Formulation Of Inverse Problem Research Articles

One-dimensional model of heat-recovery, non-recovery coke ovens Part V: Coking-bed sub-model using an inverse procedure

A one-dimensional home-made mathematical model of coke-making process in the heat-recovery (HR)/non-recovery (NR) coke oven has been developed. The model includes a series of sub-models described in associated publications (Buczynski et al., 2016 [1–4]). In this paper (Part V) a different approach for predicting the carbonization process is proposed. The coking-bed sub-model presented in Buczynski et al. (2016) [2] is based on a direct procedure (forward calculations) and the moving boundary technique. This approach is difficult to implement and calculations are time consuming. Moreover, the model is valid for one particular coal blend only. This paper describes the coking-bed sub-model based on an inverse problem formulation that adjusts itself to represent carbonization of particular charge. The new coking-bed sub-model contains completely different algorithm for predicting time-dependent processes such as: moisture evaporation and condensation, formation and condensation of tars.

Information content in data sets: A review of methods for interrogation and model comparison

Abstract In this review we discuss methodology to ascertain the amount of information in given data sets with respect to determination of model parameters with desired levels of uncertainty. We do this in the context of least squares (ordinary, weighted, iterative reweighted weighted or “generalized”, etc.) based inverse problem formulations. The ideas are illustrated with several examples of interest in the biological and environmental sciences.

Minimization Based Formulations of Inverse Problems and Their Regularization

The conventional way of formulating inverse problems such as identification of a (possibly infinite dimensional) parameter is via some forward operator, which is the concatenation of the observation operator with the parameter-to-state-map for the underlying model. Recently, all-at-once formulations have been considered as an alternative to this reduced formulation, avoiding the use of a parameter-to-state map, which would sometimes lead to overly restrictive conditions. Here the model and the observation are considered simultaneously as one large system, with the state and the parameter as unknowns. A still more general formulation of inverse problems, containing not only the reduced and all-at-once formulations but also the well-known and highly versatile variational approach (also called the Kohn--Vogelius functional approach) as special cases, is to formulate the inverse problem as a minimization problem---instead of an equation---for the state and parameter. Regularization can be incorporated via imp...

MISFIT LANDFORMS IMPOSED BY ILL-CONDITIONED INVERSE PARAMETRIC PROBLEMS

In the paper, we put forward a new topological taxonomy which allows to distinguish and separate multiple solutions to the ill-conditioned parametric inverse problems appearing in engineering, geophysics, medicine, etc. This taxonomy distinguishes the areas of insensitivity to parameters, called landforms of the misfit landscape, be it around minima (lowlands), maxima (uplands), or stationary points (shelves). We proved their important separability and completeness conditions. In particular, lowlands, uplands and shelves are pairwise disjoint and there are no other subsets of the positive measure in the admissible domain on which misfit function takes a constant value. The topological taxonomy is related to the second, 'local' one, which characterizes the types of ill-conditioning of the particular solutions. We hope that the proposed results will be helpful for a better and more precise formulation of the ill-conditioned inverse problems and for selecting and profiling complex optimization strategies used to solve these problems.

Analytical Study of Algorithms for Solving Inverse Kinematic Problems in Robot Motion Control

The given article covers the general formulations of inverse kinematic problems for robot motion control systems. We have discussed the difficulties how to solve such problems using analytical and numerical methods. We have also analyzed the convergence of iterative algorithms with the regularization on the trajectory with the points outside of the gripper reachability. The example of an iterative calculation of joint trajectories for a 3-link robot using the recursive algorithm for the Jacobi matrix calculation has been presented.

Evaluation of changes in thermodiffusion properties of mineral wool resulting from treatment with water and re-drying

The work presents the formulation of inverse problem that allows to estimate the basic parameters describing coupled thermodiffusion of moisture in porous fibrous material and construction of the measuring stand in order to obtain the necessary data to carry out calculations for this purpose. This approach was used to evaluate the changes in the thermodiffusion properties of stone wool samples for the indoor applications that were treated with water and re-dried. In the mathematical model, one introduced a simplification that coupling in the thermodiffusion process is unilateral and includes only the influence of heat transport on moisture transport.

