Use Of High-order Derivatives Research Articles

Sparse-View Spectral CT Reconstruction Based on Tensor Decomposition and Total Generalized Variation

Spectral computed tomography (CT)-reconstructed images often exhibit severe noise and artifacts, which compromise the practical application of spectral CT imaging technology. Methods that use tensor dictionary learning (TDL) have shown superior performance, but it is difficult to obtain a high-quality pre-trained global tensor dictionary in practice. In order to resolve this problem, this paper develops an algorithm called tensor decomposition with total generalized variation (TGV) for sparse-view spectral CT reconstruction. In the process of constructing tensor volumes, the proposed algorithm utilizes the non-local similarity feature of images to construct fourth-order tensor volumes and uses Canonical Polyadic (CP) tensor decomposition instead of pre-trained tensor dictionaries to further explore the inter-channel correlation of images. Simultaneously, introducing the TGV regularization term to characterize spatial sparsity features, the use of higher-order derivatives can better adapt to different image structures and noise levels. The proposed objective minimization model has been addressed using the split-Bregman algorithm. To assess the performance of the proposed algorithm, several numerical simulations and actual preclinical mice are studied. The final results demonstrate that the proposed algorithm has an enormous improvement in the quality of spectral CT images when compared to several existing competing algorithms.

A One-Parameter Family of Methods with a Higher Order of Convergence for Equations in a Banach Space

The conventional approach of the local convergence analysis of an iterative method on Rm, with m a natural number, depends on Taylor series expansion. This technique often requires the calculation of high-order derivatives. However, those derivatives may not be part of the proposed method(s). In this way, the method(s) can face several limitations, particularly the use of higher-order derivatives and a lack of information about a priori computable error bounds on the solution distance or uniqueness. In this paper, we address these drawbacks by conducting the local convergence analysis within the broader framework of a Banach space. We have selected an important family of high convergence order methods to demonstrate our technique as an example. However, due to its generality, our technique can be used on any other iterative method using inverses of linear operators along the same line. Our analysis not only extends in Rm spaces but also provides convergence conditions based on the operators used in the method, which offer the applicability of the method in a broader area. Additionally, we introduce a novel semilocal convergence analysis not presented before in such studies. Both forms of convergence analysis depend on the concept of generalized continuity and provide a deeper understanding of convergence properties. Our methodology not only enhances the applicability of the suggested method(s) but also provides suitability for applied science problems. The computational results also support the theoretical aspects.

A General Non-hydrostatic Hyperbolic Formulation for Boussinesq Dispersive Shallow Flows and Its Numerical Approximation

In this paper, we propose a novel first-order reformulation of the most well-known Boussinesq-type systems that are used in ocean engineering. This has the advantage of collecting in a general framework many of the well-known systems used for dispersive flows. Moreover, it avoids the use of high-order derivatives which are not easy to treat numerically, due to the large stencil usually needed. These first-order PDE dispersive systems are then approximated by a novel set of first-order hyperbolic equations. Our new hyperbolic approximation is based on a relaxed augmented system in which the divergence constraints of the velocity flow variables are coupled with the other conservation laws via an evolution equation for the depth-averaged non-hydrostatic pressures. The most important advantage of this new hyperbolic formulation is that it can be easily discretized with explicit and high-order accurate numerical schemes for hyperbolic conservation laws. There is no longer need of solving implicitly some linear system as it is usually done in many classical approaches of Boussinesq-type models. Here a third-order finite volume scheme based on a CWENO reconstruction has been used. The scheme is well-balanced and can treat correctly wet–dry areas and emerging topographies. Several numerical tests, which include idealized academic benchmarks and laboratory experiments are proposed, showing the advantage, efficiency and accuracy of the technique proposed here.

