Overdetermined System Research Articles

Studying the fractal properties of Ceres

Currently, the asteroid Ceres belongs to small celestial bodies with the most well-known physical parameters. The study of the structural and real properties of Ceres is an urgent and modern task, the solution of which will make it possible to develop the evolutionary theory of a minor planet. In this work, the fractal properties of the dwarf planet Ceres were analyzed using data from the Dawn space mission. Using the expansion in a harmonic series in spherical functions the height parameters of the structural model of Ceres, a 3D model of Ceres was constructed. The analysis showed that the resulting system has a complex multiparameter fractal configuration. The study of such objects requires the use of harmonic multiparameter methods. Multivariate fractal analysis allows to represent systems similar to the Ceres model in the form of a spectrum of fractal dimensions. The advantage of fractal analysis is the ability to explore local areas of the physical surface. In this work, the Minkowski algorithm was used for this purpose. At the final stage, an overdetermined system was solved for various local areas of topocentric information in order to postulate a model that takes into account external measures. Fractal dimensions D are determined for local regions and the entire model of the planet. Fractal dimensions vary from 1.37 to 1.92 depending on the longitude and latitude of Ceres. The main results are as follows: 1) the structure of the Ceres surface varies more strongly in longitude; 2) the structure of Ceres is smoother in latitude; 3) the coefficient of self-similarity changes rather quickly in longitude, which indicates that different local regions of the minor planet were formed under the influence of various physical processes. It is necessary to emphasize that the resulting fractal dimensions are significantly scattered both in longitude and latitude of Ceres. This fact confirms the presence of a complex structure in the spatial model of a minor planet. This also applies to the actual physical surface of Ceres. The results of the work allow us to conclude that fractal modeling can give independent values of the fractal dimension both for the entire model of Ceres and for its local macrostructural regions.

Solution of an Overdetermined Stationary Stokes-Type System in Half-Space

Solution of an Overdetermined Stationary Stokes-Type System in Half-Space

Geomagnetic inverse modeling to estimate the volume of magnetization in areas with topographic variations

A three-dimensional geomagnetic modeling is proposed to obtain information about a magnetization over topographic area from magnetic total force from land data survey. The model is divided into three-dimensional mesh grid of rectangular prisms that cover a finite depth below topographic elevation. The model parameter is assigned to each prism describing magnetization. A set of linear equations was formulated in terms of volume magnetization for each prism. The inverse calculation is obtained using the solution of over-determined problems. In this inverse calculation over three-dimensional mesh grid, the weighted matrix will be used to determine distribution of volume magnetization in the subsurface and keep the zero value of volume magnetization above the surface. A computer program has been tested with artificial topographic data and applied to digital elevation model of Bandung. The result of the inversion shows the comparison between synthetic model and inverse model to provide several analysis of algorithm performance. The high magnetization value can be characterize using inverse model up to 77% of volume area with average success recovery of model parameter value up to 56%.

PARAMETRIC IDENTIFICATION OF ARC-SUPPRESSION REACTOR IN THE BASIS OF INSTANTANEOUS VALUES

Single-phase earth faults occur in 6–35 kV distribution networks. One of the devices designed to eliminate earth faults is the arc suppression reactor. The inductance of the reactor is set in resonance with the capacitance of the network, which results in the suppression of the capacitive component of the current.
 A method of determining the inductance of a reactor using instantaneous values of current and voltage is proposed. A differential equation linking the measured quantities is derived. The equation is written for two moments of time, and a system from which the expression for the reactor inductance is derived. Numerical approximation of the derivatives by means of the asymmetric difference derivative was carried out. The mathematical model of the reactor was composed. The influence of the asymmetry coefficient and the main parameters of the considered transients on the accuracy of determining the reactor inductance was analyzed.
 The influence of noise on the accuracy of determining the inductance of the reactor was considered. The influence of the main parameters of the considered transients on the stability of the method to noise is analyzed. The confidence intervals of the calculated inductance for different values of relative noise are constructed.
 An upgraded method for determining the detuning based on the solution of an overdetermined system of equations is proposed. The robustness of the upgraded method to the influence of noise was analyzed. The modernized method has shown to be more robust to the influence of noise in comparison with the method based on solving a system consisting of equations written for two moments of time.
 A test of the methods on a real oscillogram of current and voltage is described. Recommendations on the practical application of the developed methods are offered. The upgraded method is appropriate when there is a high influence of noise and no limitations on the computing power of devices, the method of calculating the inductance by two readings should be used in all other cases.

