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Recently, there has been a growing interest in efficient numerical algorithms based on tensor networks and low-rank techniques to approximate high-dimensional functions and solutions to high-dimensional PDEs. In this paper, we propose a new tensor rank reduction method based on coordinate transformations that can greatly increase the efficiency of high-dimensional tensor approximation algorithms. The idea is simple: given a multivariate function, determine a coordinate transformation so that the function in the new coordinate system has smaller tensor rank. We restrict our analysis to linear coordinate transformations, which gives rise to a new class of functions that we refer to as tensor ridge functions. Leveraging Riemannian gradient descent on matrix manifolds we develop an algorithm that determines a quasi-optimal linear coordinate transformation for tensor rank reduction. The results we present for rank reduction via linear coordinate transformations open the possibility for generalizations to larger classes of nonlinear transformations. • Riemannian optimization of tensor ridge functions. • Tensor rank reduction via coordinate flows. • PDEs in curvilinear coordinate systems. • Rank-adaptive tensor methods. [ABSTRACT FROM AUTHOR]

In this paper, we evaluate the performance of the multilevel Monte Carlo method (MLMC) for deterministic and uncertain hyperbolic systems, where randomness is introduced either in the modeling parameters or in the approximation algorithms. MLMC is a well known variance reduction method widely used to accelerate Monte Carlo (MC) sampling. However, we demonstrate in this paper that for hyperbolic systems, whether MLMC can achieve a real boost turns out to be delicate. The computational costs of MLMC and MC depend on the interplay among the accuracy (bias) and the computational cost of the numerical method for a single sample, as well as the variances of the sampled MLMC corrections or MC solutions. We characterize three regimes for the MLMC and MC performances using those parameters, and show that MLMC may not accelerate MC and can even have a higher cost when the variances of MC solutions and MLMC corrections are of the same order. Our studies are carried out by a few prototype hyperbolic systems: a linear scalar equation, the Euler and shallow water equations, and a linear relaxation model, the above statements are proved analytically in some cases, and demonstrated numerically for the cases of the stochastic hyperbolic equations driven by white noise parameters and Glimm's random choice method for deterministic hyperbolic equations. • Investigate multilevel Monte Carlo for deterministic and uncertain hyperbolic systems. • Identify three regimes for the MLMC/MC performances using bias, cost and variance. • MLMC may be outperformed by MC when their variances are of the same order. • Analytical proof for prototypical scalar advection equation with white noise parameters. • Numerical justification for stochastic hyperbolic equations and Glimm's method. [ABSTRACT FROM AUTHOR]

In this work, we propose an accurate, robust (the solution remains in the set of states), and stable discretization of a nonconservative model for the simulation of compressible multicomponent flows with shocks and material interfaces. We consider the gamma-based model by Shyue (1998) [58] where each component follows a stiffened gas equation of state (EOS). We here extend the framework proposed in Renac (2019) [53] and Coquel et al. (2021) [15] for the discretization of hyperbolic systems, with both fluxes and nonconservative products, to unstructured meshes with curved elements in multiple space dimensions. The framework relies on a high-order discontinuous Galerkin spectral element method (DGSEM) using collocation of quadrature and interpolation points as proposed by Gassner (2013) [27] in the case of hyperbolic conservation laws. We modify the integrals over discretization elements where we replace the physical fluxes and nonconservative products by two-point numerical fluctuations. The contributions of this work are threefold. First, we analyze the semi-discrete DGSEM discretization of general hyperbolic systems with conservative and nonconservative terms and derive the conditions to obtain a scheme that is high-order accurate, free-stream preserving, and entropy stable when excluding material interfaces. Second, we design a three-point scheme with a HLLC solver for the gamma-based model that does not require a root-finding algorithm for the approximation of the nonconservative products. The scheme is proved to be robust and entropy stable for convex entropies, to preserve uniform profiles of pressure and velocity across material interfaces (material interface preservation), and to satisfy a discrete minimum principle on the specific entropy and maximum principles on the parameters of the EOS. Third, the HLLC solver is applied at interfaces in the DGSEM, while we consider two kinds of fluctuations in the integrals over discretization elements: the former is entropy conservative (EC), while the latter preserves material interfaces (CP). Time integration is performed using high-order strong-stability preserving Runge-Kutta schemes. The fully discrete scheme is shown to preserve material interfaces with CP fluctuations. Under a given condition on the time step, both EC and CP fluctuations ensure that the cell-averaged solution remains in the set of states; satisfy a minimum principle on any convex entropy and maximum principles on the EOS parameters. These results have allowed us to use existing limiters in order to restore positivity, and discrete maximum principles of degrees-of-freedom within elements. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and accuracy of the proposed DGSEM with either CP, or EC fluctuations, while the scheme with CP fluctuations is shown to offer better resolution capabilities. • High-order and robust discretization of a compressible nonconservative multicomponent flow model. • A DGSEM for unstructured meshes with curved elements is proposed. • Design contact-preserving and entropy conservative numerical fluxes. • Design an HLLC solver that is applied at cell interfaces. • The discrete scheme maintains positivity of the solution, preserves material interfaces and invariant domains. [ABSTRACT FROM AUTHOR]