Quantum Correction for Newton’s Law of Motion

A description of the motion in noninertial reference frames by means of the inclusion of high time derivatives is studied. Incompleteness of the description of physical reality is a problem of any theory, both in quantum mechanics and classical physics. The “stability principle” is put forward. We also provide macroscopic examples of noninertial mechanics and verify the use of high-order derivatives as nonlocal hidden variables on the basis of the equivalence principle when acceleration is equal to the gravitational field. Acceleration in this case is a function of high derivatives with respect to time. The definition of dark metrics for matter and energy is presented to replace the standard notions of dark matter and dark energy. In the Conclusion section, problem symmetry is noted for noninertial mechanics.

Numerical approximation of the Steiner problem in dimension $2$ and $3$

The aim of this work is to present some numerical computations of solutions of the Steiner Problem , based on the recent phase field approximations proposed in [12] and analyzed in [5, 4]. Our strategy consists in improving the regularity of the associated phase field solution by use of higher-order derivatives in the Cahn-Hilliard functional as in [6]. We justify the convergence of this slightly modified version of the functional, together with other technics that we employ to improve the numerical experiments. In particular, we are able to consider a large number of points in dimension 2. We finally present and justify an approximation method that is efficient in dimension 3, which is one of the major novelties of the paper.

UXO Detection by Multiscale Potential Field Methods

The choice of an effective interpretative methodology in terms of management costs, timing, reliability, safety and efficiency is a key issue for the identification of unexploded ordnance (UXO). Among geophysical methods, magnetometry plays an important role in detection and interpretation of UXO objects. We present a study aimed at the identification and characterization of buried UXO in Rohoznik, Slovakia, by a multiscale analysis on high-resolution magnetic data. The area is located close to a former army shooting range and is characterized by several buried UXO. When a UXO hits the ground but does not explode, the impact causes a re-organization of the magnetic domains within the armature (shock demagnetization). Due to shock demagnetization the object gets a magnetization oriented in the direction of the actual Earth magnetic field. Furthermore, the elongated shape of the UXO of Rohoznik causes their anomalies to be oriented along the longitudinal axis of the projectiles. Our multiscale analysis was performed through methods such as depth from extreme points (DEXP) and the multi-ridge method. The upward continuation employed by these two methods allowed reducing the influence of the high-frequency noise related to non-UXO buried iron objects. Also, the use of high-order derivatives in the multiscale approach decreased interference effects related to close-by sources and avoided any pre-filtering. Our approach was first validated on a test site located close to Rohoznik, where unexplored projectiles were buried with different orientations at a known depth. Then we focused on the real data of the former army shooting range. For all our analyses we employed a structural index value of 2.5, which was estimated through the scaling function analysis and the derivative euler deconvolution algorithm. Both the (DEXP) and multi-ridge depth-estimation methods proved to be effective and rapid with respect to more expensive and time-consuming methods such as inversion. Our analysis revealed average depths for the analysed UXOs matching rather well with what was found by subsequent excavations, with a maximum discrepancy of a few decimetres.

Improved solutions for ill-conditioned problems involved in set-membership estimation for fault detection and isolation

Set-membership (SM) estimation implies that the computed solution sets are guaranteed to contain all the feasible estimates consistent with the bounds specified in the model. Two issues often involved in the solution of SM estimation problems and their application to engineering case studies are considered in this paper. The first one is the estimation of derivatives from noisy signals, which in a bounded uncertainty framework means obtaining an enclosure by lower and upper bounds. In this paper, we improve existing methods for enclosing derivatives using Higher-Order Sliding Modes (HOSM) differentiators combining filtering. Our approach turns the use of high order derivatives more efficiently especially when the signal to differentiate has slow dynamics. The second issue of interest is solving linear interval equation systems, which is often an ill-conditioned problem. This problem is reformulated as a Constraint Satisfaction Problem and solved by the combination of the constraint propagation Forward Backward algorithm and the SIVIA algorithm. The two proposed methods are tested on illustrative examples. The two methods are then used in a fault detection and isolation algorithm based on SM parameter estimation that is applied to detect abnormal parameter values in a biological case study.