Cross‐well pressure test analysis for CO2 plume characterization based on arrival time and peak pressure change observations

AbstractCO2 migration from the injection zone into an upper zone is possible through potential migration paths such as improperly abandoned wells or other weaknesses in the caprock overlying the injection zone. Various monitoring technologies may be used to obtain information on the resulting CO2 plume in an overlying zone. Cross‐well pressure testing—where water is to be injected/produced at a well and the pressure response to be observed at monitoring well(s)—has been previously investigated to characterize the CO2 plume. However, the methods proposed in the literature are not simple and may be computationally expensive either in forward modeling, inverse modeling, or both. For the cross‐well test analysis proposed herein, both the forward solution—a closed‐form analytical model—and the inverse model—a well‐posed overdetermined system of nonlinear equations—are simple. Consequently, a simple analysis methodology is presented to invert the cross‐well test data to obtain important information on the plume. To achieve this simplified approach, the migrated CO2 plume is assumed circular in shape with constant gaseous CO2 fraction over the plume area. The analysis methodology is first validated using a single‐phase model with a circular heterogeneity. Next, the methodology is applied to synthetic cross‐well pressure test data for a two‐phase system with CO2 plume. The arrival times and peak pressure change values from multiple monitoring wells are used to estimate the location and size of the plume. The estimated values are in good agreement with the actual plume size and locations used in generating the synthetic data. © 2021 Society of Chemical Industry and John Wiley & Sons, Ltd.

TLS-EM algorithm of Mixture Density Models for exponential families

As a widely used model, Mixture Density Model (MDM) is traditionally solved by Expectation–Maximization (EM) algorithm. EM maximizes a lower bound function iteratively, especially for exponential families. This paper managed to improve EM by combining it with Total Least Squares (TLS), proposing a new algorithm called the TLS-EM algorithm. In this algorithm, parameters are divided into two groups, linear parameters and sub-model parameters. They are solved in each iteration separately. First, data set is separated in different intervals and the conditional maximizing question is transformed into the over-determined linear equations. TLS is adopted to solve these equations and calculate linear parameters, with sub-model parameters fixed. Second, sub-model parameters are solved with EM. Properties of TLS-EM have been provided with proofs. Combining the properties of TLS, EM and the properties of its own, TLS-EM not only inherits most advantages of EM but also improves it in most cases, especially in bad initial or bad model conditions. Numerical experiments confirm these properties.

Moments-based method for boundary conditions in the lattice Boltzmann framework: A comparative analysis for the lid driven cavity flow

Dealing with boundary conditions (BC) was ever considered a puzzling question in the lattice Boltzmann (LB) method. The most popular BC models are based on Ad-Hoc rules and, although these BC models were shown to be suitable for low-order LB equations, their extension to high-order LB was shown to be a very difficult problem and, at authors knowledge, never solved with satisfaction. The main question to be solved is how to deal with a problem when the number of unknowns (the particle populations coming from the outside part of the numerical domain) is greater than the number of equations at our disposal at each boundary site. Recently, BC models based on the regularization of the LB equation, or moments-based models, were proposed. These moments replace the discrete populations as unknowns, independently of the number of discrete velocities that are needed for solving a given problem. The full set of moments-based BC leads, nevertheless, to an overdetermined system of equations, and what distinguishes one model from another is the way this system is solved. In contrast with previous work, we base our approach on second-order moments. Four versions of this model are compared with previous moments-based models considering, in addition to the accuracy, some main model attributes such as global and local mass conservation, rates of convergence, and stability. For this purpose, the complex flow patterns displayed in a two-dimensional lid-driven cavity are investigated.

