This paper develops a two-level fourth-order scheme for solving time-fractional convection-diffusion-reaction equation with variable coefficients subjected to suitable initial and boundary conditions. The basis properties of the new approach are investigated and both stability and error estimates of the proposed numerical scheme are deeply analyzed in the $L^{\infty}(0,T;L^{2})$-norm. The theory indicates that the method is unconditionally stable with convergence of order $O(k^{2-\frac{\lambda}{2}}+h^{4})$, where $k$ and $h$ are time step and mesh size, respectively, and $\lambda\in(0,1)$. This result suggests that the two-level fourth-order technique is more efficient than a large class of numerical techniques widely studied in the literature for the considered problem. Some numerical evidences are provided to verify the unconditional stability and convergence rate of the proposed algorithm.
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In this paper we provide some more details on the numerical analysis and we present some enlightening numerical results related to the spectrum of a finite element least-squares approximation of the linear elasticity formulation introduced recently. We show that, although the formulation is robust in the incompressible limit for the source problem, its spectrum is strongly dependent on the Lam\'e parameters and on the underlying mesh.
This paper mainly investigates the strong convergence and stability of the truncated Euler-Maruyama (EM) method for stochastic differential delay equations with variable delay whose coefficients can be growing super-linearly. By constructing appropriate truncated functions to control the super-linear growth of the original coefficients, we present a type of the truncated EM method for such SDDEs with variable delay, which is proposed to be approximated by the value taken at the nearest grid points on the left of the delayed argument. The strong convergence result (without order) of the method is established under the local Lipschitz plus generalized Khasminskii-type conditions and the optimal strong convergence order $1/2$ can be obtained if the global monotonicity with U function and polynomial growth conditions are added to the assumptions. Moreover, the partially truncated EM method is proved to preserve the mean-square and H_\infty stabilities of the true solutions. Compared with the known results on the truncated EM method for SDDEs, a better order of strong convergence is obtained under more relaxing conditions on the coefficients, and more refined technical estimates are developed so as to overcome the challenges arising due to variable delay. Lastly, some numerical examples are utilized to confirm the effectiveness of the theoretical results.
This paper presents and analyzes an immersed finite element (IFE) method for solving Stokes interface problems with a piecewise constant viscosity coefficient that has a jump across the interface. In the method, the triangulation does not need to fit the interface and the IFE spaces are constructed from the traditional $CR$-$P_0$ element with modifications near the interface according to the interface jump conditions. We prove that the IFE basis functions are unisolvent on arbitrary interface elements and the IFE spaces have the optimal approximation capabilities, although the proof is challenging due to the coupling of the velocity and the pressure. The stability and the optimal error estimates of the proposed IFE method are also derived rigorously. The constants in the error estimates are shown to be independent of the interface location relative to the triangulation. Numerical examples are provided to verify the theoretical results.
We propose a novel method to compute a finite difference stencil for Riesz derivative for artibitrary speed of convergence. This method is based on applying a pre-filter to the Gr\"unwald-Letnikov type central difference stencil. The filter is obtained by solving for the inverse of a symmetric Vandemonde matrix and exploiting the relationship between the Taylor's series coefficients and fast Fourier transform. The filter costs O\left(N^{2}\right) operations to evaluate for O\left(h^{N}\right) of convergence, where h is the sampling distance. The higher convergence speed should more than offset the overhead with the requirement of the number of nodal points for a desired error tolerance significantly reduced. The benefit of progressive generation of the stencil coefficients for adaptive grid size for dynamic problems with the Gr\"unwald-Letnikov type difference scheme is also kept because of the application of filtering. The higher convergence rate is verified through numerical experiments.
Ill-posed linear inverse problems appear in many scientific setups, and are typically addressed by solving optimization problems, which are composed of data fidelity and prior terms. Recently, several works have considered a back-projection (BP) based fidelity term as an alternative to the common least squares (LS), and demonstrated excellent results for popular inverse problems. These works have also empirically shown that using the BP term, rather than the LS term, requires fewer iterations of optimization algorithms. In this paper, we examine the convergence rate of the projected gradient descent (PGD) algorithm for the BP objective. Our analysis allows to identify an inherent source for its faster convergence compared to using the LS objective, while making only mild assumptions. We also analyze the more general proximal gradient method under a relaxed contraction condition on the proximal mapping of the prior. This analysis further highlights the advantage of BP when the linear measurement operator is badly conditioned. Numerical experiments with both $\ell_1$-norm and GAN-based priors corroborate our theoretical results.
The main aim of this paper is to solve an inverse source problem for a general nonlinear hyperbolic equation. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the inverse problem. To find this fixed point, we define a recursive sequence with an arbitrary initial term by the same manner as in the classical proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution with the exponential rate. Therefore, our new method can be considered as an analog of the contraction principle. We rigorously study the stability of our method with respect to noise. Numerical examples are presented.
Orbit-determination programs find the orbit solution that best fits a set of observations by minimizing the RMS of the residuals of the fit. For near-Earth asteroids, the uncertainty of the orbit solution may be compatible with trajectories that impact Earth. This paper shows how incorporating the impact condition as an observation in the orbit-determination process results in a robust technique for finding the regions in parameter space leading to impacts. The impact pseudo-observation residuals are the b-plane coordinates at the time of close approach and the uncertainty is set to a fraction of the Earth radius. The extended orbit-determination filter converges naturally to an impacting solution if allowed by the observations. The uncertainty of the resulting orbit provides an excellent geometric representation of the virtual impactor. As a result, the impact probability can be efficiently estimated by exploring this region in parameter space using importance sampling. The proposed technique can systematically handle a large number of estimated parameters, account for nongravitational forces, deal with nonlinearities, and correct for non-Gaussian initial uncertainty distributions. The algorithm has been implemented into a new impact monitoring system at JPL called Sentry-II, which is undergoing extensive testing. The main advantages of Sentry-II over JPL's currently operating impact monitoring system Sentry are that Sentry-II can systematically process orbits perturbed by nongravitational forces and that it is generally more robust when dealing with pathological cases. The runtimes and completeness of both systems are comparable, with the impact probability of Sentry-II for 99% completeness being $3\times10^{-7}$.
Semi-lagrangian schemes for discretization of the dynamic programming principle are based on a time discretization projected on a state-space grid. The use of a structured grid makes this approach not feasible for high-dimensional problems due to the curse of dimensionality. Here, we present a new approach for infinite horizon optimal control problems where the value function is computed using Radial Basis Functions (RBF) by the Shepard's moving least squares approximation method on scattered grids. We propose a new method to generate a scattered mesh driven by the dynamics and the selection of the shape parameter in the RBF using an optimization routine. This mesh will help to localize the problem and approximate the dynamic programming principle in high dimension. Error estimates for the value function are also provided. Numerical tests for high dimensional problems will show the effectiveness of the proposed method.
We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a unique weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions-Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.
UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.
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