Piecewise Homogeneity Research Articles

Systematic derivation of [formula omitted] equations, its discretization using GTIN method and development of a switchable SP3 to [formula omitted] neutron transport solver

The Generalized Simplified Spherical Harmonics (GSPN) method has drawn recent interest owing to the fact that theoretically, it is equivalent to the PN and does not rely on any assumption other than piecewise homogeneity. Its level wise approach to PN offers an agreeable balance between the accuracy and computational efficiency making it a valuable mathematical tool for analyzing neutron transport problems pertaining to nuclear engineering and reactor physics. The lowest level approximation for N = 3, expressed as GSP3(0), offers reducing the complexity while still capturing the essential physics. In the current work, GSP3(0) equations are derived along with the interface and boundary conditions for linear anisotropic scattering. These equations are then discretized using the generalized transverse integration nodal method for a two-dimensional system of piecewise homogeneous rectangular regions. The discretization is done with an option kept for reducing the GSP3(0) formalism to the conventional SP3. Subsequently, a computer code has been developed to solve these discretized equations in the multigroup structure for vacuum, reflective or albedo boundaries. At present, this switchable SP3/GSP3(0) code has the capability of estimating k-eigenvalue and position dependent scalar neutron flux within a given two-dimensional rectangular geometry. The code has been verified by solving SP3, GSP3(0) and other benchmark problems from available literature. The results obtained from our code demonstrate that GSP3(0) is superior to the conventional SP3 and comparable to the recently proposed SDPN (N = 1,2,3) [M. Nazari, A. Zolfaghari, M. Abbasi, Prog. Nucl. Energy, 2023; 166, 104,933] in terms of accuracy as well as computation time.

Addressing Random Variations in MWD Penetration Rate with the DPM Algorithm

Measurement While Drilling (MWD) is an in situ technique for the assessment of ground conditions. It records the drilling parameters of a drilling machine when its drill bit is penetrating into new geomaterials below the bottom of a drill hole. Its penetration rate along drill hole depth, however, has various random variations. Such random variations are inconsistent with the fact that the geomaterials in the ground have piece-wise homogeneity. Many methods have been adopted to filter and normalize the random variations in penetration rate for ground characterization. However, they have not completely resolved the issue and obtained the piece-wise homogeneous distribution of geomaterials in the ground. This paper uses the time-series algorithm of the Drilling Process Monitoring (DPM) technique to remove the random variations in MWD penetration rate. It further obtains the piece-wise constant variations in the newly addressed DPM penetration rate along the drill hole depth. With this algorithm, the DPM penetration rate is consistent with the fact that ground geomaterials have the property of piece-wise homogeneous distribution. Consequently, the piece-wise constant variations in DPM penetration rate can be used to characterize the strength variations in geomaterials along the drill hole depth to substantially upgrade the MWD technique for ground investigation.

A Hierarchical Bayesian Inversion Method for Electromagnetic Imaging of Inhomogeneous Objects With Piecewise Homogeneities

Electromagnetic imaging of inhomogeneous objects is a nonlinear inverse scattering problem which can be effectively handled by means of linear inversion methods or nonlinear optimization methods. In addition to being nonlinear, inverse scattering problems are also ill-posed which can be overcome through a regularization scheme by taking into account some kind of a priori information. However, it is not straightforward to incorporate such inhomogeneity prior information into conventional inversion methods. In this work, we propose an innovative method to solve the problem of imaging inhomogeneous objects with piecewise homogeneity. Under Born approximation (BA), this problem is cast into a linear inversion one which is then addressed by means of a hierarchical Bayesian inversion method. To account for the piecewise homogeneity of the image, a Gaussian mixture model (GMM) is developed to enforce the feature. Specifically, a hidden label variable which is associated with each pixel is introduced for the class of materials and the contrast of each pixel is modeled as a mixture of independent Gaussian priors. Then, the proposed hierarchical Bayesian inversion method incorporates the GMM to the variational Bayesian inference framework to approximate the posterior distribution of all the unknowns (the unknown contrast, hidden label variable, and other model parameters). In addition, the damped generalized approximate message passing (GAMP) is utilized to circumvent the matrix inversion, making the overall process very computationally efficient. Extensive numerical examples will be shown to demonstrate the effectiveness of the proposed method.

