• Abstract

Full-waveform inversion is a powerful geophysical imaging technique that infers high-resolution subsurface physical parameters by solving a non-convex optimization problem. However, due to limitations in observation, e.g. limited shots or receivers, and random noise, conventional inversion methods are confronted with numerous challenges, such as the local-minimum problem. In recent years, a substantial body of work has demonstrated that the integration of deep neural networks and partial differential equations for solving full-waveform inversion problems has shown promising performance. In this work, drawing inspiration from the expressive capacity of neural networks, we provide a new deep learning approach aimed at accurately reconstructing subsurface physical velocity parameters. This method is founded on a re-parametrization technique for Bayesian inference, achieved through a deep neural network with random weights. Notably, our proposed approach does not hinge upon the requirement of the labeled training dataset, rendering it exceedingly versatile and adaptable to diverse subsurface models. Furthermore, uncertainty analysis is effectively addressed through approximate Bayesian inference. Extensive experiments show that the proposed approach performs noticeably better than existing conventional inversion methods.

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• Abstract

Numerically solving multi-marginal optimal transport (MMOT) problems is computationally prohibitive, even for moderate-scale instances involving $l≥4$ marginals with support sizes of $N ≥ 1000.$ The cost in MMOT is represented as a tensor with $N^l$ elements. Even accessing each element once incurs a significant computational burden. In fact, many algorithms require direct computation of tensor-vector products, leading to a computational complexity of $\mathcal{O}(N^l)$ or beyond. In this paper, inspired by our previous work [Comm. Math. Sci., 20, 2022], we observe that the costly tensor-vector products in the Sinkhorn Algorithm can be computed with a recursive process by separating summations and dynamic programming. Based on this idea, we propose a fast tensor-vector product algorithm to solve the MMOT problem with $L^1$ cost, achieving a miraculous reduction in the computational cost of the entropy regularized solution to $\mathcal{O}(N).$ Numerical experiment results confirm such high performance of this novel method which can be several orders of magnitude faster than the original Sinkhorn algorithm.

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We use the quantum fluid analogy [Nazarenko et al., Phys. Rev. E, 92, 2015] for coherent structures (vortices or solitons) interacting among themselves and with the random wave component. This is performed for 2D defocusing media based on quantum fluid approximation of the two-dimensional nonlinear Schrödinger equation in the statistical frame. With this, the Lundgren-Monin-Novikov infinite chain of equations for the $n$-point density function $f_n$ for the vorticity field is used. The conformal group of symmetry transformations calculated [Grebenev et. al., Theor. Math. Phys., 217(2), 2023] is applied to implement several elements of a gauge theory in the conformal transformation optics. Finally, we demonstrate how to use the variational generalized Brenier principle [Brenier, J. Am. Math. Soc., 2, 1989] together with the conformal invariance of statistics to close the infinite chain of Landgren-Monin-Novikov equations.

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Biodiesel, a sustainable and renewable energy source, is a promising alternative to fossil fuels. The transesterification process of biodiesel production effectively captures memory effects in reaction kinetics. In this study, we developed two fractional order models of the chemical catalytic transesterification reaction to explore the memory effects of the reaction kinetics utilizing two different non-singular kernel methods: Caputo-Fabrizio and Atangana-Baleanu in the Caputo sense. We compared the results with experimental data of biodiesel production and demonstrated the existence and uniqueness of the solution for the fractional system. A sensitivity analysis is performed using the Latin hypercube sampling method to evaluate the impact of various parameters on biodiesel production, followed by the computation of partial rank correlation coefficients based on Pearson’s correlation coefficient. We exhibit the dynamic behavior of all reactants corresponding to these fractional models with the variation of fractional order and the memory rate parameter. Additionally, we display the memory effect through the surface plots for biodiesel production by varying fractional order, molar ratio, and ultrasound frequency. Our numerical comparison with experimental data identifies the fractional-order value for the best fit of biodiesel production and can be increased by applying optimal control on ultrasound frequency.

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In this paper, we propose and develop a novel hue-preserving color image restoration model. In the proposed model, saturation-value total variation is used as the regularization term. In order to reduce hue loss during regularization, the huepreserving technique is incorporated into the proposed energy functional. We then rewrite the proposed model equivalently into a three-block separable convex version and apply an alternating direction method of multipliers based algorithm to solve the proposed equivalent model numerically. We also give the convergence analysis of the proposed algorithm. Numerical examples are presented to demonstrate that the performance of the proposed restoration model is better than that of other testing methods in terms of visual quality and some criteria such as peak signal-to-noise ratio, structure similarity, and spatial-CIELAB color error.

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In this paper, we analyze the superconvergence properties for spectral interpolations by Hermite polynomials and mapped Hermite functions. At the superconvergence points, the $(N−k)$-th term in the Hermite spectral interpolation remainder for the $(k+1)$-th derivatives vanish. To solve multi-point weakly singular nonlocal problems, we previously introduced mapped Hermite functions (MHFs), which are constructed by applying a mapping to the Hermite polynomials. We prove that the superconvergence points of the spectral interpolations based on MHFs for the $(k+1)$-th derivatives are the zero points of the $(N−k)$-th term. Additionally, due to the rapid growth of the logarithmic function at the endpoints 0 and 1, we further propose generalized mapped Hermite functions (GMHFs). We develop basic approximation theory for these new orthogonal functions and prove the projection error and interpolation error in the $L^2$ -weighted space using the pseudo-derivative. We demonstrate that the superconvergence points of the spectral interpolations based on both MHFs and GMHFs for the $(k+1)$-th derivative are the zero points of the $(N−k)$-th term. Numerical experiments confirm our theoretical results.

• General Summary