challenges for solving stochastic hyperbolic systems with the SG method is that the resulting deterministic system may not be hyperbolic. The lack of hyperbolicity may lead to the ill-posedness of the problem and the instability of numerical simulations. The main objective of this paper is to show that by approximating the solution in the random space with the SG method in a pseudo-spectral way with suitable quadrature rules, the SG scheme can be written as a SC scheme on a set of specific nodes. The resulting collocation scheme preserves the hyperbolicity of the original hyperbolic system, and is more efficient to implement. On the other hand, entropy conditions play an essential role in the well-posedness of hyperbolic conservation laws. Thus we approximate the resulted collocation scheme in space by the entropy stable nodal discontinuous Galerkin (DG) method Chen and Shu (J. Comput. Phys. 345:427-461, 2017), where the entropy stability is guaranteed by high order summation-by-parts operators, entropy conservative fluxes and entropy stable fluxes. Numerical experiments are performed to validate the accuracy and effectiveness of the proposed numerical red schemes. [ABSTRACT FROM AUTHOR]

Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite element method for a class of nonlinear stochastic wave equations, where the diffusion term is globally Lipschitz continuous while the drift term is only assumed to satisfy weaker conditions as in Chow (Ann Appl Probab 12(1):361–381, 2002). The novelties of this paper are threefold. First, the error estimates cannot be directly obtained if the numerical scheme in primal form is used. An equivalent numerical scheme in mixed form is therefore utilized and several Hölder continuity results of the strong solution are proved, which are used to establish the error estimates in both L 2 norm and energy norms. Second, two types of discretization of the nonlinear term are proposed to establish the L 2 stability and energy stability results of the discrete solutions. These two types of discretization and proper test functions are designed to overcome the challenges arising from the stochastic scaling in time issues and the nonlinear interaction. These stability results play key roles in proving the probability of the set on which the error estimates hold approaches one. Third, higher moment stability results of the discrete solutions are proved based on an energy argument and the underlying energy decaying property of the method. Numerical experiments are also presented to show the stability results of the discrete solutions and the convergence rates in various norms. [ABSTRACT FROM AUTHOR]

As the use of spectral/hp element methods, and high-order finite element methods in general, continues to spread, community efforts to create efficient, optimized algorithms associated with fundamental high-order operations have grown. Core tasks such as solution expansion evaluation at quadrature points, stiffness and mass matrix generation, and matrix assembly have received tremendous attention. With the expansion of the types of problems to which high-order methods are applied, and correspondingly the growth in types of numerical tasks accomplished through high-order methods, the number and types of these core operations broaden. This work focuses on solution expansion evaluation at arbitrary points within an element. This operation is core to many postprocessing applications such as evaluation of streamlines and pathlines, as well as to field projection techniques such as mortaring. We expand barycentric interpolation techniques developed on an interval to 2D (triangles and quadrilaterals) and 3D (tetrahedra, prisms, pyramids, and hexahedra) spectral/hp element methods. We provide efficient algorithms for their implementations, and demonstrate their effectiveness using the spectral/hp element library Nektar++ by running a series of baseline evaluations against the ‘standard’ Lagrangian method, where an interpolation matrix is generated and matrix-multiplication applied to evaluate a point at a given location. We present results from a rigorous series of benchmarking tests for a variety of element shapes, polynomial orders and dimensions. We show that when the point of interest is to be repeatedly evaluated, the barycentric method performs at worst 50 % slower, when compared to a cached matrix evaluation. However, when the point of interest changes repeatedly so that the interpolation matrix must be regenerated in the ‘standard’ approach, the barycentric method yields far greater performance, with a minimum speedup factor of 7 × . Furthermore, when derivatives of the solution evaluation are also required, the barycentric method in general slightly outperforms the cached interpolation matrix method across all elements and orders, with an up to 30 % speedup. Finally we investigate a real-world example of scalar transport using a non-conformal discontinuous Galerkin simulation, in which we observe around 6 × speedup in computational time for the barycentric method compared to the matrix-based approach. We also explore the complexity of both interpolation methods and show that the barycentric interpolation method requires O (k) storage compared to a best case space complexity of O (k 2) for the Lagrangian interpolation matrix method. [ABSTRACT FROM AUTHOR]