Stochastic interest rate modelling using a single or multiple curves: an empirical performance analysis of the Lévy forward price model

In current financial markets negative interest rates have become rather persistent, while in theory it is often common practice to discard such rates as incredible and irrelevant. However, from a risk management perspective, it is crucially important to financial institutions to properly account for this phenomenon in their Asset Liability Management (ALM) studies. In this paper, we develop a coherent framework on how to best incorporate negative interest rates in these studies through a single curve stochastic term structure model and compare it to its multiple curve analogue. It turns out that, from the wide range of available single curve models, especially the Lévy Forward Price model (LFPM) of Eberlein and Özkan [The Lévy LIBOR model. Financ. Stoch., 2005, 9, 327–348] seems appropriate for ALM purposes. This paper describes an optimisation routine for calibrating this LFPM under the risk-neutral measure in both the single and multiple curve framework to the market prices of interest rate caplets with different strike rates, maturities and tenors. In addition, an empirical performance analysis is made of the single and multiple curve LFPM, where we include four deterministic volatility specifications and provide an explicit parametrisation of a piecewise homogeneity restriction with both deterministic and random breakpoints. This comparative analysis indicates that both the single and multiple curve LFPM is best adopted with the Linear-Exponential Volatility (LEV) specification and that deterministic breakpoints should be included, rather than random breakpoints.

Automated gradient-based electrical properties tomography in the human brain using 7 Tesla MRI

Electrical properties of the brain tissues may yield useful biomarkers for neurological disorders and diseases, as well as contribute to safety assurance of ultra-high-field MRI. It has been reported that using B1 maps from a multi-channel RF coil, the spatial variation of the electrical properties can be robustly retrieved. The absolute electrical property values can then be obtained by spatial integration, given that an integration seed point is assigned. In this study, we propose to exploit automatically detected seed points based on tissue piece-wise homogeneity (Helmholtz equation) for spatial integration. Numerical simulations of a numerical brain model and experiments involving 12 healthy volunteers were performed to demonstrate its feasibility and robustness in various noisy conditions and head positions. For in vivo imaging, we consistently observed higher conductivity and permittivity values in the white and gray matter compared to tabulated ex vivo probe measurement results found in the literature, a discrepancy that may be attributed to ex vivo experimental constraints. Our results suggest that the proposed technique produces consistent brain electrical properties in vivo that may contribute to improving diagnostic and therapeutic decisions.

A Multiscale Hierarchical Model for Sparse Hyperspectral Unmixing

Due to the complex background and low spatial resolution of the hyperspectral sensor, observed ground reflectance is often mixed at the pixel level. Hyperspectral unmixing (HU) is a hot-issue in the remote sensing area because it can decompose the observed mixed pixel reflectance. Traditional sparse hyperspectral unmixing often leads to an ill-posed inverse problem, which can be circumvented by spatial regularization approaches. However, their adoption has come at the expense of a massive increase in computational cost. In this paper, a novel multiscale hierarchical model for a method of sparse hyperspectral unmixing is proposed. The paper decomposes HU into two domain problems, one is in an approximation scale representation based on resampling the method’s domain, and the other is in the original domain. The use of multiscale spatial resampling methods for HU leads to an effective strategy that deals with spectral variability and computational cost. Furthermore, the hierarchical strategy with abundant sparsity representation in each layer aims to obtain the global optimal solution. Both simulations and real hyperspectral data experiments show that the proposed method outperforms previous methods in endmember extraction and abundance fraction estimation, and promotes piecewise homogeneity in the estimated abundance without compromising sharp discontinuities among neighboring pixels. Additionally, compared with total variation regularization, the proposed method reduces the computational time effectively.

