Inverse Formulation Research Articles

Sets in ℝd determining k taxicab distances

We address an analog of a problem introduced by Erd\H{o}s and Fishburn, itself an inverse formulation of the famous Erd\H{o}s distance problem, in which the usual Euclidean distance is replaced with the metric induced by the $\ell^1$-norm, commonly referred to as the $\textit{taxicab metric}$. Specifically, we investigate the following question: given $d,k\in \mathbb{N}$, what is the maximum size of a subset of $\mathbb{R}^d$ that determines at most $k$ distinct taxicab distances, and can all such optimal arrangements be classified? We completely resolve the question in dimension $d=2$, as well as the $k=1$ case in dimension $d=3$, and we also provide a full resolution in the general case under an additional hypothesis.

Preconditioner for estimation of multipole sources via full waveform inversion

Accurate representation and estimation of seismic sources is crucial to the joint medium-source full waveform inversion problem. We focus on the source estimation subproblem and its difficulties, where seismic sources are modeled by truncated series of multipoles. The source full waveform inversion formulation results in a highly ill-conditioned (potentially ill-posed) linear least squares problem which we attempt to solve iteratively via conjugate gradient. Our main contribution lies in developing a preconditioner to accelerate the performance of conjugate gradient on multipole source inversion. The proposed preconditioner consists of (fractional) time derivative/integral operators based on analytical solutions to the wave equation with multipole sources in an unbounded, homogeneous medium. Numerical results in 2D demonstrate that the conjugate gradient iterations are accelerated when incorporating the proposed preconditioning scheme.

Application of the Trefftz method in an original variational formulation to the analysis of the current transformer field

The previous papers by the author propose a new classification scheme of the boundary methods based on variational formulations. They contain initial numerical experiments conducted in order to compare nonsingular methods from the group generated by the original and inverse variational formulations. This paper is a continuation of that research and its aim is to apply one of the nonsingular Trefftz methods for modelling a magnetic field in a current transformer. It presents theoretical considerations and derivation of a mathematical model of the current transformer. A straightforward boundary approach analysed previously is now developed into a hybrid Trefftz method. Based on that, 2D numerical models are formulated and implemented in the Matlab environment. Some numerical tests are conducted in order to validate the performance of the method. Finally, some solutions concerning current transformer problems are presented. These results are compared to the ones obtained by Matlab's PDE Toolbox.

Reflection-Based Source Inversion for Sparse Imaging of Low-Loss Composite Panels

This article presents a bistatic reflection-based imaging technique for electromagnetic testing of conductor-backed composite panels, which is based on an inverse formulation of the numerical problem at hand, and technically extends a previously demonstrated transmission-based system. The technique exploits the a priori assumption of pixel-based sparsity and retrieves the final image using data from only a single measurement, thereby removing the necessity of a reference measurement to be conducted. To demonstrate the capability of the technique, retrieved images are presented for a synthetic proof-of-concept device under test and an industrially manufactured composite panel. The presented technique can be considered as the initial model in the development of more complex bistatic imaging systems and has been developed for the purpose of extending the opportunities to conduct reflection-based electromagnetic nondestructive testing on aircraft structural components.

Thermal Solutions for a Plate With an Arbitrary Temperature Transient on One Surface and Convection on the Other: Direct and Inverse Formulations

Abstract Thick plates that are thermally loaded on one surface with convection on the other are often encountered in engineering practice. Given this wide utility and the limitations of most existing solutions to an adiabatic boundary condition, generalized direct thermal solutions were first derived for an arbitrary surface loading as modeled by a polynomial and its coefficients on the loaded surface with convection on the other. Once formulated, the temperature solutions were then used with elasticity relationships to determine the resulting thermal stresses. Additionally, the inverse thermal problem was solved using a least-squares type determination of the aforementioned polynomial coefficients based on the direct-solution and temperatures measured at the surface with convection. Previously published relationships for a thick-walled cylinder with internal heating/cooling and external convection are also included for comparison. Given the versatility of the polynomial solutions advocated, the method appears well suited for complicated thermal scenarios provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties do not vary with temperature.

Second-order Volterra Filter based on DFP Technique

There have some problems caused by selection of improper parameters when LMS, NLMS or RLS algorithms are used to evaluate coefficients of a second-order Volterra filter. We propose a Davidon-Fletcher-Powell-based second-order Volterra filter (DFP-SOVF). The proposed filter is based on a posteriori error assumption and has a variable convergence factor. We give the recursive inverse auto-correlation matrix formulation and present computational complexity analysis. Then applying the proposed DFP-SOVF filter to single step predictions for pure Lorenz chaotic series, prediction results show that the DFP-SOVF filter can guarantee its convergence and stability and there have not divergence problems when using LMS and NLMS algorithms.