A multigrid multilevel Monte Carlo (MGMLMC) method is developed for the stochastic Stokes–Darcy interface model with random hydraulic conductivity both in the porous media domain and on the interface. Three interface conditions with randomness are considered on the interface between Stokes and Darcy equations, especially the Beavers–Joesph interface condition with random hydraulic conductivity. Because the randomness through the interface affects the flow in the Stokes domain, we investigate the coupled stochastic Stokes–Darcy model to improve the fidelity. Under suitable assumptions on the random coefficient, we prove the existence and uniqueness of the weak solution of the variational form. To construct the numerical method, we first adopt the Monte Carlo (MC) method and finite element method, for the discretization in the probability space and physical space, respectively. In order to improve the efficiency of the classical single-level Monte Carlo (SLMC) method, we adopt the multilevel Monte Carlo (MLMC) method to dramatically reduce the computational cost in the probability space. A strategy is developed to calculate the number of samples needed in MLMC method for the stochastic Stokes–Darcy model. In order to accomplish the strategy for MLMC method, we also present a practical method to determine the variance convergence rate for the stochastic Stokes–Darcy model with Beavers–Joseph interface condition. Furthermore, MLMC method naturally provides the hierarchical grids and sufficient information on these grids for multigrid (MG) method, which can in turn improve the efficiency of MLMC method. In order to fully make use of the dynamical interaction between this two methods, we propose a multigrid multilevel Monte Carlo (MGMLMC) method with finite element discretization for more efficiently solving the stochastic model, while additional attention is paid to the interface and the random Beavers–Joesph interface condition. The computational cost of the proposed MGMLMC method is rigorously analyzed and compared with the SLMC method. Numerical examples are provided to verify and illustrate the proposed method and the theoretical conclusions. [ABSTRACT FROM AUTHOR]

Compared with the well-known classical Allen–Cahn equation, the modified Allen–Cahn equation, which is equipped with a nonlocal Lagrange multiplier or a local-nonlocal Lagrange multiplier, enforces the mass conservation for modeling phase transitions. In this work, a class of up to third-order explicit structure-preserving schemes is proposed for solving these two modified conservative Allen–Cahn equations. Based on second-order finite-difference space discretization, we investigate the newly developed improved stabilized integrating factor Runge–Kutta (isIFRK) schemes for conservative Allen–Cahn equations. We prove that the original stabilized integrating factor Runge–Kutta schemes fail to preserve the mass conservation law when the stabilizing constant κ > 0 and the initial mass does not equal zero, while isIFRK schemes not only preserve the maximum principle unconditionally, but also conserve the mass to machine accuracy without any restriction on the time-step size. Convergence of the proposed schemes are also presented. At last, a series of numerical experiments validate that each reformulation of the conservative Allen–Cahn equations has it own advantage, and isIFRK schemes can reach the expected high-order accuracy, conserve the mass, and preserve the maximum principle unconditionally. [ABSTRACT FROM AUTHOR]

The basis generation in reduced order modeling usually requires multiple high-fidelity large-scale simulations that could take a huge computational cost. In order to accelerate these numerical simulations, we introduce a FOM/ROM hybrid approach in this paper. It is developed based on an a posteriori error estimation for the output approximation of the dynamical system. By controlling the estimated error, the method dynamically switches between the full-order model and the reduced-oder model generated on the fly. Therefore, it reduces the computational cost of a high-fidelity simulation while achieving a prescribed accuracy level. Numerical tests on the non-parametric and parametric PDEs illustrate the efficacy of the proposed approach. [ABSTRACT FROM AUTHOR]