Meshing strategies in the absolute nodal coordinate formulation-based Euler–Bernoulli beam elements

The absolute nodal coordinate formulation (ANCF) is a finite-element-based formulation developed for large deformation analysis in multibody applications. During the course of progress of the ANCF, a number of beam, plate and shell elements are introduced. Numerous different kinematic and strain energy definitions have been considered. However, the convergence behavior of ANCF-based beam elements has been investigated in the past primarily only with uniform mesh refinement. The objective of this paper is to study numerically the effects of different mesh refinement strategies within an Euler–Bernoulli beam-type problem solved with two-dimensional ANCF-based beam elements. To this end, h- and p-refinement meshing strategies are implemented for two-dimensional ANCF beam elements. The p-refinement is achieved by adding internal degrees of freedom to the elements. In addition, another p-refinement strategy is studied that employs higher-order derivatives as nodal coordinates. The rates of convergence for the different meshing strategies are calculated based on the known iteratively solved extensible elastica analytical solution. The numerical results indicate that the p-refinement strategy, with increasing number of higher-order derivatives at the element interfaces, in conjunction with h-refinement improves the rate of convergence. For the studied numerical example, this mesh strategy leads to better absolute error of displacement convergence than the uniform h-refinement or the p-refinement with internal nodes. The improvement in rate of convergence results from the use of higher-order derivatives that impose continuity to the bending and axial deformations across the elements, which may smooth the oscillations of bending and the axial strain fields.

Improving the local wavenumber method by automatic DEXP transformation

In this paper we present a new method for source parameter estimation, based on the local wavenumber function. We make use of the stable properties of the Depth from EXtreme Points (DEXP) method, in which the depth to the source is determined at the extreme points of the field scaled with a power-law of the altitude. Thus the method results particularly suited to deal with local wavenumber of high-order, as it is able to overcome its known instability caused by the use of high-order derivatives. The DEXP transformation enjoys a relevant feature when applied to the local wavenumber function: the scaling-law is in fact independent of the structural index. So, differently from the DEXP transformation applied directly to potential fields, the Local Wavenumber DEXP transformation is fully automatic and may be implemented as a very fast imaging method, mapping every kind of source at the correct depth. Also the simultaneous presence of sources with different homogeneity degree can be easily and correctly treated. The method was applied to synthetic and real examples from Bulgaria and Italy and the results agree well with known information about the causative sources.

Depth to Basement calculation in Southern Thomson, QLD.

Depth to crystalline basement is a major consideration when conducting minerals exploration. The thickness of sedimentary cover concealing mineralised basement rock is a primary factor in determining whether economic greenfields exploration is viable. A good knowledge of depth to basement is therefore critical for tenement selection and drill hole targeting. New airborne magnetic data released in 2012 allows the calculation of a depth to basement surface in the southern Thomson area adjacent to the Queensland - New South Wales border in south Queensland. Available drill hole data shows basement depths from about 100m to over 3km in the study area but is too sparse to create a reliable surface. A combination of different automatic depth to basement techniques, including Euler deconvolution and Naudy, were used. These techniques were preferred due to a regionally extensive high frequency, low amplitude magnetic signature attributed to shallow sources in one of the cover sequences in the study region. This signature created difficulties for depth to basement calculations which involved the use of higher order derivatives. Geosoft's Located Euler calculation was preferred in areas where remanence was evident as it uses the analytical signal grid. This technique was used in combination with standard Euler. Naudy depths were also compared to the Euler solutions, particularly in areas where the drill holes indicated the basement was relatively shallow (where smoothing associated gridding can cause overestimation of source depths). Seismic data available in the north of the study area was interpreted and used as a secondary (along with the drill hole data) quality control check on the basement surface. The calculated depth to basement surface was used to create voxet model for further analysis using gravity and magnetic inversion. The final depth to basement surface defines an area of shallow basement in the south-west of the study area.