The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems on one- and two-dimensional rectangular domains. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems. • A fast approximation algorithm for variable-order spectral fractional Laplacian. • Construction of solution landscapes of space-fractional phase field models. • Reconfiguration of fractional Laplacian problems by integer-order problems. [ABSTRACT FROM AUTHOR]

This article presents a fast solver for the dense “frontal” matrices that arise from the multifrontal sparse elimination process of 3D elliptic PDEs. The solver relies on the fact that these matrices can be efficiently represented as a hierarchically off-diagonal low-rank (HODLR) matrix. To construct the low-rank approximation of the off-diagonal blocks, we propose a new pseudo-skeleton scheme, the boundary distance low-rank approximation, that picks rows and columns based on the location of their corresponding vertices in the sparse matrix graph. We compare this new low-rank approximation method to the adaptive cross approximation (ACA) algorithm and show that it achieves better speedup specially for unstructured meshes. Using the HODLR direct solver as a preconditioner (with a low tolerance) to the GMRES iterative scheme, we can reach machine accuracy much faster than a conventional LU solver. Numerical benchmarks are provided for frontal matrices arising from 3D finite element problems corresponding to a wide range of applications. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a novel algorithm to approximate the long-range electrostatic potential field in the Cartesian coordinates applicable to 3D coarse-grained simulations of biopolymers. In such models, coarse-grained clusters are formed via treating groups of atoms as rigid and/or flexible bodies connected together via kinematic joints. Therefore, multibody dynamic techniques are used to form and solve the equations of motion of such coarse-grained systems. In this article, the approximations for the potential fields due to the interaction between a highly negatively/positively charged pseudo-atom and charged particles, as well as the interaction between clusters of charged particles, are presented. These approximations are expressed in terms of physical and geometrical properties of the bodies such as the entire charge, the location of the center of charge, and the pseudo-inertia tensor about the center of charge of the clusters. Further, a novel substructuring scheme is introduced to implement the presented far-field potential evaluations in a binary tree framework as opposed to the existing quadtree and octree strategies of implementing fast multipole method. Using the presented Lagrangian grids, the electrostatic potential is recursively calculated via sweeping two passes: assembly and disassembly. In the assembly pass, adjacent charged bodies are combined together to form new clusters. Then, the potential field of each cluster due to its interaction with faraway resulting clusters is recursively calculated in the disassembly pass. The method is highly compatible with multibody dynamic schemes to model coarse-grained biopolymers. Since the proposed method takes advantage of constant physical and geometrical properties of rigid clusters, improvement in the overall computational cost is observed comparing to the tradition application of fast multipole method. [Copyright &y& Elsevier]

Abstract: The optimal selection of experimental conditions is essential to maximizing the value of data for inference and prediction, particularly in situations where experiments are time-consuming and expensive to conduct. We propose a general mathematical framework and an algorithmic approach for optimal experimental design with nonlinear simulation-based models; in particular, we focus on finding sets of experiments that provide the most information about targeted sets of parameters. Our framework employs a Bayesian statistical setting, which provides a foundation for inference from noisy, indirect, and incomplete data, and a natural mechanism for incorporating heterogeneous sources of information. An objective function is constructed from information theoretic measures, reflecting expected information gain from proposed combinations of experiments. Polynomial chaos approximations and a two-stage Monte Carlo sampling method are used to evaluate the expected information gain. Stochastic approximation algorithms are then used to make optimization feasible in computationally intensive and high-dimensional settings. These algorithms are demonstrated on model problems and on nonlinear parameter inference problems arising in detailed combustion kinetics. [Copyright &y& Elsevier]

Abstract: The multiscale finite volume (MsFV) method has been developed to efficiently solve large heterogeneous problems (elliptic or parabolic); it is usually employed for pressure equations and delivers conservative flux fields to be used in transport problems. The method essentially relies on the hypothesis that the (fine-scale) problem can be reasonably described by a set of local solutions coupled by a conservative global (coarse-scale) problem. In most cases, the boundary conditions assigned for the local problems are satisfactory and the approximate conservative fluxes provided by the method are accurate. In numerically challenging cases, however, a more accurate localization is required to obtain a good approximation of the fine-scale solution. In this paper we develop a procedure to iteratively improve the boundary conditions of the local problems. The algorithm relies on the data structure of the MsFV method and employs a Krylov-subspace projection method to obtain an unconditionally stable scheme and accelerate convergence. Two variants are considered: in the first, only the MsFV operator is used; in the second, the MsFV operator is combined in a two-step method with an operator derived from the problem solved to construct the conservative flux field. The resulting iterative MsFV algorithms allow arbitrary reduction of the solution error without compromising the construction of a conservative flux field, which is guaranteed at any iteration. Since it converges to the exact solution, the method can be regarded as a linear solver. In this context, the schemes proposed here can be viewed as preconditioned versions of the Generalized Minimal Residual method (GMRES), with a very peculiar characteristic that the residual on the coarse grid is zero at any iteration (thus conservative fluxes can be obtained). [ABSTRACT FROM AUTHOR]