The necessity of dealing with uncertainties is growing in many different fields of science and engineering. Due to the constant development of computational capabilities, current solvers must satisfy both statistical accuracy and computational efficiency. The aim of this work is to introduce an asynchronous framework for Monte Carlo and Multilevel Monte Carlo methods to achieve such a result. The proposed approach presents the same reliability of state of the art techniques, and aims at improving the computational efficiency by adding a new level of parallelism with respect to existing algorithms: between batches, where each batch owns its hierarchy and is independent from the others. Two different numerical problems are considered and solved in a supercomputer to show the behavior of the proposed approach. [ABSTRACT FROM AUTHOR]

Associated to a finite measure on the real line with finite moments are recurrence coefficients in a three-term formula for orthogonal polynomials with respect to this measure. These recurrence coefficients are frequently inputs to modern computational tools that facilitate evaluation and manipulation of polynomials with respect to the measure, and such tasks are foundational in numerical approximation and quadrature. Although the recurrence coefficients for classical measures are known explicitly, those for nonclassical measures must typically be numerically computed. We survey and review existing approaches for computing these recurrence coefficients for univariate orthogonal polynomial families and propose a novel “predictor–corrector” algorithm for a general class of continuous measures. We combine the predictor–corrector scheme with a stabilized Lanczos procedure for a new hybrid algorithm that computes recurrence coefficients for a fairly wide class of measures that can have both continuous and discrete parts. We evaluate the new algorithms against existing methods in terms of accuracy and efficiency. [ABSTRACT FROM AUTHOR]

We re-visit the semi-Lagrangian Runge–Kutta exponential integrators presented in Celledoni et al. (J Sci Comput 41:139–164, 2009) for Convection-dominated convection-diffusion problems. Third order accurate methods of this class consist of subsets of coefficients of commutator-free exponential integrators including some free parameters. We consider an optimal choice of parameters for the third order accurate methods based on the least-squares methods applied on fourth order commutator-free conditions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a numerical solution of a stochastic partial differential equation (SPDE) of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. These Sinc points are defined by a conformal map and when mixed with the polynomial interpolation, it yields an accurate approximation. The first step to solve SPDE is to use stochastic Galerkin method in conjunction with polynomial chaos, which implies a system of deterministic partial differential equations to be solved. The main difficulty is the higher dimensionality of the resulting system of partial differential equations. The idea here is to solve this system using a small number of collocation points in space. This collocation technique is called Poly-Sinc and is used for the first time to solve high-dimensional systems of partial differential equations. Two examples are presented, mainly using Legendre polynomials for stochastic variables. These examples illustrate that we require to sample at few points to get a representation of a model that is sufficiently accurate. [ABSTRACT FROM AUTHOR]

Identifying unknown differential equations from a given set of discrete time dependent data is a challenging problem. A small amount of noise can make the recovery unstable. Nonlinearity and varying coefficients add complexity to the problem. We assume that the governing partial differential equation (PDE) is a linear combination of few differential terms in a prescribed dictionary, and the objective of this paper is to find the correct coefficients. We propose a new direction based on the fundamental convergence principle of numerical PDE schemes. We utilize Lasso for efficiency, and a performance guarantee is established based on an incoherence property. The main contribution is to validate and correct the results by time evolution error (TEE). A new algorithm, called identifying differential equations with numerical time evolution (IDENT), is explored for data with non-periodic boundary conditions, noisy data and PDEs with varying coefficients. Based on the recovery theory of Lasso, we propose a new definition of Noise-to-Signal ratio, which better represents the level of noise in the case of PDE identification. The effects of data generations and downsampling are systematically analyzed and tested. For noisy data, we propose an order preserving denoising method called least-squares moving average (LSMA), to preprocess the given data. For the identification of PDEs with varying coefficients, we propose to add Base Element Expansion (BEE) to aid the computation. Various numerical experiments from basic tests to noisy data, downsampling effects and varying coefficients are presented. [ABSTRACT FROM AUTHOR]