Study of the effect of the spiral geometry on wheel/rail contact forces

Spiral sections are used in the construction of railroad tracks in order to achieve a smooth change in the curvature. In order to satisfy the kinematic conditions at the position, velocity, and acceleration levels in railroad vehicle dynamics computational algorithms based on the constraint contact formulations; third-order derivatives of the wheel/rail contact constraint functions with respect to the wheel and rail geometric surface parameters must be evaluated. Discontinuities in these higher derivatives can produce jumps in the wheel/rail contact forces at the spiral entries and exits. This paper develops a simple procedure that can be applied online to ensure that the curvature and its derivative of the rail space curve are continuous at the spiral entries and exits. Because continuity of the space curve geometric variables does not ensure continuity of these variables on the rail surface, the effect of the proposed procedure on the contact forces is examined. The constraint and elastic wheel/rail contact formulations are first reviewed in order to show the degree of continuity required and the basic kinematic conditions used to search online for the wheel/rail contact points. The rigid track geometry is described using the absolute nodal coordinate formulation (ANCF), which employs a geometric description that ensures the continuity of the position and gradient vectors at the track nodes. In order to ensure the continuity of the curvature vector and its derivative at the intersection of the spirals with the tangent and curve segments, a set of linear conditions are developed and solved online, allowing for implicit elimination of track nodal variables. These conditions are not considered as kinematic constraints imposed on the system motion since they are mainly used to improve the rigid track geometric description and do not involve the system generalized coordinates or their time derivatives. The procedure described in this investigation does not require making changes in the description of the track geometry and does not require the use of higher order derivatives of the track node angles in order to achieve the desired degree of continuity. Because higher degree of continuity achieved for the rail space curve does not ensure a similar degree of continuity on the surface where the wheel/rail contact occurs; the effect of the change of the degree of continuity of the space curve at the spiral entries and exits on the rail surface geometry is also discussed in this paper. The results presented in this study show that the proposed procedure can lead to improvement in the wheel/rail contact force results in some simulation scenarios. The results, obtained using the proposed procedure, are compared with the results obtained using local mesh refinements.

An approximation method for high-order fractional derivatives of algebraically singular functions

The fractional derivative D q f ( s ) ( 0 ≤ s ≤ 1 ) of a given function f ( s ) with a positive non-integer q is defined in terms of an indefinite integral. We propose a uniform approximation scheme to D q f ( s ) for algebraically singular functions f ( s ) = s α g ( s ) ( α > − 1 ) with smooth functions g ( s ) . The present method consists of interpolating g ( s ) at sample points t j in [ 0 , 1 ] by a finite sum of the Chebyshev polynomials. We demonstrate that for the non-negative integer m such that m < q < m + 1 , the use of high-order derivatives g ( i ) ( 0 ) and g ( i ) ( 1 ) ( 0 ≤ i ≤ m ) at both ends of [ 0 , 1 ] as well as g ( t j ) , t j ∈ [ 0 , 1 ] in interpolating g ( s ) , is essential to uniformly approximate D q { s α g ( s ) } for 0 ≤ s ≤ 1 when α ≥ q − m − 1 . Some numerical examples in the simplest case 1 < q < 2 are included.

Parameter estimation from gravitational waves generated by nonspinning binary black holes with laser interferometers: Beyond the Fisher information

In this paper we apply to gravitational waves from non-spinning binary systems a recently intro- duced frequentist methodology to calculate analytically the error for a maximum likelihood estimate (MLE) of physical parameters. While existing literature focuses on using the Cramer Rao Lower bound (CRLB) and Monte Carlo simulations, we use a power expansion of the bias and covariance in inverse powers of the signal to noise ratio. The use of higher order derivatives of the likelihood function in the expansions makes the prediction also sensitive to the secondary lobes of the MLE probability distribution. We discuss conditions for validity of the CRLB and predict new features in regions of the parameter space currently not explored. For example, we see how the bias can become the most important contributor to the parameters' errors for high mass systems (200M and above).

Image decomposition combining staircase reduction and texture extraction

This paper proposes a natural and efficient way to achieve staircase reduction in texture extraction models of image processing. Moreover, we propose a precise framework for this amalgamation. In a sense, we utilize the best of both worlds: (I) the use of higher order derivatives through a variant of the Chambolle–Lions inf convolution energy (an image decomposition model in itself) along with (II) approximations to Meyer’s G and E norms including the H −1 negative norm for ameliorating staircasing in image decomposition and restoration problems.

