查看更多>>摘要：In this article, we develop a residual-driven adaptive Gaussian mixture approximation (RD-AGMA) for Bayesian inverse problems. The posterior distribution is often non-Gaussian in practical Bayesian inference. To obtain a good approximation of the posterior, we provide the adaptive Gaussian mixture approximation (GMA) based on a residual. For GMA, the clustering of ensemble samples provides the predictor of means, covariances and weights by smoothed expectation-maximization (SmEM). SmEM can overcome the singularity of covariance matrix for small ensemble size. Then the parameters of GMA are updated by an iterative ensemble smoother (IES). To enhance clustering efficiency, the ensemble samples for the clustering are updated by IES as well. Since the goal of inverse problems is to minimize the residual between the observation data and the model response, the adaptive GMA of the posterior is constructed through a residual threshold. The mixture components with large residuals will be discarded in the adaptive procedure. When the prior is incorporated into the likelihood model, small residuals can drive the AGMA close to the true posterior. In the proposed method, a large number of samples can be efficiently drawn from the posterior distribution of GMA. A few numerical examples are presented to demonstrate the efficacy of RD-AGMA with applications in multimodal inversion and channel identification for subsurface flow problems in porous media. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：The time distributed-order diffusion-wave equation describes radial groundwater flow to or from a well. In the paper, an alternating direction implicit (ADI) Legendre-Laguerre spectral scheme is proposed for the two-dimensional time distributed-order diffusion wave equation on a semi-infinite domain. The Gauss quadrature formula has a higher computational accuracy than the Composite Trapezoid formula and Composite Simpson formula, which is presented to approximate the distributed order time derivative so that the considered equation is transformed into a multi-term fractional equation. Moreover, the transformed equation is solved by discretizing in space by the ADI Legendre-Laguerre spectral scheme to avoid introducing the artificial boundary and in time using the weighted and shifted Grunwald-Letnikov difference (WSGD) method. A stability and convergence analysis is performed for the numerical approximation. Some numerical results are illustrated to justify the theoretical analysis. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：This paper proposes a class of randomized Kaczmarz algorithms for obtaining isolated solutions of large-scale well-posed or overdetermined nonlinear systems of equations. This type of algorithm improves the classic Newton method. Each iteration only needs to calculate one row of the Jacobian instead of the entire matrix, which greatly reduces the amount of calculation and storage. Therefore, these algorithms are called matrix-free algorithms. According to the different probability selection patterns of choosing a row of the Jacobian matrix, the nonlinear Kaczmarz (NK) algorithm, the nonlinear randomized Kaczmarz (NRK) algorithm and the nonlinear uniformly randomized Kaczmarz (NURK) algorithm are proposed. In addition, the NURK algorithm is similar to the stochastic gradient descent (SGD) algorithm in nonlinear optimization problems. The only difference is the choice of step size. In the case of noise-free data, theoretical analysis and the results of numerical based on the classical tangential cone conditions show that the algorithms proposed in this paper are superior to the SGD algorithm in terms of iterations and calculation time. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：We introduce a tamed exponential time integrator which exploits linear terms in both the drift and diffusion for Stochastic Differential Equations (SDEs) with a one sided globally Lipschitz drift term. Strong convergence of the proposed scheme is proved, exploiting the boundedness of the geometric Brownian motion (GBM) and we establish order 1 convergence for linear diffusion terms. In our implementation we illustrate the efficiency of the proposed scheme compared to existing fixed step methods and utilize it in an adaptive time stepping scheme. Furthermore we extend the method to nonlinear diffusion terms and show it remains competitive. The efficiency of these GBM based approaches is illustrated by considering some well-known SDE models. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：We study the problem of minimizing the first eigenvalue of the p-Laplacian operator for a two-phase material in a bounded open domain Omega subset of R-N, N >= 2 assuming that the amount of the best material is limited. We provide a relaxed formulation of the problem and prove some smoothness results for these solutions. As a consequence we show that if Omega is of class C-1,C-1, simply connected with connected boundary, then the unrelaxed problem has a solution if and only if Omega is a ball. We also provide an algorithm to approximate the solutions of the relaxed problem and perform some numerical simulations. (C) 2021 Elsevier B.V. All rights reserved.