In this paper, we present a new nonintrusive reduced basis method when a cheap low-fidelity model and an expensive high-fidelity model are available. The method employs proper orthogonal decomposition method to generate the high-fidelity reduced basis and a shallow multilayer perceptron to learn the high-fidelity reduced coefficients. In contrast to previously proposed methods, besides the model parameters, we also augmented the features extracted from the data generated by an efficient bi-fidelity surrogate developed in Narayan et al. (SIAM J Sci Comput 36(2):A495–A521, 2014) and Zhu et al. (SIAM/ASA J Uncertain Quantif 2(1):444–463, 2014) as the input feature of the proposed neural network. By incorporating relevant bi-fidelity features, we demonstrate that such an approach can improve the predictive capability and robustness of the neural network via several benchmark examples. Due to its nonintrusive nature, it is also applicable to general parameterized problems. [ABSTRACT FROM AUTHOR]

We present a distributed active subspace method for training surrogate models of complex physical processes with high-dimensional inputs and function valued outputs. Specifically, we represent the model output with a truncated Karhunen–Loève (KL) expansion, screen the structure of the input space with respect to each KL mode via the active subspace method, and finally form an overall surrogate model of the output by combining surrogates of individual output KL modes. To ensure scalable computation of the gradients of the output KL modes, needed in active subspace discovery, we rely on adjoint-based gradient computation. The proposed method combines benefits of active subspace methods for input dimension reduction and KL expansions used for spectral representation of the output field. We provide a mathematical framework for the proposed method and conduct an error analysis of the mixed KL active subspace approach. Specifically, we provide an error estimate that quantifies errors due to active subspace projection and truncated KL expansion of the output. We demonstrate the numerical performance of the surrogate modeling approach with an application example from biotransport. [ABSTRACT FROM AUTHOR]

We develop error estimates for the finite element approximation of elliptic partial differential equations on perturbed domains, i.e. when the computational domain does not match the real geometry. The result shows that the error related to the domain can be a dominating factor in the finite element discretization error. The main result consists of H 1 - and L 2 -error estimates for the Laplace problem. Theoretical considerations are validated by a computational example. [ABSTRACT FROM AUTHOR]

The key to most successful applications of Chebyshev and Fourier spectral methods in high space dimension are a combination of a Smolyak sparse grid together with so-called “hyperbolic cross” truncation. It is easy to find counterexamples for which the hyperbolic cross truncation is far from optimal. An important question is: what characteristics of a function make it “crossy”, that is, suitable for the hyperbolic crosss truncation? We have not been able to find a complete answer to this question. However, by combining low-rank SVD approximation, Poisson summation and imbricate series, hyperbolic coordinates and numerical experimentation, we are, to borrow from Fermi, “confused at a higher level”. For rank-one (separable) functions, which are the product of two univaraiate functions, we show that the hyperbolic cross truncation is indeed the best if the functions have weak singularities on the domain or boundaries so that the spectral series has a finite order of power-law convergence. For functions smooth on the domain, and therefore blessed with exponentially convergent spectral series, we have failed to find any reasonable examples where the hyperbolic cross truncation is best. [ABSTRACT FROM AUTHOR]

In this paper, we propose, analyze, and implement a new reduced basis method (RBM) tailored for the linear (ordinary and partial) differential equations driven by arbitrary (i.e. not necessarily Gaussian) types of noise. There are four main ingredients of our algorithm. First, we propose a new space-time-like treatment of time in the numerical schemes for ODEs and PDEs. The second ingredient is an accurate yet efficient compression technique for the spatial component of the space-time snapshots that the RBM is adopting as bases. The third ingredient is a non-conventional "parameterization" of a non-parametric problem. The last is a RBM that is free of any dedicated offline procedure yet is still efficient online. The numerical experiments verify the effectiveness and robustness of our algorithms for both types of differential equations. [ABSTRACT FROM AUTHOR]

We propose an alternative approach to the direct polynomial chaos expansion in order to approximate one-dimensional uncertain field exhibiting steep fronts. The principle of our non-intrusive approach is to decompose the level points of the quantity of interest in order to avoid the spurious oscillations encountered in the direct approach. This method is more accurate and less expensive than the direct approach since the regularity of the level points with respect to the input parameters allows achieving the convergence with low-order polynomial series. The additional computational cost induced in the post-processing phase is largely offset by the use of low-level sparse grids that require a weak number of direct model evaluations in comparison with high-level sparse grids. We apply the method to subsurface flows problem with uncertain hydraulic conductivity. Infiltration test cases having different levels of complexity are presented. [ABSTRACT FROM AUTHOR]

