A Stochastic Collocation Approach to Bayesian Inference in Inverse Problems

We present an efficient numerical strategy for the Bayesian solution of inverse problems. Stochastic collocation methods, based on generalized polynomial chaos (gPC), are used to construct a polynomial approximation of the forward solution over the support of the prior distribution. This approximation then defines a surrogate posterior probability density that can be evaluated repeatedly at minimal computational cost. The ability to simulate a large number of samples from the posterior distribution results in very accurate estimates of the inverse solution and its associated uncertainty. Combined with high accuracy of the gPC-based forward solver, the new algorithm can provide great efficiency in practical applications. A rigorous error analysis of the algorithm is conducted, where we establish convergence of the approximate posterior to the true posterior and obtain an estimate of the convergence rate. It is proved that fast (exponential) convergence of the gPC forward solution yields similarly fast (exponential) convergence of the posterior. The numerical strategy and the predicted convergence rates are then demonstrated on nonlinear inverse problems of varying smoothness and dimension.     

Moment-Based Multi-Resolution HWENO Scheme for Hyperbolic Conservation Laws

In this paper, a high-order moment-based multi-resolution Hermiteweighted essentially non-oscillatory (HWENO) scheme is designed for hyperbolic conservation laws. The main idea of this scheme is derived from our previous work [J.Comput. Phys., 446 (2021) 110653], in which the integral averages of the function andits first order derivative are used to reconstruct both the function and its first orderderivative values at the boundaries. However, in this paper, only the function values atthe Gauss-Lobatto points in the one or two dimensional case need to be reconstructedby using the information of the zeroth and first order moments. In addition, an extramodification procedure is used to modify those first order moments in the troubled-cells, which leads to an improvement of stability and an enhancement of resolutionnear discontinuities. To obtain the same order of accuracy, the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as thegeneral HWENO scheme and is more compact than the general WENO scheme. Moreover, the linear weights are not unique and are independent of the node position, andthe CFL number can still be 0.6 whether for the one or two dimensional case, which hasto be 0.2 in the two dimensional case for other HWENO schemes. Extensive numericalexamples are given to demonstrate the stability and resolution of such moment-basedmulti-resolution HWENO scheme.     

On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws

This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix. The resulting scheme is much simpler than the original one and is applicable to general hyperbolic conservation laws, while retaining the attractive features of the original solver: the method is entropy-satisfying, differentiable and complete in the sense that it attributes a different numerical viscosity to each characteristic field, in particular to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system is used. To illustrate the potential of the proposed scheme we show applications to the following hyperbolic conservation laws: Euler equations of compressible gasdynamics with ideal gas and real gas equation of state, classical and relativistic MHD equations as well as the equations of nonlinear elasticity. To the knowledge of the authors, apart from the Euler equations with ideal gas, an Osher-type scheme has never been devised before for any of these complicated PDE systems. Since our new general Riemann solver can be directly used as a building block of high order finite volume and discontinuous Galerkin schemes we also show the extension to higher order of accuracy and multiple space dimensions in the new framework of P_NP_M schemes on unstructured meshes recently proposed in [9].     

An Adaptive Surrogate Modeling Based on Deep Neural Networks for Large-Scale Bayesian Inverse Problems

In Bayesian inverse problems, surrogate models are often constructed tospeed up the computational procedure, as the parameter-to-data map can be veryexpensive to evaluate. However, due to the curse of dimensionality and the nonlinear concentration of the posterior, traditional surrogate approaches (such us thepolynomial-based surrogates) are still not feasible for large scale problems. To thisend, we present in this work an adaptive multi-fidelity surrogate modeling framework based on deep neural networks (DNNs), motivated by the facts that the DNNscan potentially handle functions with limited regularity and are powerful tools forhigh dimensional approximations. More precisely, we first construct offline a DNN-based surrogate according to the prior distribution, and then, this prior-based DNN-surrogate will be adaptively & locally refined online using only a few high-fidelitysimulations. In particular, in the refine procedure, we construct a new shallow neuralnetwork that views the previous constructed surrogate as an input variable – yieldinga composite multi-fidelity neural network approach. This makes the online computational procedure rather efficient. Numerical examples are presented to confirm that theproposed approach can obtain accurate posterior information with a limited numberof forward simulations.     