Material‐based process‐chain optimization in metal forming

AbstractThe characteristics of a manufacturing product are influenced by a variety of different factors, such as the material properties of the base product. The prediction of properties that give optimal results in metal forming applications is a complex task but of high interest for the manufacturer. To realize such a prediction scheme, the process chain is split up into individual process steps and for each of them an inverse modeling is required. The specific aim of this work is to present an approach for the inverse problem formulation of a process step and to solve it using methods of machine learning. Moreover, the challenges that often arise due to the ill‐posed nature of inverse problems will be discussed. The main focus is on the crystallographic texture of metals, which strongly affects the deformation behavior during a process step and highly influences the characteristics of the final product. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Inverse Problem of Ultrasound Beamforming With Sparsity Constraints and Regularization.

Ultrasound (US) beamforming is the process of reconstructing an image from acquired echo traces on several transducer elements. Typical beamforming approaches, such as delay-and-sum, perform simple projection operations, while techniques using statistical information also exist, e.g., adaptive, phase coherence, delay-multiply-and-sum, and sparse coding approaches. Inspired by the feasibility and success of inverse problem (IP) formulations in several image reconstruction problems, such as computed tomography, we herein devise an IP approach for US beamforming. We define a linear forward model for the synthesis of the beamformed image, and solve its IP thanks to several intuitive and physics-based constraints and regularization terms proposed. These reflect the prior knowledge about the spectra of carrier signal and spatial coherence of modulated signal. These constraints admit effective formulation through sparse representations. Our proposed method was evaluated for plane-wave imaging (PWI) that transmits unfocused waves, enabling high frame rates with large field of view at the expense of much lower image quality with conventional beamforming techniques. Results are evaluated in numerical simulations, as well as tissue-mimicking phantoms and in vivo data provided by PWI challenge in medical US. The best results achieved by our proposed techniques are 0.39-mm full-width at half-maximum for spatial resolution and 16.3-dB contrast-to-noise ratio, using a single plane-wave transmit.

Etiology of ankle osteoarthritis in an Asian population

The paper presents a seismic data interpretation approach to reconstruction of stress fields in rocks at any stage of coal mining. The approach involves: thee-axial compression testing of coal in order to find an empirical relation between elastic wave velocity and stresses; coal bed tomography using standard monitoring systems and taking signals generated by dynamic events (with the energy higher than the background level) as the probing signals; formulation and solving of a boundary inverse problem within a geomechanical model of an object under analysis where the input data is the velocity field reconstructed in the illuminated area of coal bed by tomography. The laboratory data of the coal specimens tested by the von Karman scheme have been approximated to plot analytical relationships between P-waves velocity, axial stress and confining pressure. The numerical experiments performed for a typical underground layout at coal producer “Vorkutaugol” have shown that given the accepted array of seismic pickups, the single-valued solution of the inverse problem requires good illumination of coal bed areas with the spatial stress gradient.

Joint sparse and low rank recovery algorithm for compressive hyperspectral imaging.

Compressive spectral imaging techniques encode and disperse a hyperspectral image (HSI) to sense its spatial and spectral information with few bidimensional (2D) multiplexed projections. Recovering the original HSI from the 2D projections is carried by traditional compressive sensing-based techniques that exploit the sparsity property of natural HSI as they are represented in a proper orthonormal basis. Nevertheless, HSIs also exhibit a low rank property inasmuch only a few numbers of spectral signatures are present in the images. Specifically, when an HSI is rearranged as a matrix whose columns represent vectorized 2D spatial images in a different wavelength, this matrix is said to be low rank. Therefore, this paper proposes an HSI recovering algorithm from compressed measurements involving a joint sparse and low rank optimization problem, which seeks to jointly minimize the ℓ2-, ℓ1-, and ℓ*-norm, leading the solution to fit the given projections, and be simultaneously sparse and low rank. Several simulations, along different data sets and optical sensing architectures, show that when the low rank property is included in the inverse problem formulation, the reconstruction quality increases up to four (dB) in terms of peak signal to noise ratio.