We consider a stochastic analysis of non-linear viscous fluid flow problems with smooth and sharp gradients in stochastic space. As a representative example we consider the viscous Burgers' equation and compare two typical intrusive and non-intrusive uncertainty quantification methods. The specific intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The specific non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. The two methods are compared in terms of error in the estimated variance, computational efficiency and accuracy. This comparison, although not general, provide insight into uncertainty quantification of problems with a combination of sharp and smooth variations in stochastic space. It suggests that combining intrusive and non-intrusive methods could be advantageous. [ABSTRACT FROM AUTHOR]

In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L 2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to assess the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and high dimensional problems. [ABSTRACT FROM AUTHOR]

We study a pressure-correction ensemble scheme for fast calculation of thermal flow ensembles. The proposed scheme (1) decouples the Boussinesq system into two smaller subphysics problems; (2) decouples the nonlinearity from the incompressibility condition in the Navier–Stokes equations and linearizes the momentum equation so that it reduces to a system of scalar equations; (3) results in linear systems with the same coefficient matrix for all realizations. This reduces the size of linear systems to be solved at each time step and allows efficient direct/iterative linear solvers for fast computation. We prove the scheme is long time stable and first order in time convergent under a time step condition. Numerical tests are provided to confirm the theoretical results and demonstrate the efficiency of the scheme. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the stochastic Galerkin methods for time-dependent Maxwell's equations with random input. The generalized polynomial chaos approach is first adopted to convert the original random Maxwell's equation into a system of deterministic equations for the expansion coefficients (the Galerkin system). It is shown that the stochastic Galerkin approach preserves the energy conservation law. Then, we propose a finite element approach in the physical space to solve the Galerkin system, and error estimates is presented. For the time domain approach, we propose two discrete schemes, namely, the Crank–Nicolson scheme and the leap-frog type scheme. For the Crank–Nicolson scheme, we show the energy preserving property for the fully discrete scheme. While for the classic leap-frog scheme, we present a conditional energy stability property. It is well known that for the stochastic Galerkin approach, the main challenge is how to efficiently solve the coupled Galerkin system. To this end, we design a modified leap-frog type scheme in which one can solve the coupled system in a decouple way—yielding a very efficient numerical approach. Numerical examples are presented to support the theoretical finding. [ABSTRACT FROM AUTHOR]

In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained, e.g, by Gram–Schmidt orthogonalization. However, in most cases, they are not eigenfunctions of some singular Sturm–Liouville problem, as is the case for known polynomials, such as the Jacobi polynomials. Therefore, the standard convergence theorems do not apply. Furthermore, since in general multidimensional cases the weight functions are not a tensor product of one-dimensional functions, the orthogonal polynomials are not a product of one-dimensional orthogonal polynomials, as well. This work provides a way of estimating the convergence rate using a comparison lemma. We also present a spectrally convergent, multidimensional, integration method. Numerical examples demonstrate the efficacy of the proposed method. We also show that the use of non-standard weight functions can allow for efficient integration of singular functions. We demonstrate the use of this method to uncertainty quantification problem using Generalized Polynomial Chaos Expansions in the case of dependent random variables, as well. [ABSTRACT FROM AUTHOR]

Surrogate modeling has become a critical component of scientific computing in situations involving expensive model evaluations. However, training a surrogate model can be remarkably challenging and even computationally prohibitive in the case of intensive simulations and large-dimensional systems. We develop a systematic approach for surrogate model construction in reduced input parameter spaces. A sparse set of model evaluations in the original input space is used to approximate derivative based global sensitivity measures (DGSMs) for individual uncertain inputs of the model. An iterative screening procedure is developed that exploits DGSM estimates in order to identify the unimportant inputs. The screening procedure forms an integral part of an overall framework for adaptive construction of a surrogate in the reduced space. The framework is tested for computational efficiency through an initial implementation in simple test cases such as the classic Borehole function, and a semilinear elliptic PDE with a random source function. The framework is then deployed for a realistic application from chemical kinetics, where we study the ignition delay in an H 2 / O 2 reaction mechanism with 19 and 33 uncertain rate-controlling parameters. It is observed that significant computational gains can be attained by constructing accurate low-dimensional surrogates using the proposed framework. [ABSTRACT FROM AUTHOR]

