Least-squares solutions of matrix inverse problem for bi-symmetric matrices with a submatrix constraint

2007 ◽  
Vol 14 (5) ◽  
pp. 425-444 ◽  
Author(s):  
An-ping Liao ◽  
Yuan Lei
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Symmetric Matrices ◽  
Matrix Inverse

Matrix inverse problem and its optimal approximation problem for R-skew symmetric matrices

2010 ◽  
Vol 216 (12) ◽  
pp. 3515-3521
Author(s):  
Guang-Xin Huang ◽  
Feng Yin ◽  
Hua-Fu Chen ◽  
Ling Chen ◽  
Ke Guo
Keyword(s):  
Inverse Problem ◽  
Approximation Problem ◽  
Optimal Approximation ◽  
Symmetric Matrices ◽  
Matrix Inverse

scholarly journals An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology

2021 ◽  
Author(s):  
Karl Kunisch ◽  
Philip Trautmann
Keyword(s):  
Inverse Problem ◽  
Least Squares ◽  
Eikonal Equation ◽  
State Equation ◽  
High Accuracy ◽  
Well Posedness ◽  
Numerical Examples ◽  
Marquardt Method ◽  
Math Biol ◽  
Levenberg Marquardt

AbstractIn this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

Angle-domain common-image gathers from plane-wave least-squares reverse time migration

Geophysics ◽  
2021 ◽  
pp. 1-60
Author(s):  
Chuang Li ◽  
Zhaoqi Gao ◽  
Jinghuai Gao ◽  
Feipeng Li ◽  
Tao Yang
Keyword(s):  
Inverse Problem ◽  
Plane Wave ◽  
Least Squares ◽  
Migration Velocity ◽  
Velocity Analysis ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Migration Velocity Analysis ◽  
Amplitude Versus

Angle-domain common-image gathers (ADCIGs) that can be used for migration velocity analysis and amplitude versus angle analysis are important for seismic exploration. However, because of limited acquisition geometry and seismic frequency band, the ADCIGs extracted by reverse time migration (RTM) suffer from illumination gaps, migration artifacts, and low resolution. We have developed a reflection angle-domain pseudo-extended plane-wave least-squares RTM method for obtaining high-quality ADCIGs. We build the mapping relations between the ADCIGs and the plane-wave sections using an angle-domain pseudo-extended Born modeling operator and an adjoint operator, based on which we formulate the extraction of ADCIGs as an inverse problem. The inverse problem is iteratively solved by a preconditioned stochastic conjugate gradient method, allowing for reduction in computational cost by migrating only a subset instead of the whole dataset and improving image quality thanks to preconditioners. Numerical tests on synthetic and field data verify that the proposed method can compensate for illumination gaps, suppress migration artifacts, and improve resolution of the ADCIGs and the stacked images. Therefore, compared with RTM, the proposed method provides a more reliable input for migration velocity analysis and amplitude versus angle analysis. Moreover, it also provides much better stacked images for seismic interpretation.

Periodic plane-wave least-squares reverse time migration for diffractions

Geophysics ◽  
2020 ◽  
Vol 85 (4) ◽  
pp. S185-S198
Author(s):  
Chuang Li ◽  
Jinghuai Gao ◽  
Zhaoqi Gao ◽  
Rongrong Wang ◽  
Tao Yang
Keyword(s):  
Inverse Problem ◽  
Image Quality ◽  
Plane Wave ◽  
Least Squares ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Diffraction Imaging ◽  
Periodic Plane

Diffraction imaging is important for high-resolution characterization of small subsurface heterogeneities. However, due to geometry limitations and noise distortion, conventional diffraction imaging methods may produce low-quality images. We have adopted a periodic plane-wave least-squares reverse time migration method for diffractions to improve the image quality of heterogeneities. The method reformulates diffraction imaging as an inverse problem using the Born modeling operator and its adjoint operator derived in the periodic plane-wave domain. The inverse problem is implemented for diffractions separated by a plane-wave destruction filter from the periodic plane-wave sections. Because the plane-wave destruction filter may fail to eliminate hyperbolic reflections and noise, we adopt a hyperbolic misfit function to minimize a weighted residual using an iteratively reweighted least-squares algorithm and thereby reduce residual reflections and noise. Synthetic and field data tests show that the adopted method can significantly improve the image quality of subsalt and deep heterogeneities. Compared with reverse time migration, it produces better images with fewer artifacts, higher resolution, and more balanced amplitude. Therefore, the adopted method can accurately characterize small heterogeneities and provide a reliable input for seismic interpretation in the prediction of hydrocarbon reservoirs.

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