Stochastic Interest Rate Modeling: An Empirical Performance Analysis of the L vy Forward Price Model

In current financial markets negative interest rates have become rather persistent, while in theory it is often common practice to discard such rates as incredible and irrelevant. However, from a risk management perspective, it is crucially important to financial institutions to properly account for this phenomenon in their Asset Liability Management (ALM) studies. In this paper, we develop a coherent framework on how to best incorporate negative interest rates in these studies through a single curve stochastic term structure model and compare it to its multiple curve analogue. It turns out that, from the wide range of available single curve models, especially the Levy Forward Price model (LFPM) of Eberlein and Ozkan [The Levy LIBOR model. Financ. Stoch., 2005, 9, 327–348] seems appropriate for ALM purposes. This paper describes an optimisation routine for calibrating this LFPM under the risk-neutral measure in both the single and multiple curve framework to the market prices of interest rate caplets with different strike rates, maturities and tenors. In addition, an empirical performance analysis is made of the single and multiple curve LFPM, where we include four deterministic volatility specifications and provide an explicit parametrisation of a piecewise homogeneity restriction with both deterministic and random breakpoints. This comparative analysis indicates that both the single and multiple curve LFPM is best adopted with the Linear-Exponential Volatility (LEV) specification and that deterministic breakpoints should be included, rather than random breakpoints.

A new and rigorous SPN theory – Part II: Generalization to GSPN

A new SPN theory valid for piecewise homogeneous regions was rigorously derived in a paper published last year by the author. The new theory does not need the assumption of locally 1D behavior near a surface. The resulting interface and the boundary conditions, different from the ad hoc conditions conventionally assumed, contain higher order derivatives of the solution functions to the SPN equations.This new SPN theory was presented in the previous paper for the mono-energetic case with generic isotropic neutron sources in piecewise homogeneous regions. In this sequel paper the theory is generalized to the generic case of multi-energy groups including anisotropic neutron sources as well. Although the multi-group extension is straightforward, the inclusion of anisotropic sources poses subtle problems. To be valid for any arbitrary linear anisotropic external source a Generalized SP3 (GSP3) theory is proposed. GSP3 clarifies how to decompose a linearly anisotropic source via Helmholtz decomposition so that its irrotational component can be properly incorporated into SP3. GSP3 is more accurate than SP3. It includes two additional diffusion type equations decoupled from the two original SP3 equations, with in total eight instead of four interface or boundary conditions. The GSP3 equations are all of diffusion type and the additional interface and boundary conditions are manageable for practical engineering applications.By generalizing the Helmholtz decomposition to higher order tensors and using a powerful lemma embedded in a Boris Davison’s 1958 paper, we can formulate a comprehensive theory for the generic GSPN, which can accommodate any spatially differentiable anisotropic neutron sources of any order in anisotropy. The only restriction to GSPN is piecewise homogeneity. GSPN theory provides various levels of approximations to PN, from the simplest SPN all the way to the rigorous equivalency to PN. GSPN contains decomposed independent sets of diffusion type second order differential equations without any crossed variable derivatives, which are very easy to solve. But the interface and boundary conditions are more complicated where the solutions to the independent sets of equations do get coupled. An interesting byproduct of GSPN is that it provides the recipe of how to decompose and incorporate arbitrary anisotropic neutron sources into the SPN equations via the use of the generalized Helmholtz decomposition. The general formulation for the GSPN theory is presented, although specific details are given only for GSP3.

An efficient layered finite element method with domain decomposition for simulations of resistivity well logging

A novel layered finite element scheme combined with domain decomposition method is proposed for fast simulations of resistivity well logging. The recently developed layered finite element scheme employs a 2D discretization to the cross section of a 3D layered structure, and exploits the piecewise homogeneity within each layer in an analytical way with high accuracy. Combining this layered finite element method with domain decomposition strategy leads to a new scheme with further improved modeling flexibility and efficiency. Each layer of a stratified structure will be treated as a subdomain and discretized independently. Nonconforming meshes are allowed between adjacent layers, which makes the discretization very flexible, and saves a great number of unknowns compared with conventional finite element discretizations with conforming meshes. The proposed method can be easily hybridized with conventional finite element schemes, and this combination is especially suitable for modeling layered problems with local inhomogeneities, such as resistivity logging with borehole washouts in some layers. Numerical examples including a 28-layer Oklahoma formation and a washed-out layered borehole model are given to demonstrate the validity and efficiency of this method.