In this paper, we propose a mass conservative semi-Lagrangian finite difference scheme for multi-dimensional problems without dimensional splitting. The semi-Lagrangian scheme, based on tracing characteristics backward in time from grid points, does not necessarily conserve the total mass. To ensure mass conservation, we propose a conservative correction procedure based on a flux difference form. Such procedure guarantees local mass conservation, while introducing time step constraints for stability. We theoretically investigate such stability constraints from an ODE point of view by assuming exact evaluation of spatial differential operators and from the Fourier analysis for linear PDEs. The scheme is tested by classical two dimensional linear passive-transport problems, such as linear advection, rotation and swirling deformation. The scheme is applied to solve the nonlinear Vlasov–Poisson system and guiding center Vlasov model using high order tracing schemes. The effectiveness of the proposed conservative semi-Lagrangian scheme is demonstrated numerically by extensive numerical tests. [ABSTRACT FROM AUTHOR]

In this paper, we develop two ensemble time-stepping algorithms to solve the convection-diffusion equation with random diffusion coefficients, forcing terms and initial conditions based on the pseudo-spectral stochastic collocation method. The key step of the pseudo-spectral stochastic collocation method is to solve a number of deterministic problems derived from the original stochastic convection-diffusion equation. In general, a common way to solve the set of deterministic problems is by using the backward differentiation formula, which requires us to store the coefficient matrix and right-hand-side vector multiple times, and solve them one by one. However, the proposed algorithm only need to solve a single linear system with one shared coefficient matrix and multiple right-hand-side vectors, reducing both storage required and computational cost of the solution process. The stability and error analysis of the first- and second-order ensemble time-stepping algorithms are provided. Several numerical experiments are presented to confirm the theoretical analyses and verify the feasibility and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper introduces a novel boundary integral approach of shape uncertainty quantification for the Helmholtz scattering problem in the framework of the so-called parametric method. The key idea is to construct an integration grid whose associated weight function encompasses the irregularities and non-smoothness imposed by the random boundary. Thus, the solution can be evaluated accurately with relatively low number of grid points. The integration grid is obtained by employing a low-dimensional spatial embedding using the coarea formula. The proposed method can handle large variation as well as non-smoothness of the random boundary. For the ease of presentation the theory is restricted to star-shaped obstacles in low-dimensional setting. Higher spatial and parametric dimensional cases are discussed, though, not extensively explored in the current study. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a stochastic Galerkin approximation scheme for an optimal control problem governed by an elliptic PDE with random field in its coefficients. The optimal control minimizes the expectation of a cost functional with mean-state constraints. We first represent the stochastic elliptic PDE in terms of the generalized polynomial chaos expansion and obtain the parameterized optimal control problems. By applying the Slater condition in the subdifferential calculus, we obtain the necessary and sufficient optimality conditions for the state-constrained stochastic optimal control problem for the first time in the literature. We then establish a stochastic Galerkin scheme to approximate the optimality system in the spatial space and the probability space. Then the a priori error estimates are derived for the state, the co-state and the control variables. A projection algorithm is proposed and analyzed. Numerical examples are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we deal with an optimal control problem governed by the convection diffusion equations with random field in its coefficients. Mathematically, we prove the necessary and sufficient optimality conditions for the optimal control problem. Computationally, we establish a scheme to approximate the optimality system through the discretization by the upwind finite volume element method for the physical space, and by the sparse grid stochastic collocation algorithm based on the Smolyak construction for the probability space, which leads to the discrete solution of uncoupled deterministic problems. Moreover, the existence and uniqueness of the discrete solution are given. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]