Hardcopy Extraction Considerations for Vector Product Format (VPF) Products.

We develop a spectral theory for the equation (∇ + ieA) × u = ±mu on Minkowski 3-space (one time variable and two space variables); here, A is a real vector potential and the vector product is defined with respect to the Minkowski metric. This equation was formulated by Elton and Vassiliev, who conjectured that it should have properties similar to those of the two-dimensional Dirac equation. Our equation contains a large parameter c (speed of light), and this motivates the study of the asymptotic behaviour of its spectrum as c → +∞. We show that the essential spectrum of our equation is the same as that of Dirac (theorem 3.1), whereas the discrete spectrum agrees with Dirac to a relative accuracy δλ/mc2 ~ O(c−4) (theorem 3.3). In other words, we show that our equation has the same accuracy as the two-dimensional Pauli equation, its advantage over Pauli being relativistic invariance.

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DMRG Approach to Fast Linear Algebra in the TT-Format

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2011-0021 ◽

2011 ◽

Vol 11 (3) ◽

pp. 382-393 ◽

Cited By ~ 26

Author(s):

Ivan Oseledets

Keyword(s):

Linear Systems ◽

Linear Algebra ◽

Direct Method ◽

Fast Algorithms ◽

Approximate Solutions ◽

Vector Product ◽

The Matrix ◽

Minimization Scheme

AbstractIn this paper, the concept of the DMRG minimization scheme is extended to several important operations in the TT-format, like the matrix-by-vector product and the conversion from the canonical format to the TT-format. Fast algorithms are implemented and a stabilization scheme based on randomization is proposed. The comparison with the direct method is performed on a sequence of matrices and vectors coming as approximate solutions of linear systems in the TT-format. A generated example is provided to show that randomization is really needed in some cases. The matrices and vectors used are available from the author or at http://spring.inm.ras.ru/osel

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Parsimonious truncated Newton method for time-domain full waveform inversion based on Fourier-domain full-scattered-field approximation

Geophysics ◽

10.1190/geo2021-0164.1 ◽

2021 ◽

pp. 1-147

Author(s):

Peng Yong ◽

Romain Brossier ◽

Ludovic Métivier

Keyword(s):

Frequency Domain ◽

Time Domain ◽

Cross Correlation ◽

Waveform Inversion ◽

Field Approximation ◽

Second Order ◽

Vector Product ◽

First Order ◽

Full Waveform ◽

Truncated Newton

In order to exploit Hessian information in Full Waveform Inversion (FWI), the matrix-free truncated Newton method can be used. In such a method, Hessian-vector product computation is one of the major concerns due to the huge memory requirements and demanding computational cost. Using the adjoint-state method, the Hessian-vector product can be estimated by zero-lag cross-correlation of the first-order/second-order incident wavefields and the second-order/first-order adjoint wavefields. Different from the implementation in frequency-domain FWI, Hessian-vector product construction in the time domain becomes much more challenging as it is not affordable to store the entire time-dependent wavefields. The widely used wavefield recomputation strategy leads to computationally intensive tasks. We present an efficient alternative approach to computing the Hessian-vector product for time-domain FWI. In our method, discrete Fourier transform is applied to extract frequency-domain components of involved wavefields, which are used to compute wavefield cross-correlation in the frequency domain. This makes it possible to avoid reconstructing the first-order and second-order incident wavefields. In addition, a full-scattered-field approximation is proposed to efficiently simplify the second-order incident and adjoint wavefields computation, which enables us to refrain from repeatedly solving the first-order incident and adjoint equations for the second-order incident and adjoint wavefields (re)computation. With the proposed method, the computational time can be reduced by 70% and 80% in viscous media for Gauss-Newton and full-Newton Hessian-vector product construction, respectively. The effectiveness of our method is also verified in the frame of a 2D multi-parameter inversion, in which the proposed method almost reaches the same iterative convergence of the conventional time-domain implementation.

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