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Abstract: Abstract The sharp pointwise-in-time error estimate of the Grünwald-Letnikov scheme for the initial-boundary value problem of a multi-term time fractional diffusion equation is considered, where the solutions exhibit typical weak singularity at initial time. The Grünwald-Letnikov scheme on uniform mesh is used to discretize the multi-term time fractional Caputo derivative and finite difference method is adopted for spatial discretization. A bound for the stability multipliers is deduced using complete monotonicity, by which stability and α-robust error estimate of the fully discrete scheme are rigorously established. Numerical examples are presented to show the sharpness of the error estimate. PubDate: 2022-11-29

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Abstract: Abstract In this paper, we propose a new positive finite volume scheme for degenerate parabolic equations with strongly anisotropic diffusion tensors. The key idea is to approximate the fluxes thanks to a weighted-centered scheme depending on the sign of the stiffness coefficients. More specifically, we employ a centered discretization for the mobility-like function when the transmissibilities are positive and a weighted harmonic scheme in regions where the negative transmissibilities occur. This technique prevents the formation of undershoots which entails the positivity-preserving of the approach. Also, the scheme construction retains the main elements, namely the coercivity and compactness estimates, allowing the existence and the convergence of the nonlinear finite volume scheme under general assumptions on the physical inputs, the nonlinearities, and the mesh. The numerical implementation of the methodology shows that the lower bound on the discrete solution is respected and optimal convergence rates are recovered on strongly anisotropic various test-cases. PubDate: 2022-11-18

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Abstract: Abstract Often the easiest way to discretize an ordinary or partial differential equation is by a rectangular numerical method, in which n basis functions are sampled at m ≫ n collocation points. We show how eigenvalue problems can be solved in this setting by QR reduction to square matrix generalized eigenvalue problems. The method applies equally in the limit “ \(m=\infty \) ” of eigenvalue problems for quasimatrices. Numerical examples are presented as well as pointers to related literature. PubDate: 2022-11-16

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Abstract: Abstract Physics-informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely Kolmogorov equations that include the heat equation and Black-Scholes equation of option pricing, as examples. We construct neural networks, whose PINN residual (generalization error) can be made as small as desired. We also prove that the total L2-error can be bounded by the generalization error, which in turn is bounded in terms of the training error, provided that a sufficient number of randomly chosen training (collocation) points is used. Moreover, we prove that the size of the PINNs and the number of training samples only grow polynomially with the underlying dimension, enabling PINNs to overcome the curse of dimensionality in this context. These results enable us to provide a comprehensive error analysis for PINNs in approximating Kolmogorov PDEs. PubDate: 2022-11-15

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Abstract: Abstract In this paper, we discuss a framework for the polynomial approximation to the solution of initial value problems for differential equations. The framework is based on an expansion of the vector field along an orthonormal basis, and relies on perturbation results for the considered problem. Initially devised for the approximation of ordinary differential equations, it is here further extended and, moreover, generalized to cope with constant delay differential equations. Relevant classes of Runge-Kutta methods can be derived within this framework. PubDate: 2022-11-14

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Abstract: Abstract We are concerned with an inverse problem of simultaneously recovering the initial value and source strength in a transmission problem for a parabolic equation from extra measurements. First, we establish a conditional stability of the inverse problem by combining the Carleman estimate with the logarithmic convexity theory. Second, we construct a regularized minimizing functional and transform the inverse problem into an optimization problem. The existence and stability of solutions to the optimization problem are analyzed rigorously. Then, we introduce a surrogate functional and propose an iterative thresholding algorithm for solving the optimization problem. The algorithm is cheap and easy to be implemented numerically. Finally, we present several numerical examples for the two-dimensional (2D) and three-dimensional (3D) cases to show the validity of the proposed algorithm. PubDate: 2022-11-14

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Abstract: Abstract In this paper, we analyze the virtual element method for the quasilinear convection-diffusion-reaction equation. The most important part in the analysis is the proof of existence and uniqueness of the branch of solution of the discrete problem. We extend the explicit analysis given by Lube (Numer. Math. 61, 335–357, 1992) for the finite element discretization to virtual element framework. We prove the optimal rate of convergence in the energy norm. In order to reduce the overall computational cost incurred for the nonlinear equations, we have performed the numerical experiments using a two-grid method. We validate the theoretical estimates with the computed numerical results. PubDate: 2022-11-14

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Abstract: Abstract In this article, we consider transport networks with uncertain demands. Network dynamics are given by linear hyperbolic partial differential equations and suitable coupling conditions, while demands are incorporated as solutions to stochastic differential equations. For the demand satisfaction, we solve a constrained optimal control problem. Controls in terms of network inputs are then calculated explicitly for different assumptions. Numerical simulations are performed to underline the theoretical results. PubDate: 2022-11-10