Optical diffraction tomography within a variational Bayesian framework

Optical diffraction tomography techniques aim to find an image of an unknown object (e.g. a map of its refraction index) using measurements of the scattered field that results from its interaction with a known interrogating wave. We address this issue as a nonlinear inverse scattering problem. The forward model is based upon domain integral representations of the electric field whose discrete counterparts are obtained by means of a method of moments. The inverse problem is tackled in a Bayesian estimation framework involving a hierarchical prior model that accounts for the piece-wise homogeneity of the object. A joint unsupervised estimation approach is adopted to estimate the induced currents, the contrast and all the other parameters introduced in the prior model. As an analytic expression for the joint maximum a posteriori (MAP) and posterior mean (PM) estimators is hard to obtain, a tractable approximation of the latter is proposed. This approximation is based upon a variational Bayesian technique and consists in the best separable distribution that approximates the true posterior distribution in the Kullback–Leibler sense. This leads to an implicit parametric optimization scheme which is solved iteratively.

Effects of dispersion and dissipation on the optical rotation and ellipticity of chiral sculptured thin films

The optical rotation and ellipticity spectra from a chiral sculptured thin film were calculated using the piecewise homogeneity approximation method and Bruggeman homogenization formalism. The effects of dispersion and dissipation on the spectra were investigated using experimental dielectric functions for bulk zirconia (ZrO2) and titania (TiO2). The results showed that, in the event of weak frequency dependence of the optical properties of the raw material, effects on the transmission optical activity spectra can be discounted.

Spectral remittances of chiral sculptured zirconia thin films in non-axial propagation

The transmission and reflection spectra from a right-handed chiral sculptured zirconia thin film are calculated using the piecewise homogeneity approximation method and the Bruggeman homogenization formalism and by considering that the dispersion of the dielectric function happens in a non-axial propagation state. First and second Bragg peaks were observed in cross-polarized reflectances and it became clear that increasing the azimuthal angle affects the spectra of the linearly polarized state significantly. This is opposite to the circularly polarized state.

Comparative study between dispersive and non-dispersive dielectric permittivity in spectral remittances of chiral sculptured zirconia thin films

The transmission and reflection spectra from a right-handed chiral sculptured zirconia thin film are calculated using the piecewise homogeneity approximation method and the Bruggeman homogenization formalism using both dispersive and non-dispersive functions for axial and non-axial states. The comparison of spectral results shows that the dispersion of the dielectric function has a considerable effect on the results. In axial excitation of cross-polarized reflectances and co-polarized transmittances the dispersion effect becomes more pronounced at wavelengths further away from the homogenization wavelength. This is also true in the case of non-axial excitation of circular transmittances, while there are considerable differences for cross-polarized reflectances where (wavelength) the first Bragg peak occurs. At wavelengths in the vicinity of the homogenization wavelength the dispersion effect of the dielectric function in RRR becomes more significant.

Computer simulation of intracardiac potential with whole-heart model

Computer simulation of intracardiac potentials was performed based on a whole-heart model, emphasising applications to Electrophysiology Study (EPS) and catheter ablation. A numerical method was proposed based on a volume conductor model with multiple interfaces and piecewise homogeneity. A model of a normal heart and a model of Wolff-Parkinson-White (WPW) Syndrome with a right free-wall Accessory Pathway (AP) were employed to simulate intracardiac electrograms. The study was validated by comparing the simulated electrograms with those in catheter-based EPS and ablations. Applying whole-heart modelling to the EPS and ablation is regarded as a new direction of electrocardiologic forward problem.

