Feasibility-based fixed point networks

Inverse problems consist of recovering a signal from a collection of noisy measurements. These problems can often be cast as feasibility problems; however, additional regularization is typically necessary to ensure accurate and stable recovery with respect to data perturbations. Hand-chosen analytic regularization can yield desirable theoretical guarantees, but such approaches have limited effectiveness recovering signals due to their inability to leverage large amounts of available data. To this end, this work fuses data-driven regularization and convex feasibility in a theoretically sound manner. This is accomplished using feasibility-based fixed point networks (F-FPNs). Each F-FPN defines a collection of nonexpansive operators, each of which is the composition of a projection-based operator and a data-driven regularization operator. Fixed point iteration is used to compute fixed points of these operators, and weights of the operators are tuned so that the fixed points closely represent available data. Numerical examples demonstrate performance increases by F-FPNs when compared to standard TV-based recovery methods for CT reconstruction and a comparable neural network based on algorithm unrolling. Codes are available on Github: github.com/howardheaton/feasibility_fixed_point_networks.

A New Conjugate Gradient Method and Application to Dynamic Load Identification Problems

In this paper, a modified conjugate gradient (MCG) algorithm is proposed for solving the force reconstruction problems in practical engineering. This new method is derived from a stable regularization operator and is also strictly proved using the mathematical theory. Moreover, we also prove the sufficient descent and global convergence characteristic of the newly developed algorithm. Finally, the proposed algorithm is applied to force reconstruction for the airfoil structure and composite laminated cylindrical shell. Numerical simulations show that the proposed method is highly efficient and has robust convergence performances. Additionally, the accuracy of the proposed algorithm in identifying the expected loads is satisfactory and acceptable in practical engineering.

Impulse noise treatment in magnetotelluric inversion

The problem of model recovering in the presence of impulse noise on the data is considered for the magnetotelluric (MT) inverse problem. The application of total variation regularization along with L1-norm penalized data fitting (TVL1) is the usual approach for the impulse noise treatment in image recovery. This combination works poorly when a high level of impulse noise is present on the data. A nonconvex operator named smoothly clipped absolute deviation (TVSCAD) was recently applied to the image recovery problem. This operator is solved using a sequence of TVL1 equivalent problems, providing a significant improvement over TVL1. In practice, TVSCAD requires the selection of several parameters, a task that can be very difficult to attain. A more simple approach to the presence of impulse noise in data is presented here. A nonconvex function is also considered in the data fitness operator, along with the total variation regularization operator. The nonconvex operator is solved by following a half-quadratic procedure of minimization. Results are presented for synthetic and also for field data, assessing the proposed algorithm’s capacity in model recovering under the influence of impulse noise on data for the MT problem.

Vector-Matrix Implementation of the Integrated Method of Recovery of Input Signals of Nonlinear Dynamic Systems

In the article the method of signal reconstruction at the input of nonlinear dynamic objects based on the application of vector-matrix approach to solving polynomial Volterra integral equations of the first kind of the second degree using a differential regularization operator is studied.

A lowest-order staggered DG method for the coupled Stokes–Darcy problem

In this paper we propose a locally conservative, lowest-order staggered discontinuous Galerkin method for the coupled Stokes–Darcy problem on general quadrilateral and polygonal meshes. This model is composed of Stokes flow in the fluid region and Darcy flow in the porous media region, coupling together through mass conservation, balance of normal forces and the Beavers–Joseph–Saffman condition. Stability of the proposed method is proved. A new regularization operator is constructed to show the discrete trace inequality. Optimal convergence estimates for all the approximations covering low regularity are achieved. Numerical experiments are given to illustrate the performances of the proposed method. The numerical results indicate that the proposed method can be flexibly applied to rough grids such as the trapezoidal grid and $h$-perturbation grid.

Prestack structurally constrained impedance inversion

Classical prestack impedance inversion methods are based on performing a common-depth point (CDP) by CDP inversion using Tikhonov-type regularization. We refer to it as lateral unconstrained inversion (1D-LUI). Prestack seismic data usually have a low signal-to-noise ratio, and the 1D-LUI approach is sensitive to noise. The inversion results can be noisy and lead to an unfocused transition between vertical formation boundaries. The lateral constrained inversion (1D-LCI) can suppress the noise and provide sharp boundaries between inverted 1D models in regions where the layer dips are less than 20°. However, in complex geology, the disadvantage of using the 1D-LC approach is the lateral smearing of the steeply dipping layers. We have developed a structurally constrained inversion (1D-SCI) approach to mitigate the smearing associated with 1D-LCI. SCI involves simultaneous inversion of all seismic CDPs using a regularization operator that forces the solution to honor the local structure. The results of the 1D-SCI were superior compared with the 1D-LUI and 1D-LCI approaches. The steeply dipping layers are clearly visible on the SCI inverted results.