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Abstract: Abstract We propose a Bayesian approach for parameter uncertainty quantification in macroscopic traffic flow models from cross-sectional data. We consider both a simple first order model consisting in the mass conservation equation and its second order version including a speed evolution equation. A bias term is introduced and modeled as a Gaussian process to account for the traffic flow models limitations. We validate the results comparing the error in the macroscopic variables (flow, speed, density) for both models, showing that second order models globally perform better in reconstructing traffic quantities of interest. PubDate: 2022-11-10

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Abstract: Abstract In this contribution, we develop an efficient surrogate modeling framework for simulation-based optimization of enhanced oil recovery, where we particularly focus on polymer flooding. The computational approach is based on an adaptive training procedure of a neural network that directly approximates an input-output map of the underlying PDE-constrained optimization problem. The training process thereby focuses on the construction of an accurate surrogate model solely related to the optimization path of an outer iterative optimization loop. True evaluations of the objective function are used to finally obtain certified results. Numerical experiments are given to evaluate the accuracy and efficiency of the approach for a heterogeneous five-spot benchmark problem. PubDate: 2022-11-09

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Abstract: Abstract We introduce a new domain decomposition strategy for time harmonic Maxwell’s equations that is valid in the case of automatically generated subdomain partitions with possible presence of cross-points. The convergence of the algorithm is guaranteed and we present a complete analysis of the matrix form of the method. The method involves transmission matrices responsible for imposing coupling between subdomains. We discuss the choice of such matrices, their construction and the impact of this choice on the convergence of the domain decomposition algorithm. Numerical results and algorithms are provided. PubDate: 2022-11-09

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Abstract: Abstract Mixed and hybrid finite element discretizations for distributed optimal control problems governed by an elliptic equation are analyzed. A cost functional keeping track of both the state and its gradient is studied. A priori error estimates and super-convergence properties for the continuous and discrete optimal states, adjoint states, and controls will be given. The approximating finite-dimensional systems will be solved by adding penalization terms for the state and the associated Lagrange multipliers. In general, performing optimization, discretization, hybridization, and penalization in any order lead to the same optimality system. Numerical examples based on the Raviart–Thomas finite elements will be presented. PubDate: 2022-11-02

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Abstract: Abstract In a previous paper (Huang et al., Advances in Computational Mathematics 47(5):1–34, 2021), we presented the fundamentals of a new hierarchical algorithm for computing the expectation of a N-dimensional function \(H(\mathbf {X})\) where \(\mathbf {X}\) satisfies the truncated multi-variate normal (TMVN) distribution. The algorithm assumes that \(H(\mathbf {X})\) is low-rank and the covariance matrix \(\Sigma\) and precision matrix \(A=\Sigma ^{-1}\) have low-rank blocks with low-dimensional features. Analysis and numerical results were presented when A is tridiagonal or given by the exponential model. In this paper, we first demonstrate how the hierarchical algorithm structure allows the simultaneous calculations of all the order M and less moments \(E(H(\mathbf {X})=X_1^{m_1}\cdots X_N^{m_N} a_i<X_i<b_i, \; i=1,\ldots ,N)\) , \(\sum _{i} m_i \le M\) using asymptotically optimal \(O(N^M)\) operations when \(M\ge 2\) and \(O(N\log (N))\) operations when \(M=1\) . These \(O(N^M)\) moments are often required in the Expectation Maximization (EM) algorithms. We illustrate the algorithm ideas using the case when A is tridiagonal or the exponential model where the off-diagonal matrix block has rank \(K=1\) and number of effective variables \(P \le 2\) for each function associated with a hierarchical tree node. The smaller K and P values allow the use of existing FFT and Non-uniform FFT (NuFFT) solvers to accelerate the computation of the compressed features in the system. To handle cases with higher K and P values, we introduce the sparse grid technique aimed at problems with \(K+P \approx 5 \sim 20\) . We present numerical results for computing both the moments and higher K and P values to demonstrate the accuracy and efficiency of the algorithms. Finally, we summarize our results and discuss the limitations and generalizations, in particular, our algorithm capability is limited by the availability of mathematical tools in higher dimensions. When \(K+P\) is greater than 20, as far as we know, there are no practical tools available for problems with 20 truly independent variables. PubDate: 2022-11-02

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Abstract: Abstract In this work we extend some ideas about greedy algorithms, which are well-established tools for, e.g., kernel bases, and exponential-polynomial splines whose main drawback consists in possible overfitting and consequent oscillations of the approximant. To partially overcome this issue, we develop some results on theoretically optimal interpolation points. Moreover, we introduce two algorithms which perform an adaptive selection of the spline interpolation points based on the minimization either of the sample residuals (f-greedy), or of an upper bound for the approximation error based on the spline Lebesgue function ( \(\lambda\) -greedy). Both methods allow us to obtain an adaptive selection of the sampling points, i.e., the spline nodes. While the f-greedy selection is tailored to one specific target function, the \(\lambda\) -greedy algorithm enables us to define target-data-independent interpolation nodes. PubDate: 2022-10-27