Inverse Design of Artificial Materials Based Lens Antennas through the Scattering Matrix Method

The design of spatially varying lens antennas based on artificial materials is of high interest for their wide range of applicability. In this paper, we propose a novel design procedure relying on an inverse formulation of the scattering matrix method (SMM). Differently from many adopted approaches, which resort to global optimizations or homogenization procedures, the inverse SMM (I-SMM) allows the synthesis of optimal parameters (geometrical and/or electromagnetic) for the inclusions realizing the overall device in a very effective manner. With reference to the 2D TM case, the proposed tool has been successfully assessed through the synthesis of different kinds of lenses radiating a pencil beam.

A multi-focusing contrast source Bayesian compressive method for solving inverse scattering problems

This work presents a novel inverse scattering (IS) methodology to deal wit.li the retrieval of the electromagnetic (EM) properties of unknown scat.t.erers. The proposed technique is based on the effective combination of a customized Bayesian compressive sensing (BCS) solver wit.li the iterative multi-scaling approach (IMSA). Accordingly, a-priori information on the class of imaged targets as well as progressively acquired information on their location and size is exploited to yield accurate and robust reconstructions. Moreover, a contrast source inversion (CSI) formulation is adopted in order to enable the retrieval of lion-Born scat.t.erers. Numerical results are shown to verify the effectiveness of the proposed IMSA-BCS-CSI method, as well as to compare it with state-of-the-art alternatives.

NDT/NDE by means of a probabilistic differential compressive sensing method

A novel microwave imaging (MI) method is proposed to deal with the nondestructive testing and evaluation (NDT/NDE) of dielectric structures. The proposed technique exploits a probabilistic differential contrast source inversion (D-CSI) formulation of the inverse scattering (IS) problem, which is then effectively solved by means of a customized multi-task Bayesian compressive sensing (MT-BCS) solver. Numerical results are shown to assess the effectiveness and robustness of the proposed NDT/NDE method, as well as to compare it. with a single-task (ST-BCS) implementation within the same framework.

Improved formulation of the latent variable model inversion–based optimization problem for quality by design applications

AbstractLatent variable regression model (LVRM) inversion is a relevant tool for finding, if they exist, different combinations of manufacturing conditions that yield the desired process outputs. Finding the best manufacturing conditions can be done by optimizing an appropriately formulated objective function using nonlinear programming. To this end, different formulations of the optimization problem based on LVRM inversion have been proposed in the literature that allow the use of happenstance data (eg, historical data) for this purpose, present lower computational costs than optimizing in the space of the original variables, and guarantee that the solution will conform to the correlation structure of available data from the past. However, these approaches, as presented, suffer from some limitations, such as having to actively modify the constraints imposed on the solution to achieve different sets of conditions to those available in the LVRM calibration dataset, or the lack of a standardized approach for optimizing a linear combination of variables. Furthermore, when minimizing or maximizing one or more outputs, a severe handicap is also present related to the definition of arbitrarily low or high “desired” values. This paper aims at tackling all of these issues. The resulting proposed formulation of the optimization problem is illustrated with three case studies.

A novel method for the identification of the unloaded configuration of a deformed hyperelastic body

ABSTRACT The unloaded configuration of a body without residual stresses is generally used as the starting point to compute the displacement field but may be unknown for some applications. In this paper, we propose a novel inverse formulation to identify the unloaded configuration of a deformed hyperelastic body using finite element based discretization schemes. The inverse problem consists of finding a domain that is geometrically defined by its boundary such that it deforms in a way consistent with the measurements taken on its deformed configuration. The coordinates of boundary nodes in the deformed configuration can be measured and are assumed to be known for the inverse problem. Since only the coordinates of boundary nodes in the unloaded configuration are updated in each step of the inverse analysis, re-meshing is essential to continue the optimization process. The Akin’s method is employed to map the mesh used for the unloaded configuration in the previous step for the updated one. Both numerical and experimental data sets are used in 2D/3D to demonstrate the effectiveness of the proposed inverse technique. This work could have practical applications in the design of elastomeric parts or to recover the unloaded configuration of soft tissues in vivo.