Data-driven Estimation of Origin-Destination Demand and User Cost Functions for the Optimization of Transportation Networks

In earlier work (Zhang et al., 2016) we used actual traffic data from the Eastern Massachusetts transportation network in the form of spatial average speeds and road segment flow capacities in order to estimate Origin-Destination (OD) flow demand matrices for the network. Based on a Traffic Assignment Problem (TAP) formulation (termed “forward problem”), in this paper we use a scheme similar to our earlier work to estimate initial OD demand matrices and then propose a new inverse problem formulation in order to estimate user cost functions. This new formulation allows us to efficiently overcome numerical difficulties that limited our prior work to relatively small subnetworks and, assuming the cost functions are available, to adjust the values of the OD demands accordingly so that the flow observations are as close as possible to the solutions of the forward problem. Finally, using the same actual traffic data from the Eastern Massachusetts transportation network, we quantify the Price of Anarchy (PoA) for a much larger network than that in Zhang et al. (2016).

Uncertainty Quantification in Waveform-based Imaging Methods - a Sleipner CO2 Monitoring Study

Full Waveform Inversion (FWI) and Controlled Source Electro-Magnetic (CSEM) inversion can provide quantitative estimates of selected subsurface properties. In the context of CO2 monitoring, FWI has the capability to locate and image a CO2 plume, seen as velocity changes caused by the injected CO2, while CSEM can be useful to determine the CO2 saturation. Quantifying the uncertainty of the reconstructed parameters is however challenging. In this study, preliminary work has been performed to assess the reliability of the results obtained using CSEM and FWI when monitoring CO2 storage at the Sleipner field offshore Norway. This paper begins with a brief review of the imaging techniques used. Then the uncertainty quantification methodology is described. Finally, preliminary results of uncertainty quantification for two CO2 monitoring examples, one synthetic case using CSEM and one real data case using FWI, are presented. For the synthetic CSEM case, the inversion of the EM data has a very positive effect on the uncertainty, clearly reducing the covariance of the probability distribution for possible conductivity models as compared to the prior knowledge of these models. For the real, rather large-scale FWI case, computationally efficient strategies had to be used. The covariance in this case is not reduced to the same extent as for the CSEM case, but an improvement is observed. In both cases, the quantification approach works as expected providing a means to assess the quality of obtained subsurface models. For future studies, a more formal way of providing prior model knowledge and a stricter formulation of the Bayesian inverse problem will be employed.

Airborne gamma-ray spectrometry data processing using 1.5D inversion

Standard processing of Airborne Gamma-Ray Spectrometry data generally gives good results when the measurement conditions are almost constant within several footprint area sizes, with the possible exception of flight height variations in a small range. In practice, deviations, such as large or abrupt changes of flight height and/or rugged terrain are not so rare and lead to certain problems. This article proposes a different approach where the solutions of inverse problems are used for data processing. The approach is quite natural in the processing of field data measured along the flight lines: it explicitly takes into account 1.5D survey models and flight parameters – from topography to sources distribution on the surface. Also, it clearly demonstrates that the inverse problem of the Airborne Gamma-Ray Spectrometry does not have a unique solution. This feature can be used in accordance with the underlying geological problem since various formulations of inverse problems can lead to various geological solutions. The use of the approach is illustrated by several examples given for flight lines and survey areas. This approach can be particularly useful in situations where geological, geophysical and/or geographic survey conditions are far from the standard assumptions.

The Bayesian formulation and well-posedness of fractional elliptic inverse problems

We study the inverse problem of recovering the order and the diffusion coefficient of an elliptic fractional partial differential equation from a finite number of noisy observations of the solution. We work in a Bayesian framework and show conditions under which the posterior distribution is given by a change of measure from the prior. Moreover, we show well-posedness of the inverse problem, in the sense that small perturbations of the observed solution lead to small Hellinger perturbations of the associated posterior measures. We thus provide a mathematical foundation to the Bayesian learning of the order—and other inputs—of fractional models.

AIC under the framework of least squares estimation

Abstract In this note we explain the use of the Akiake Information Criterion and its related model comparison indices (usually derived for maximum likelihood estimator inverse problem formulations) in the context of least squares (ordinary, weighted, iterative weighted or “generalized”, etc.) based inverse problem formulations. The ideas are illustrated with several examples of interest in biology.