A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws: Application to Shallow Water Equations

We construct a new first-order central-upwind numerical method for solving systems of hyperbolic equations in conservative form. It applies in multidimensional structured and unstructured meshes. The proposed method is an extension of the UFORCE method developed by Stecca, Siviglia and Toro [25], in which the upwind bias for the modification of the staggered mesh is evaluated taking into account the smallest and largest wave of the entire Riemann fan. The proposed first-order method is shown to be identical to the Godunov upwind method in applications to a 2×2 linear hyperbolic system. The method is then extended to non-linear systems and its performance is assessed by solving the two-dimensional inviscid shallow water equations. Extension to second-order accuracy is carried out using an ADER-WENO approach in the finite volume framework on unstructured meshes. Finally, numerical comparison with current competing numerical methods enables us to identify the salient features of the proposed method.     

Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws

An entropy stable fully discrete shock capturing space-time Discontinuous Galerkin (DG) method was proposed in a recent paper [20] to approximate hyperbolic systems of conservation laws. This numerical scheme involves the solution of a very large nonlinear system of algebraic equations, by a Newton-Krylov method, at every time step. In this paper, we design efficient preconditioners for the large, nonsymmetric linear system, that needs to be solved at every Newton step. Two sets of preconditioners, one of the block Jacobi and another of the block Gauss-Seidel type are designed. Fourier analysis of the preconditioners reveals their robustness and a large number of numerical experiments are presented to illustrate the gain in efficiency that results from preconditioning. The resulting method is employed to compute approximate solutions of the compressible Euler equations, even for very high CFL numbers.     

A Multi-Domain Hybrid DG and WENO Method for Hyperbolic Conservation Laws on Hybrid Meshes

In [SIAM J. Sci. Comput., 35(2)(2013), A1049–A1072], a class of multi-domain hybrid DG and WENO methods for conservation laws was introduced. Recent applications of this method showed that numerical instability may encounter if the DG flux with Lagrangian interpolation is applied as the interface flux during the moment of conservative coupling. In this continuation paper, we present a more robust approach in the construction of DG flux at the coupling interface by using WENO procedures of reconstruction. Based on this approach, such numerical instability is overcome very well. In addition, the procedure of coupling a DG method with a WENO-FD scheme on hybrid meshes is disclosed in detail. Typical testing cases are employed to demonstrate the accuracy of this approach and the stability under the flexibility of using either WENO-FD flux or DG flux at the moment of requiring conservative coupling.     

Parameter Identification in Uncertain Scalar Conservation Laws Discretized with the Discontinuous Stochastic Galerkin Scheme

We study an identification problem which estimates the parameters of theunderlying random distribution for uncertain scalar conservation laws. The hyperbolic equations are discretized with the so-called discontinuous stochastic Galerkinmethod, i.e., using a spatial discontinuous Galerkin scheme and a Multielement stochastic Galerkin ansatz in the random space. We assume an uncertain flux or uncertaininitial conditions and that a data set of an observed solution is given. The uncertainty is assumed to be uniformly distributed on an unknown interval and we focus onidentifying the correct endpoints of this interval. The first-order optimality conditionsfrom the discontinuous stochastic Galerkin discretization are computed on the time-continuous level. Then, we solve the resulting semi-discrete forward and backwardschemes with the Runge-Kutta method. To illustrate the feasibility of the approach,we apply the method to a stochastic advection and a stochastic equation of Burgers' type. The results show that the method is able to identify the distribution parametersof the random variable in the uncertain differential equation even if discontinuities arepresent.     