Taylor impact test revisited: Determination of plasticity parameters for metals at high strain rate

Taylor impact test is the simplest experimental approach for characterizing plastic behaviour of metals under high strain rate (up to 105 s−1). Herein, a cylindrical rod is impacted upon a rigid stationary anvil and the dynamic yield strength is estimated by comparing the experimentally obtained rod deformation with its theoretical counterpart. Despite being simple and potent concept, the lack of a comprehensive theoretical model (which relates the material constitutive with the deformation pattern of the rod) has so far restricted the use of the Taylor impact test in material characterization. In the present study the Taylor impact test is revisited and an attempt is made to formulate an inverse framework for efficient determination of constitutive parameters for metals. The inverse analysis is performed via the Extended Kalman Filter technique where the forward model is constituted based on the derivation given in Chakraborty et al. (2015). Unlike the common practice, for better estimation of material constants, the deformation time histories are also included in the present inverse formulation. For demonstration five different plasticity models viz Johnson–Cook model, Zerilli–Armstrong model, Steinberg–Cochran–Guinan–Lund model, Mechanical Threshold Stress model and Preston-Tonks-Wallace model are considered. The estimated parameters are found to be in good agreement with their reference values. It is shown that the most distinguished feature of the present attempt is that a single Taylor test is sufficient to determine the parameters with reasonable accuracy.

An inverse problem formulation of the immersed‐boundary method

AbstractWe formulate the immersed‐boundary method (IBM) as an inverse problem. A control variable is introduced on the boundary of a larger domain that encompasses the target domain. The optimal control is the one that minimizes the mismatch between the state and the desired boundary value along the immersed target‐domain boundary. We begin by investigating a naïve problem formulation that we show is ill‐posed: in the case of the Laplace equation, we prove that the solution is unique, but it fails to depend continuously on the data; for the linear advection equation, even solution uniqueness fails to hold. These issues are addressed by two complimentary strategies. The first strategy is to ensure that the enclosing domain tends to the true domain, as the mesh is refined. The second strategy is to include a specialized parameter‐free regularization that is based on penalizing the difference between the control and the state on the boundary. The proposed inverse IBM is applied to the diffusion, advection, and advection‐diffusion equations using a high‐order discontinuous Galerkin discretization. The numerical experiments demonstrate that the regularized scheme achieves optimal rates of convergence and that the reduced Hessian of the optimization problem has a bounded condition number, as the mesh is refined.

Intravoxel B0 inhomogeneity corrected reconstruction using a low-rank encoding operator.

To present a general and efficient method for macroscopic intravoxel inhomogeneity corrected reconstruction from multi-TE acquisitions. A signal encoding model for multi-TE gradient echo (GRE) acquisitions that incorporates 3D intravoxel field variations is derived, and a low-rank approximation to the encoding operator is introduced under piecewise linear assumption. The low-rank approximation enables very efficient computation and memory usage, and allows the proposed signal model to be integrated into general inverse problem formulations that are compatible with multi-coil and undersampling acquisitions as well as different regularization functions. Experimental multi-echo GRE data were acquired to evaluate the proposed method. Effective reduction of macroscopic intravoxel inhomogeneity induced artifacts was demonstrated. Improved estimation from the corrected reconstruction over standard Fourier reconstruction has also been obtained. The proposed method can effectively correct the effects of intravoxel inhomogeneity, and can be useful for various imaging applications involving GRE-based acquisitions, including fMRI, quantitative and susceptibility mapping, and MR spectroscopic imaging.

Robust wavefield inversion via phase retrieval

SUMMARY Extended formulation of full waveform inversion (FWI), called wavefield reconstruction inversion (WRI), offers potential benefits of decreasing the non-linearity of the inverse problem by replacing the explicit inverse of the wave-equation operator of classical FWI (the oscillating Green functions) with a suitably defined data-driven regularized inverse. This regularization relaxes the wave-equation constraint to reconstruct wavefields that match the data, hence mitigating the risk of cycle skipping. The subsurface model parameters are then updated in a direction that reduces these constraint violations. However, in the case of a rough initial model, the phase errors in the reconstructed wavefields may trap the waveform inversion in a local minimum leading to inaccurate subsurface models. In this paper, in order to avoid matching such incorrect phase information during the early WRI iterations, we design a new cost function based upon phase retrieval, namely a process which seeks to reconstruct a signal from the amplitude of linear measurements. This new formulation, called wavefield inversion with phase retrieval (WIPR), further improves the robustness of the parameter estimation subproblem by a suitable phase correction. We implement the resulting WIPR problem with an alternating-direction approach, which combines the majorization-minimization (MM) algorithm to linearise the phase-retrieval term and a variable splitting technique based upon the alternating direction method of multipliers (ADMM). This new workflow equipped with Tikhonov-Total variation (TT) regularization, which is the combination of second-order Tikhonov and total variation regularizations and bound constraints, successfully reconstructs the 2004 BP salt model from a sparse fixed-spread acquisition using a 3 Hz starting frequency and a homogeneous initial velocity model.