Estimating the rate of prion aggregate amplification in yeast with a generation and structured population model

Prions are a special class of proteins capable of adopting multiple (misfolded) conformations, some of which have been associated with fatal diseases in mammals such as bovine spongiform encephalopathy or Creutzfeldt–Jakob Disease. Prion diseases, like protein misfolding diseases in general, are caused by the formation and amplification of ordered aggregates of proteins called amyloids. While such diseases in mammals can take decades to form, yeast have a variety of prion phenotypes that occur over a few hours, making this system an ideal model for protein misfolding disease in general. Most experimental assays of colonies with yeast prions provide steady-state population observations which complicate the inference of biochemical parameters both by the inability to directly measure aggregate amplification and by obscuring heterogeneity between cells. We develop a mathematical and inverse problem formulation to determine the amplification rate with prion aggregates from single-cell measurements observed in propagon amplification experiments. We demonstrate the ability of our formulation to determine heterogeneous amplification rates on simulated and experimental data. Our results show that aggregate amplification rates for two prion variants are strongly bimodal, suggesting that the generational structure in the yeast population impacts the ability of prion aggregates to amplify.

Designing pairwise interactions that stabilize open crystals: Truncated square and truncated hexagonal lattices.

Using a recently introduced formulation of the ground-state inverse design problem for a targeted lattice [W. Piñeros et al., J. Chem. Phys. 144, 084502 (2016)], we discover purely repulsive and isotropic pair interactions that stabilize low-density truncated square and truncated hexagonal crystals, as well as promote their assembly in Monte Carlo simulations upon isochoric cooling from a high-temperature fluid phase. The results illustrate that the primary challenge to stabilizing very open two-dimensional lattices is to design interactions that can favor the target structure over competing stripe microphases.

Regularizing Biomechanical Maps for Partially Known Material Properties

We present the solution of the inverse problem for partially known elastic modulus values, e.g., the elastic modulus is known in some small region on the boundary of the domain from measurements. The inverse problem is posed as a constrained minimization problem and regularized with two different regularization types. In particular, the total variation diminishing (TVD) and the total contrast diminishing (TCD) regularizations are employed. We test both regularization strategies with theoretical diseased tissues, such as a stiff tumor surrounded by healthy background tissue and an atherosclerotic plaque having a soft inclusion surrounded by a stiff cap. In the present study, it is assumed that no traction data is available and the absolute elastic modulus distribution is calibrated from partially known elastic modulus values. We observe that this calibration fails with TVD regularization, while TCD regularization yields well-recovered absolute elastic modulus reconstructions in the presence of high noise levels in the displacement data. Finally, we investigate this problem analytically and provide an explanation for these observations. This work will advance efforts in parameter identification of heterogeneous materials as it provides a methodology to incorporate partially known parameters into the inverse problem formulation that will ultimately drive the inverse solution to an absolute and unique parameter distribution. This has great importance in classifying breast tumors based on their elastic modulus values and in planning surgical interventions of atherosclerotic plaques.

Determination of two unknowns simultaneously for stochastic Euler–Bernoulli beam equations

In this paper, we are concerned with inverse problems with two unknowns for stochastic dynamic Euler–Bernoulli beam equations. By some suitable observations, one can determine the random force intensity and the initial velocity uniquely. The uniqueness result is proved by means of global Carleman estimates for the stochastic beam equations. It shows that the formulation of inverse problems for the stochastic Euler–Bernoulli beam equations is different from the deterministic counterpart.

Stochastic inversion for the roll gyradius second moment mass property in ships at full-scale and model-scale

In current Naval Architecture practice, employing static considerations is an important and necessary step in assessing ship stability and seakeeping properties (e.g. inclining experiments, load line regulations, range of stability calculations). However, damaged vessels and vessels operating in heavy weather or in conditions where topside icing is a concern may require an additional assessment of stability that considers dynamic effects. Within such contexts, the actual (i.e. current) second moment properties of the vessel mass become very important in the associated equations of motion for a given ship. One such critical second mass moment property is the roll gyradius, as it is closely related to the occurrence of capsizing. The present paper furnishes a means for reckoning the actual roll gyradius of a given ship operating within a seaway. The approach hinges on the formulation and solution of a stochastic inverse problem that leverages existing seakeeping software against the shipboard inertial measurement unit (IMU) telemetry. The method is demonstrated at full-scale and validated at model scale.