Yilmaz, BilgiHekimoglu, A. AlperSelcuk-Kestel, A. Sevtap
15页
查看更多>>摘要：In this paper, a new approach, the Variance Gamma (VG) model, which is used to capture unexpected shocks (e.g., Covid-19) in housing markets, is proposed to contribute to the standard option-based mortgage valuation methods. Based on the VG model, the closed-form solutions are performed for pricing mortgage default and prepayment options. It solves the options pricing equations explicitly and illustrates numerical results for both mortgage default and prepayment options' prices. Furthermore, the study enables researchers to monitor the default probability of mortgagors. Analyzing the effect of risks on default and prepayment options using simulations shows that the VG model captures the systematic and systemic (idiosyncratic) risks of default and prepayment options prices with closed-form solutions and computes the mortgage default probabilities. Therefore, it allows lenders a more advanced decision process compared to the standard option-based mortgage valuation method. (C) 2021 Elsevier B.V. All rights reserved.
Yoon, Hyun ChulLee, SanghyunMallikarjunaiah, S. M.
21页
查看更多>>摘要：We investigate a quasi-static tensile fracture in nonlinear strain-limiting solids by coupling with the phase-field approach. A classical model for the growth of fractures in an elastic material is formulated in the framework of linear elasticity for deformation systems. This linear elastic fracture mechanics (LEFM) model is derived based on the assumption of small strain. However, the boundary value problem formulated within the LEFM and under traction-free boundary conditions predicts large singular crack-tip strains. Fundamentally, this result is directly in contradiction with the underlying assumption of small strain. In this work, we study a theoretical framework of nonlinear strain-limiting models, which are algebraic nonlinear relations between stress and strain. These models are consistent with the basic assumption of small strain. The advantage of such framework over the LEFM is that the strain remains bounded even if the crack-tip stress tends to the infinity. Then, employing the phase-field approach, the distinct predictions for tensile crack growth can be governed by the model. Several numerical examples to evaluate the efficacy and the performance of the model and numerical algorithms structured on finite element method are presented. Detailed comparisons of the strain, fracture energy with corresponding discrete propagation speed between the nonlinear strain-limiting model and the LEFM for the quasi-static tensile fracture are discussed. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：The Bernstein operators (BO) are not orthogonal, but they have duals, which are obtained by a linear combination of BO. In recent years dual BO have been adopted in computer graphics, computer aided geometric design, and numerical analysis. This paper presents a numerical method based on the Bernstein operational matrices to solve the time-space fractional convection-diffusion equation. A generalization of the derivative matrix operator of fractional order and the error analysis are discussed. Numerical examples compare the proposed approach with previous works, showing that the method is more accurate and efficient. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：Jacobian-free Newton-Krylov (JFNK) method is a popular approach to solve nonlinear algebraic equations arising from computational physics. The key issue is the calculation of Jacobian-vector product, commonly done through finite difference methods. However, these approaches suffer from both truncation error and round-off error, and the accuracy heavily depends on a sophisticated choice of the difference step size. In some extreme cases, even with the best choice of the difference step size, the accuracy may still not meet the requirement for the inner Krylov iteration. In this paper, we extend the complex step finite difference (CSFD) method to the JFNK method. Some tips are presented for accelerating the method. Multiple examples are presented to reveal the performance of the JFNK with the CSFD, and different methods for approximating the Jacobian-vector product are compared. It is demonstrated with a relatively easy way of implementation that the CSFD method is well-suited for the JFNK method, leading to extremely accurate and stable numerical performance. In strong contrast to traditional finite difference approaches, it frees us from the disturbing choice for the difference step size, and one can fully rely on the method without any accuracy concerns. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：The study of planar and spherical geometric subdivision schemes was done in Dyn and Hormann (2012); Bellaihou and Ikemakhen (2020). In this paper we complete this study by examining the hyperbolic case. We define general interpolatory geometric subdivision schemes generating curves on the hyperbolic plane by using geodesic polygons and the hyperbolic trigonometry. We show that a hyperbolic interpolatory geometric subdivision scheme is convergent if the sequence of maximum edge lengths is summable and the limit curve is G(1)-continuous if in addition the sequence of maximum angular defects is summable. In particular, we study the case of bisector interpolatory schemes. Some examples are given to demonstrate the properties of these schemes and some fascinating images on Poincare disk are produced from these schemes. (C) 2021 Elsevier B.V. All rights reserved.
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