High Order Fixed-Point Sweeping WENO Methods for Steady State of Hyperbolic Conservation Laws and Its Convergence Study

Fixed-point iterative sweeping methods were developed in the literature to efficiently solve static Hamilton-Jacobi equations. This class of methods utilizes the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. They take advantage of the properties of hyperbolic partial differential equations (PDEs) and try to cover a family of characteristics of the corresponding Hamilton-Jacobi equation in a certain direction simultaneously in each sweeping order. Different from other fast sweeping methods, fixed-point iterative sweeping methods have the advantages such as that they have explicit forms and do not involve inverse operation of nonlinear local systems. In principle, it can be applied to solving very general equations using any monotone numerical fluxes and high order approximations easily. In this paper, based on the recently developed fifth order WENO schemes which improve the convergence of the classical WENO schemes by removing slight post-shock oscillations, we design fifth order fixed-point sweeping WENO methods for efficient computation of steady state solution of hyperbolic conservation laws. Especially, we show that although the methods do not have linear computational complexity, they converge to steady state solutions much faster than regular time-marching approach by stability improvement for high order schemes with a forward Euler time-marching.     

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

In this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solving nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say, the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique not only reduces the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD isless expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements.     

On Arbitrary-Lagrangian-Eulerian One-Step WENO Schemes for Stiff Hyperbolic Balance Laws

In this article we present a new family of high order accurate Arbitrary Lagrangian-Eulerian one-step WENO finite volume schemes for the solution of stiff hyperbolic balance laws. High order accuracy in space is obtained with a standard WENO reconstruction algorithm and high order in time is obtained using the local space-time discontinuous Galerkin method recently proposed in [20]. In the Lagrangian framework considered here, the local space-time DG predictor is based on a weak formulation of the governing PDE on a moving space-time element. For the space-time basis and test functions we use Lagrange interpolation polynomials defined by tensor-product Gauss-Legendre quadrature points. The moving space-time elements are mapped to a reference element using an isoparametric approach, i.e. the space-time mapping is defined by the same basis functions as the weak solution of the PDE. We show some computational examples in one space-dimension for non-stiff and for stiff balance laws, in particular for the Euler equations of compressible gas dynamics, for the resistive relativistic MHD equations, and for the relativistic radiation hydrodynamics equations. Numerical convergence results are presented for the stiff case up to sixth order of accuracy in space and time and for the non-stiff case up to eighth order of accuracy in space and time.     

Two-Dimensional Multi-Domain Hybrid Spectral-WENO Methods for Conservation Laws

The multi-domain hybrid Spectral-WENO method (Hybrid) is introduced for the numerical solution of two-dimensional nonlinear hyperbolic systems in a Cartesian physical domain which is partitioned into a grid of rectangular subdomains. The main idea of the Hybrid scheme is to conjugate the spectral and WENO methods for solving problems with shock or high gradients such that the scheme adapts its solver spatially and temporally depending on the smoothness of the solution in a given subdomain. Built as a multi-domain method, an adaptive algorithm is used to keep the solutions parts exhibiting high gradients and discontinuities always inside WENO subdomains while the smooth parts of the solution are kept inside spectral ones, avoiding oscillations related to the well-known Gibbs phenomenon and increasing the numerical efficiency of the overall scheme. A higher order version of the multi-resolution analysis proposed by Harten is used to determine the smoothness of the solution in each subdomain. We also discuss interface conditions for the two-dimensional problem and the switching procedure between WENO and spectral subdomains. The Hybrid method is applied to the two-dimensional Shock-Vortex Interaction and the Richtmyer-Meshkov Instability (RMI) problems.     