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Abstract: Abstract An implicit numerical method for a fractional diffusion problem in the presence of an absorbing boundary is analyzed. The discretization chosen for the spatial fractional differential operator is known to be second-order accurate, when the problem is defined in the real line. The main purpose of this work is to show how the presence of the boundary can change the properties of the scheme, namely its consistency and convergence. We establish that the order of accuracy of the spatial truncation error can be lower than two in the presence of the boundary and in some cases we have inconsistency, depending not necessarily on the regularity of the solution but on the values of its derivatives at the boundary. Furthermore, we prove the rate of convergence will be higher than the order of accuracy given by the consistency analysis and sometimes we can recover the order two. In particular, the convergence is achieved for some of the inconsistent cases. PubDate: 2022-10-12

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Abstract: Abstract The stabilized linear Crank-Nicolson (SL-CN) scheme is a very important time discretization for the Cahn-Hilliard (CH) equation since it is an unconditionally energy stable method of second order, and allows to use time steps as large as possible to reduce the total calculation. Though, it still requires a very large amount of calculations for simulating the CH equation because of the essential nature of the CH equation. To accelerate the simulation process, we propose in this paper to solve the differential system resulting from the time discretization by an optimized Schwarz method using a newly proposed Ventcel transmission condition. For a setting of two-subdomain domain decomposition with or without overlap, we derive using Fourier analysis the convergence factor, which takes on two different forms according to the size of the time steps. By solving the hard min-max problem of convergence factors using asymptotic analysis, we rigorously optimized the convergence factors for each case and obtained the optimized transmission parameters in explicit form, and the estimate for the corresponding convergence rates. The theoretical results are illustrated by several numerical examples. PubDate: 2022-10-07

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Abstract: Abstract In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to accomplish the tensor compression, provide a detailed analysis of the computational costs, provide insight into the error of the resulting approximations, and discuss the benefits of the proposed approaches. We also apply the tensor-based function approximation to develop low-rank matrix approximations to kernel matrices that describe pairwise interactions between two sets of points; the resulting low-rank approximations are efficient to compute and store (the complexity is linear in the number of points). We present an adaptive version of the function and kernel approximation that determines an approximation that satisfies a user-specified relative error over a set of random points. We extend our approach to the case where the kernel requires repeated evaluations for many values of (hyper)parameters that govern the kernel. We give detailed numerical experiments on example problems involving multivariate function approximation, low-rank matrix approximations of kernel matrices involving well-separated clusters of sources and target points, and a global low-rank approximation of kernel matrices with an application to Gaussian processes. We observe speedups up to 18X over standard matrix-based approaches. PubDate: 2022-10-04

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Abstract: Abstract This paper deals with the construction of the Algebraic Trigonometric Pythagorean Hodograph (ATPH) cubic-like Hermite interpolant. A characterization of solutions according to the tangents at both ends and a global free shape parameter α is performed. Since this degree of freedom can be used for adjustments, we study how the curve evolves with respect to α. Several examples illustrating the construction process and a simple fitting method to determine the unique ATPH curve passing through a given point are proposed. PubDate: 2022-10-03

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Abstract: Abstract In this report we present a second-order, stabilized SAV based, Crank–Nicolson leap-frog (CNLF) ensemble method, and perform a comprehensive numerical study of it as well as the Crank–Nicolson ensemble method with a linear extrapolation (CNLE) presented in Jiang and Yang (SIAM J. Sci. Comput. 43:A2869–A2896, 2021). Both methods are extremely efficient as only one linear system with multiple right hands needs to be solved at each time for a (potentially large) number of realizations of the flow problems. In particular the coefficient matrix of the fully discretized system is a constant matrix that does not change from one time step to another. We present extensive testing of these two methods and demonstrate the advantages of each. We also present long time stability analysis for both methods. PubDate: 2022-10-03

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Abstract: Abstract In transient simulations of particulate Stokes flow, to accurately capture the interaction between the constituent particles and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the particles. Consequently, standard fast direct solvers lose their efficiency since the linear system changes at each time step. This manuscript presents a new computational approach that avoids this issue by pre-constructing a fast direct solver for the wall ahead of time, computing a low-rank factorization to capture the changes due to the refinement, and solving the problem on the refined discretization via a Woodbury formula. Numerical results illustrate the efficiency of the solver in accelerating particulate Stokes simulations. PubDate: 2022-09-27 DOI: 10.1007/s10444-022-09974-y