Variational Multiscale error estimator for anisotropic adaptive fluid mechanic simulations: Application to convection–diffusion problems
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In seismic waveform inversion using the adjoint operator, using the pseudo-Hessian matrix as a pre-conditioner can allow us to achieve computation efficiency. However, the former pseudo-Hessian matrices (i.e., the original and new pseudo-Hessian matrices) have a limitation to simulate the features of the approximate Hessian matrix for the deeper parts of given models, in particular in elastic waveform inversion. To compensate for this limitation, we propose the improved pseudo-Hessian matrix, which is obtained introducing an auxiliary matrix in the original pseudo-Hessian matrix. Comparing the values of the improved pseudo-Hessian matrix with those of the two former pseudo-Hessian matrices indicates that the improved pseudo-Hessian matrix is much closer to the approximate Hessian matrix. Inversion results for the elastic Marmousi-2 model show that the improved pseudo-Hessian matrix increases the convergence rate and yields better results, in particular for the deeper part.
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The authors show in this chapter, two different approaches for efficiently estimating the second-order derivatives (Hessian matrix) of a given objective function. The cost of evaluating the Hessian using classical finite difference approach is O(n2) where n is the number of parameters. The first adjoint approach reduces the cost of estimating all components of the Hessian matrix to only 2n extra simulations. This approach is simple, and it uses the algorithms developed in previous chapters. A second approach for estimating the complete Hessian is also presented. This approach is more complex than the first approach and requires extra memory storage. This approach requires only n + 1 extra simulations per Hessian evaluation. It follows that the computational cost is approximately one half of the first adjoint approach. This saving comes at the cost of a more complex algorithm and more extensive storage.
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Hessian matrices are important in inversion algorithms due to their potential of use in full Newton schemes, analysis of reconstruction results, and in experimental design. We present for the first time an adjoint based evaluation of the Hessian matrix for the SPN-approximation modeled forward operator in optical tomography. The Hessians so calculated are numerically validated with respect to finite difference calculations. We present comparisons between computational requirements of the present scheme with a mixed scheme which evaluates the Hessian as the first difference of the adjoint based Jacobians.
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We propose an improved Hessian estimation scheme for the second-order random directions stochastic approximation (2RDSA) algorithm [1]. The proposed scheme, inspired by [2], reduces the error in the Hessian estimate by (i) incorporating a zero-mean feedback term; and (ii) optimizing the step-sizes used in the Hessian recursion of 2RDSA.We prove that 2RDSA with our Hessian improvement scheme (2RDSA-IH) converges asymptotically to the true Hessian. The advantage with 2RDSA-IH is that it requires only 75% of the simulation cost per-iteration for 2SPSA with improved Hessian estimation (2SPSA-IH) [2]. Numerical experiments show that 2RDSA-IH outperforms both 2SPSA-IH and 2RDSA without the improved Hessian estimation scheme.
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In this paper, we present a family of generally applicable schemes for updating the Hessian from electronic structure calculations based on an equation derived with compact finite difference (CFD). The CFD-based equation is of higher accuracy than the quasi-Newton equation on which existing generally applicable Hessian update schemes are based. Direct tests of Hessian update schemes, as well as dynamics simulations using an integrator incorporating Hessian update schemes, have shown four of the new schemes produce reliably higher accuracy than existing Hessian update schemes.
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Implementation of the standard full waveform inversion (FWI) poses difficulties as the initial model offsets from the true model. The wavefield reconstruction inversion (WRI) was proposed to mitigate these difficulties by relaxing the waveequation constraint. In this abstract, working on the nonlinear term in the Hessian matrix of FWI, we develop a new approximate Hessian as an Augmented Gauss-Newton (AGN) Hessian including second-order derivative information. Moreover, we establish an intimate connection between an updating formula which results from approximate solve of the Newton's method with the AGN Hessian on the FWI problem and the WRI method. Our analysis opens new perspectives for developing efficient algorithms for FWI based on the Newton's method and highlights the importance of the nonlinear term in the Hessian matrix, which is ignored in most cases. Note: This paper was accepted into the Technical Program but was not presented at IMAGE 2022 in Houston, Texas.
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The truncated Newton method uses information contained in the exact Hessian in full-waveform inversion (FWI). The exact Hessian physically contains information regarding doubly scattered waves, especially prismatic events. These waves are mainly caused by the scattering at steeply dipping structures, such as salt flanks and vertical or nearly vertical faults. We have systematically investigated the properties and applications of the exact Hessian. We begin by giving the formulas for computing each term in the exact Hessian and numerically analyzing their characteristics. We show that the second term in the exact Hessian may be comparable in magnitude to the first term. In particular, when there are apparent doubly scattered waves in the observed data, the influence of the second term may be dominant in the exact Hessian and the second term cannot be neglected. Next, we adopt a migration/demigration approach to compute the Gauss-Newton-descent direction and the Newton-descent direction using the approximate Hessian and the exact Hessian, respectively. In addition, we determine from the forward and the inverse perspectives that the second term in the exact Hessian not only contributes to the use of doubly scattered waves, but it also compensates for the use of single-scattering waves in FWI. Finally, we use three numerical examples to prove that by considering the second term in the exact Hessian, the role of prismatic waves in the observed data can be effectively revealed and steeply dipping structures can be reconstructed with higher accuracy.
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Implementation of the standard full waveform inversion (FWI) poses difficulties as the initial model offsets from the true model. The wavefield reconstruction inversion (WRI) was proposed to mitigate these difficulties by relaxing the wave-equation constraint. In this abstract, working on the nonlinear term in the Hessian matrix of FWI, we develop a new approximate Hessian as an Augmented Gauss-Newton (AGN) Hessian including second-order derivative information. Moreover, we establish an intimate connection between an updating formula which results from approximate solve of the Newton's method with the AGN Hessian on the FWI problem and the WRI method. Our analysis opens new perspectives for developing efficient algorithms for FWI based on the Newton's method and highlights the importance of the nonlinear term in the Hessian matrix, which is ignored in most cases.
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