A Compact High Order Space-Time Method for Conservation Laws

This paper presents a novel high-order space-time method for hyperbolic conservation laws. Two important concepts, the staggered space-time mesh of the space-time conservation element/solution element (CE/SE) method and the local discontinuous basis functions of the space-time discontinuous Galerkin (DG) finite element method, are the two key ingredients of the new scheme. The staggered space-time mesh is constructed using the cell-vertex structure of the underlying spatial mesh. The universal definitions of CEs and SEs are independent of the underlying spatial mesh and thus suitable for arbitrarily unstructured meshes. The solution within each physical time step is updated alternately at the cell level and the vertex level. For this solution updating strategy and the DG ingredient, the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme (DG-CVS). The high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each SE. The present DG-CVS exhibits many advantageous features such as Riemann-solver-free, high-order accuracy, point-implicitness, compactness, and ease of handling boundary conditions. Several numerical tests including the scalar advection equations and compressible Euler equations will demonstrate the performance of the new method.     

A Modified Discontinuous Galerkin Method for Solving Efficiently Helmholtz Problems

A new solution methodology is proposed for solving efficiently Helmholtz problems. The proposed method falls in the category of the discontinuous Galerkin methods. However, unlike the existing solution methodologies, this method requires solving (a) well-posed local problems to determine the primal variable, and (b) a global positive semi-definite Hermitian system to evaluate the Lagrange multiplier needed to restore the continuity across the element edges. Illustrative numerical results obtained for two-dimensional interior Helmholtz problems are presented to assess the accuracy and the stability of the proposed solution methodology.     

Application of a Bayesian simulation model to a database for split liver transplantation on two adult recipients in the environment of WinBUGS (Bayesian Inference Using Gibbs Sampling)

A Bayesian simulation model has been applied to a database developed for split liver transplantation on two adult recipients (SLT A/A) in the context of a macroregional project funded by the Italian Ministry of Health. The model was entered within Bayesian inference Using Gibbs Sampling (WinBUGS), a free software for Bayesian analysis of complex statistical models using Markov chain Monte Carlo techniques developed by the MRC Biostatistics Unit Cambridge jointly with the Imperial College School of Medicine at St Mary's, London. The model was built by using data entry performed from January 1, 2005 to August 5, 2005. In that period, 20 potential donors suitable for the SLT A/A procedure were entered into the database. We only selected the continuous and dichotomous donor-related variables (DRV, n = 62) for which almost one data entry procedure. The model assumed that a database user learned during data entry procedures for each donor, and that the probability of a successful input may depend on the number of previous errors and corrections. After binary transformation of the DRV (value 0 for each input record, value 1 for each no input record), we calculated an overall value of 0.28 +/- 0.27 (median: 0.3; 95% confidence interval: from 0.18 to 0.629). The transformed DRV were entered within the WinBUGS environment after model specification, assuming as success (y = 1) each procedure of input record, and as failure (y = 0) each procedure of no input record. A unequivocal convergence was obtained after 10,000 iterations, and a simulation run was launched for a further 10,000 updates. We obtained a negligible Monte Carlo error and a fine profile in the kernel density plot. This study supported the application of simulation models to databases concerning liver transplantation as a useful strategy to identify a critical state in the data entry process.     

An Adaptive Moving Mesh Discontinuous Galerkin Method for the Radiative Transfer Equation

The radiative transfer equation models the interaction of radiation with scattering and absorbing media and has important applications in various fields in scienceand engineering. It is an integro-differential equation involving time, frequency, spaceand angular variables and contains an integral term in angular directions while being hyperbolic in space. The challenges for its numerical solution include the needsto handle with its high dimensionality, the presence of the integral term, and the development of discontinuities and sharp layers in its solution along spatial directions.Its numerical solution is studied in this paper using an adaptive moving mesh discontinuous Galerkin method for spatial discretization together with the discrete ordinatemethod for angular discretization. The former employs a dynamic mesh adaptationstrategy based on moving mesh partial differential equations to improve computational accuracy and efficiency. Its mesh adaptation ability, accuracy, and efficiency aredemonstrated in a selection of one- and two-dimensional numerical examples.     

