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This paper provides a computational framework for stochastic shape optimization problems. The primary control objective involves minimizing the expected value of a tracking cost function while adhering to constraints posed by a stochastic elliptic interface equation. This study incorporates stochastic shape variation and shape derivatives, enabling the establishment of a diminishing sequence of admissible interfaces. The finite element method is employed for discretizing both state and adjoint systems, yielding mesh displacement directions. Notably, the mesh manipulation technique utilized within the optimization process is subject to regularization through cubic spline interpolation employing a centripetal scheme. To ensure the fidelity of the interface curve and prevent undesirable twisting, the convex hull technique is incorporated, thus upholding mesh displacement consistency. To mitigate the computational demands associated with uncertainty quantification, the sparse grid collocation method is enlisted. This technique facilitates the matching of probability distributions, particularly in scenarios involving relatively high-dimensional problems. Moreover, for extensive-scale sampling optimization, a stochastic sampling-based descent method is integrated. The effectiveness and efficiency of the proposed algorithms are underscored through a series of numerical experiments, which serve to validate their performance. [ABSTRACT FROM AUTHOR]

• A short-memory operator splitting scheme for constant-Q wave equation is proposed. • The short memory principle is based on the scaling in the Laguerre spectral method. • A solid theoretical analysis of the Yuan-Agrawal method is provided. • An efficient algorithm for calculating the Mainardi function is given. We propose a short-memory operator splitting scheme for solving the constant-Q wave equation, where the fractional stress-strain relation contains multiple Caputo fractional derivatives with order much smaller than 1. The key is to exploit its extension problem by converting the flat singular kernels into strongly localized ones, so that the major contribution of weakly singular integrals over a semi-infinite interval can be captured by a few Laguerre functions with proper asymptotic behavior. Despite its success in reducing both memory requirement and arithmetic complexity, we show that numerical accuracy under prescribed memory variables may deteriorate in time due to the dynamical increments of projection errors. Fortunately, it can be considerably alleviated by introducing a suitable scaling factor β > 1 and pushing the collocation points closer to origin. An operator splitting scheme is introduced to solve the resulting set of equations, where the auxiliary dynamics can be solved exactly, so that it gets rid of the numerical stiffness and discretization errors. Numerical experiments on both 1-D diffusive wave equation and 2-D constant-Q P - and S -wave equations are presented to validate the accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

The requirement for high computational efficiency in inversion imaging poses challenges to further enhancing the accuracy of imaging large elastic and viscoelastic models. Considering that forward modeling serves as a prerequisite for inversion imaging, it is essential to introduce a wave propagation simulation method that can enhance simulation accuracy without significantly compromising computational speed. In our paper, cubic B-spline is introduced into the forward modeling of the wave equation. The B-spline collocation method offers a linear complexity for computation and exhibits fourth-order convergence in terms of computational errors. We also find that it can perfectly adapt to the boundary conditions of the perfectly matched layer in the forward modeling solution of the wave simulation. Several typical numerical examples, including 2-D acoustic wave equation, 2-D elastic wave propagation, and 2-D viscoelastic propagation in homogeneous media, are provided to validate its convergence, accuracy, and computational efficiency. • The B-spline collocation method (BSCM) is applied for solving wave equations in geophysics. • Unconditional stability of BSCM is proved and its numerical dispersion curves are provided. • Suitability of BSCM for ADE-PML absorption boundary is confirmed. [ABSTRACT FROM AUTHOR]