Selective Reflection and Transmission from Perturbed Chiral Sculptured Thin Films

The effects of varying the angle of rise as a function of depth are investigated for chiral sculptured thin films. A matrix differential equation describing the propagation of light in the film which may result from plane waves at normal as well as oblique angles of incidence is developed. Using the piecewise homogeneity approximation, reflectance and transmittance spectrums are then determined for chiral thin films of TaO2 with periodic perturbations in the angle of rise as a function of depth into the film. Available empirical data are used to realistically model the dielectric and structural characteristics of the perturbed films. The effects on Bragg reflections of various orders are described for each perturbation.

Sculptured nematic thin films with periodically modulated tilt angle as rugate filters

The reflectances and transmittances of linearly polarized light are calculated using the piecewise homogeneity approximation for a sculptured nematic thin film (SNTF) with a sinusoidally varying (local) columnar tilt angle. Empirical data on TiO 2, available in the literature, are used to realistically model the variation of the permittivity dyadic with the tilt angle. Wave propagation is taken to occur in the morphologically significant plane of the SNTF. Several Bragg peaks occur, with the fundamental peak making a tilt-modulated SNTF suitable for exploitation similarly to a rugate filter. Additionally, the appearance of a spectral hole in the fundamental Bragg peak is demonstrated with the inclusion of a central defect layer.

Tilt-modulated chiral sculptured thin films: an alternative to quarter-wave stacks

The reflectances and transmittances of light incident normally on a chiral sculptured thin film with a periodically modulated tilt angle are calculated using the piecewise homogeneity approximation method. The necessary concurrent modulation of the local permittivity dyadic are accounted for by using empirically determined parameters. For modulation lengths equal to half the unmodulated structural period of the film, the circular Bragg phenomenon displayed by unmodulated chiral STFs is destroyed. It is demonstrated that the chosen nanomaterial then acts as a conventional dielectric mirror.

Comparison of two methods for oblique propagation in helicoidal bianisotropic mediums

Two methods for oblique propagation in two dielectric examples of helicoidal bianisotropic mediums – chiral sculptured thin films and chiral smectic liquid crystals – are compared: (i) the series solution method, and (ii) the piecewise homogeneity approximation method. Both have to be numerically implemented, although only the former relies on an exact analytic solution of the frequency-domain Maxwell postulates. Convergence in the series solution method is investigated via calculations for a variety of planewave-illumination parameters and constitutive properties. Likewise, the maximum slice thickness for convergent results from the piecewise homogeneity approximation method is also studied. The computational efficiencies of both methods are compared.

Electromagnetic plane–wave response characteristics of non–axially excited slabs of dielectric thin–film helicoidal bianisotropic mediums

Thin–film helicoidal bianisotropic mediums (TFHBMs) are rotationally non–homogeneous mediums whose microstructure consists of parallel helical columns. We characterize here the plane–wave response of non–axially excited dielectric TFHBM slabs. The constitutive relations for the dielectric TFHBMs are substituted into the time–harmonic Maxwell curl equations, and a 4 × 4 matrix ordinary differential equation is obtained. The piecewise homogeneity approximation method is shown to be reasonably robust to solve the differential equation. Bragg reflection by non–axially excited dielectric TFHBM slabs is investigated, and transmission characteristics are examined. The effects of several key illumination and constitutive parameters on the response of non–axially excited dielectric TFHBM slabs are studied to elucidate trends in functional relationships. A TFHBM bilayer is presented as an example of a device based on non–axially excited dielectric TFHBMs.

Estimation of epicardial strain using the motions of coronary bifurcations in biplane cinéangiography.

A quantitative method is described for estimating epicardial deformation from the motion of the superficial arteries. A structural model of the time-varying surface is constructed using tensor product basis functions which are bicubic Hermite in the spatial domain and sinusoidal in the temporal domain. The loci of the superficial coronary arteries are reconstructed interactively at diastasis and the bifurcations are tracked semiautomatically throughout a cardiac cycle. An initial surface is fitted to the vessels at diastasis and is subsequently deformed under the influence of the bifurcations. The Lagrange-Green strain tensor is used to obtain a complete description of surface strain over the entire region spanned by the model. The calculated deformation field varies smoothly over space and time and is not constrained by assumptions of isotropy or piecewise homogeneity. Results are presented for a single cycle of a human heart.