A theory-guided deep-learning formulation and optimization of seismic waveform inversion

Deep-learning techniques appear to be poised to play very important roles in our processing flows for inversion and interpretation of seismic data. The most successful seismic applications of these complex pattern-identifying networks will, presumably, be those that also leverage the deterministic physical models on which we normally base our seismic interpretations. If this is true, algorithms belonging to theory-guided data science, whose aim is roughly this, will have particular applicability in our field. We have developed a theory-designed recurrent neural network (RNN) that allows single- and multidimensional scalar acoustic seismic forward-modeling problems to be set up in terms of its forward propagation. We find that training such a network and updating its weights using measured seismic data then amounts to a solution of the seismic inverse problem and is equivalent to gradient-based seismic full-waveform inversion (FWI). By refining these RNNs in terms of optimization method and learning rate, comparisons are made between standard deep-learning optimization and nonlinear conjugate gradient and limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimized algorithms. Our numerical analysis indicates that adaptive moment (or Adam) optimization with a learning rate set to match the magnitudes of standard FWI updates appears to produce the most stable and well-behaved waveform inversion results, which is reconfirmed by a multidimensional 2D Marmousi experiment. Future waveform RNNs, with additional degrees of freedom, may allow optimal wave propagation rules to be solved for at the same time as medium properties, reducing modeling errors.

Simultaneous Estimation of Pr and Ra in a Natural Convective Flow Using Inverse Technique

Abstract The present research proposes an inverse natural convection algorithm for simultaneous estimation of correct values of Prandtl number (Pr) and Rayleigh number (Ra). The inverse problem is formulated as the minimization of appropriate functional and is solved iteratively using conjugate gradient algorithm. Since the inverse technique requires temperature data as the input parameter, this work uses schlieren experiments to measure the temperature in a water-filled, differentially heated, cubic cavity. While the proposed inverse technique may be applied for a variety of thermofluidic problems, the simulations are restricted to two-dimensional (2D), laminar-free convective flows in a 50 × 50 × 50 mm3 cavity. The present experiments restrict the wall temperature difference to 6.3 K, allowing the use of Boussinesq approximation in the inverse formulation. The proposed inverse algorithm is found to be highly accurate with an estimation error less than 10% when the measurement data contain about 5% noise.

Improved Primal–Dual Interior-Point Method Using the Lawson-Norm for Inverse Problems

Electrical impedance tomography (EIT), geophysics and undersea target reconstruction are typical non-linear ill-posed inverse problems, and in many cases, the anomalous bodies feature with a clear boundary. Thus, it is suitable to obtain sharp boundaries and blocky features with the Total Variation (TV) functional regularization. However, the TV function regularization leads to a non-differentiable objective function at zero in the inverse formulation and reduces the algorithm robustness. In this paper, we propose an improved primal-dual interior-point method (PD-IPM) based on the Lawson norm to get sharp spatial profiles of the anomalous bodies. Furthermore, the impact of the smooth parameter is investigated to get the inverse results. Numerical experiment using simulated data is setup to support our claim.

Dynamic Modeling and Performance Analysis of a New Redundant Parallel Rehabilitation Robot

The inverse dynamic model and performance analysis of a new redundant parallel rehabilitation robot are presented in this paper. First, the kinematics of each part is analyzed based on a closed-loop vector chain method. Then, the dynamic model of each part is formulated based on the D’Alembert principle and Euler’s equation. After that, the inverse dynamic formulation of the new redundant parallel robot is established by utilizing the principle of virtual work and the concept of linked Jacobian matrices. To validate the approach, dynamic simulations are implemented in ADAMS and MATLAB. The results demonstrate the correctness of this dynamic formulation. Then, the actuating forces are optimized by utilizing the weighted Moore-Penrose generalized inverse method, which determines the minimum norm of the least quadratic sum among the possible actuating force vectors. Finally, two novel dynamic performance indices are defined. The first index reflects the coupling effect of other neighboring limbs to the dominant limb; the second reflects the coupling effect of each neighboring limb to the dominant limb. The proposed approach can be used for the dynamic performance analysis in other types of redundant parallel robots and provides a reference for dynamic control strategies.

Joint Acoustic Localization and Dereverberation Through Plane Wave Decomposition and Sparse Regularization

Acoustic source localization and dereverberation are formulated jointly as an inverse problem. The inverse problem consists of the approximation of the sound field measured by a set of microphones. The recorded sound pressure is matched with that of a particular acoustic model based on a collection of plane waves arriving from different directions at the microphone positions. In order to achieve meaningful results, spatial and spatio-spectral sparsity can be promoted in the weight signals controlling the plane waves. The large-scale optimization problem resulting from the inverse problem formulation is solved using a first order optimization algorithm combined with a weighted overlap-add procedure. It is shown that once the weight signals capable of effectively approximating the sound field are obtained, they can be readily used to localize a moving sound source in terms of direction of arrival (DOA) and to perform dereverberation in a highly reverberant environment. Results from simulation experiments and from real measurements show that the proposed algorithm is robust against both localized and diffuse noise exhibiting a noise reduction in the dereverberated signals.