Nonlinear codes from algebraic curves improving the Tsfasman-Vladut-Zink bound

In the present paper, we construct a class of nonlinear codes by making use of higher order derivatives of certain functions of algebraic curves. It turns out that the asymptotic bound derived from the Goppa geometry codes can be improved for the entire interval (0,1). In particular, the Tsfasman-Vladut-Zink (TVZ) bound is ameliorated for the entire interval (0,1).

High-Order Radiation Boundary Conditions for the Convective Wave Equation in Exterior Domains

We construct and apply radiation boundary conditions of arbitrary order for wave propagation with subsonic convection in exterior domains. The conditions generalize those we constructed earlier for the isotropic wave equation. As in the well-known work of Bayliss and Turkel, the construction is based on the progressive wave (or multipole) expansion of outgoing solutions. Using auxiliary functions defined only on the artificial boundary, problems associated with the use of high-order derivatives, as in the original Bayliss-Turkel formulation, are avoided. The order of the conditions is thus increased simply by using more auxiliary variables. By generalizing to the convective case, we have developed conditions which can be used in aeroacoustics applications whenever the Euler equations linearized about a uniform flow are sufficiently accurate in the far field. Numerical experiments in two space dimensions are presented to illustrate the stability and accuracy of the proposed technique.

Optimization of digital filters for the detection of trace elements in EELS. III – Gaussian, homomorphic and adaptive filters

This paper is the third part of a series of papers dealing with the development of digital filters for the extraction of information from electron energy loss spectra, when the signal-to-background is very weak, as it is the case for elements in weak concentration. Three new filtering approaches are investigated: the use of high-order derivatives (typically, the fourth order derivative of a Gaussian), the use of homomorphic filtering (working on logarithmically-transformed spectra) and the use of adaptive filtering, taking into account the local background shape. These signal processing methods are applied to real spectra to demonstrate and comment their effects in three different cases of signal and background shapes related to calcium, phosphorus and boron detection in definite thickness organic matrix.

Higher-order derivatives in linear and quadratic programming

An interior method for linear and quadratic programming which makes use of higher derivatives of the logarithm barrier function is presented. A convergence analysis is considered too. Our computational experience shows that the method considered performs quite well and seems to be more reliable and robust than the standard method. Scope and purpose Linear programming problems were, for many years, the exclusive domain of the simplex algorithm developed by G.B. Dantzing in 1947. With the introduction of a new algorithm, developed by N.K. Karmarkar in 1984, an alternative computational approach became available for solving such problems. This algorithm established a new class of algorithms: interior point methods for linear programming. In this paper we introduce a barrier method for solving a linear and quadratic programming problem which [9–15] makes use of higher-order derivatives. We note that a different approach used to construct higher-order interior point methods is presented in [1–4]. We think that making use of an approximation of higher-order we may obtain a faster convergence and an algorithm more robust than a method obtained using a second-order approximation.

A logarithm barrier method for linear programming

In a previous paper [6] we introduce an algorithm for solving an unconstrained optimization problem based on an iterative process which makes use of higher order derivatives. In this paper we consider the iterative process presented in [6] and prove the cubic convergence of this process. We, moreover, introduce an interior point method for solving a linear programming problem making use of a Taylor series expansion related to the gradient of the penalty function. Finally some numerical results are reported.

Mathematical Problems in Fitting Kinetic Models. Some New Perspectives.

Fitting kinetic data with kinetic rate expressions has been a challenge for many years. In hopes of providing some guidelines for tackling this problem, kinetic parameter estimation is revisited, covering such topics as data transformations, nonlinear single and multi-response techniques, optimization techniques, and parameter confidence measures. Among the optimization techniques discussed are direct and indirect methods, use of higher-order derivatives, and global optimization. New perspectives on statistical tests for model and parameter confidence are discussed, paying particular attention to exact, nonlinear methods and pointing out the distinct lack of these tests throughout the kinetics literature. In each section, either a single-site or a dual-site Langmuir-Hinshelwood (L-H) example model is shown for illustration of the aforementioned concepts.