Convolution Neural Network Shock Detector for Numerical Solution of Conservation Laws

We propose a universal discontinuity detector using convolution neural network (CNN) and apply it in conjunction of solving nonlinear conservation laws in both1D and 2D. The CNN detector is trained offline with synthetic data. The training dataare generated using randomly constructed piecewise functions, which are then processed using randomized linear advection solver to count for the cases of numericalerrors in practice. The detector is then paired with high-order numerical solvers. Inparticular, we combined high-order WENO in troubled cells with high-order centraldifference in smooth region. Extensive numerical examples are presented. We observethat the proposed method produces notably sharper and cleaner signals near the discontinuities, when compared to other well known troubled cell detector methods.     

Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning

Conservation laws are considered to be fundamental laws of nature. It hasbroad applications in many fields, including physics, chemistry, biology, geology, andengineering. Solving the differential equations associated with conservation laws is amajor branch in computational mathematics. The recent success of machine learning,especially deep learning in areas such as computer vision and natural language processing, has attracted a lot of attention from the community of computational mathematics and inspired many intriguing works in combining machine learning with traditional methods. In this paper, we are the first to view numerical PDE solvers as anMDP and to use (deep) RL to learn new solvers. As proof of concept, we focus on1-dimensional scalar conservation laws. We deploy the machinery of deep reinforcement learning to train a policy network that can decide on how the numerical solutions should be approximated in a sequential and spatial-temporal adaptive manner.We will show that the problem of solving conservation laws can be naturally viewedas a sequential decision-making process, and the numerical schemes learned in such away can easily enforce long-term accuracy. Furthermore, the learned policy networkis carefully designed to determine a good local discrete approximation based on thecurrent state of the solution, which essentially makes the proposed method a meta-learning approach. In other words, the proposed method is capable of learning how todiscretize for a given situation mimicking human experts. Finally, we will provide details on how the policy network is trained, how well it performs compared with somestate-of-the-art numerical solvers such as WENO schemes, and supervised learningbased approach L3D and PINN, and how well it generalizes.     

A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems

In this paper, we present an efficient computational methodology for diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem. It is well known that the numerical computation for these problems requires a significant amount of computer memory and time. Nevertheless, the solutions to these problems typically contain a coarse component, which is usually the quantity of interest and can be represented with a small number of degrees of freedom. There are many methods that aim at the computation of the coarse component without resolving the full details of the solution. Our proposed method falls into the framework of interior penalty discontinuous Galerkin method, which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations. A distinctive feature of our method is that the solution space contains two components, namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse component. In addition, stability of the method is proved. The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems.     

A Class of Hybrid DG/FV Methods for Conservation Laws III: Two-Dimensional Euler Equations

A concept of "static reconstruction" and "dynamic reconstruction" was introduced for higher-order (third-order or more) numerical methods in our previous work. Based on this concept, a class of hybrid DG/FV methods had been developed for one-dimensional conservation law using a "hybrid reconstruction" approach, and extended to two-dimensional scalar equations on triangular and Cartesian/triangular hybrid grids. In the hybrid DG/FV schemes, the lower-order derivatives of the piecewise polynomial are computed locally in a cell by the traditional DG method (called as "dynamic reconstruction"), while the higher-order derivatives are reconstructed by the "static reconstruction" of the FV method, using the known lower-order derivatives in the cell itself and in its adjacent neighboring cells. In this paper, the hybrid DG/FV schemes are extended to two-dimensional Euler equations on triangular and Cartesian/triangular hybrid grids. Some typical test cases are presented to demonstrate the performance of the hybrid DG/FV methods, including the standard vortex evolution problem with exact solution, isentropic vortex/weak shock wave interaction, subsonic flows past a circular cylinder and a three-element airfoil (30P30N), transonic flow past a NACA0012 airfoil. The accuracy study shows that the hybrid DG/FV method achieves the desired third-order accuracy, and the applications demonstrate that they can capture the flow structure accurately, and can reduce the CPU time and memory requirement greatly than the traditional DG method with the same order of accuracy.