Identification of the inertial parameters of underactuated Cable-Driven Parallel Robots

Cable-Driven Parallel Robots (CDPR) employ extendable cables to control the pose of an end-effector (EE). If the number of cables is smaller than the degrees of freedom of the EE, and cables have no special arrangement reducing the EE freedoms, the robot is underactuated, and the EE is underconstrained: as a consequence, the EE preserves some freedoms even when all actuators are locked, which may lead to undesirable free motions. This paper proposes a novel methodology for the identification of the EE inertial parameters of these robots. Inertial parameters are useful, for example, in the application of feedforward control techniques. The main merit of our approach is that it does not require force or torque measurements, and only a subset of the robot kinematic variables needs to be measured. The method consists in the application of the EE internal-dynamics equations along a free-motion trajectory, also referred to as self-motion zero dynamics. This results in an over-determined system of equations that are linear in the EE inertial parameters (the Free-motion Internal-Dynamics Identification Model, FIDIM); the said system is solved according to the Total-Least-Square technique. Free-motion trajectories that are optimal for identification purposes are investigated and experimentally tested on a 4-cable robot. FIDIM is then applied, statistical analysis is performed, and the experimental results are cross-validated against additional free-motion trajectories.

A moving discontinuous Galerkin finite element method with interface condition enforcement for compressible flows

A variation of moving discontinuous Galerkin finite element method with interface condition enforcement (MDG-ICE) is developed for solving the compressible Euler equations. The MDG-ICE method, originating from the work of Corrigan et al. [1–4], is based on the space-time DG formulation, where both flow field and grid geometry are considered as independent variables and the conservation laws are enforced both on discrete elements and element interfaces. The element conservation laws are solved in the standard discontinuous solution space to determine conservative quantities, while the interface conservation is enforced using a variational formulation in a continuous space to determine discrete grid geometry. The resulting over-determined system of nonlinear equations arising from the MDG-ICE formulation can then be solved in a least-squares sense, leading to an unconstrained nonlinear least-squares problem that is regularized and solved by Levenberg-Marquardt method. A number of numerical experiments for both 1D unsteady and 2D steady state compressible flow problems are conducted to assess the accuracy and robustness of the MDG-ICE method. Numerical results obtained indicate that the MDG-ICE method is able to implicitly detect and track all types of discontinuities via interface conservation enforcement and satisfy the conservation law on both elements and interfaces via grid movement and grid management, demonstrating that an exponential rate of convergence for Sod and Lax-Harden shock tube problems can be achieved and highly accurate solutions without overheating to both double-rarefaction wave and Noh problems can be obtained.

Full-field DIC-based model updating for localized parameter identification

Identifying the local properties of a structure, either to perform structural health monitoring or to fine tune a numerical model, requires the updating of a large number of parameters. With a high spatial density, but a low dynamic range response information, high-speed-camera measurements have the potential to identify a large number of localized parameters. In contrast, accelerometer measurements provide low-spatial-density modal shapes, but a high dynamic range, and introduce the problem of mass loading. In this research, modal shapes from a high-speed camera are used, providing full-field response information about the observed structure and an over-determined optimization problem. Since the high-speed camera has a lower dynamic range than the accelerometer and the signal-to-noise ratio is low where the displacement amplitude is small, location-specific weighting methods were introduced. The numerical and real experiments showed that the accelerometer’s positioning is important for successful updating, while with a high-speed-camera measurement this is not relevant. This research showed that due to the spatial over-determination, the model updating based on high-speed-camera data, was significantly better than the low-spatial-resolution, accelerometer-based approach.

Exact Solutions to the Navier–Stokes Equations with Couple Stresses

This article discusses the possibility of using the Lin–Sidorov–Aristov class of exact solutions and its modifications to describe the flows of a fluid with microstructure (with couple stresses). The presence of couple shear stresses is a consequence of taking into account the rotational degrees of freedom for an elementary volume of a micropolar liquid. Thus, the Cauchy stress tensor is not symmetric. The article presents exact solutions for describing unidirectional (layered), shear and three-dimensional flows of a micropolar viscous incompressible fluid. New statements of boundary value problems are formulated to describe generalized classical Couette, Stokes and Poiseuille flows. These flows are created by non-uniform shear stresses and velocities. A study of isobaric shear flows of a micropolar viscous incompressible fluid is presented. Isobaric shear flows are described by an overdetermined system of nonlinear partial differential equations (system of Navier–Stokes equations and incompressibility equation). A condition for the solvability of the overdetermined system of equations is provided. A class of nontrivial solutions of an overdetermined system of partial differential equations for describing isobaric fluid flows is constructed. The exact solutions announced in this article are described by polynomials with respect to two coordinates. The coefficients of the polynomials depend on the third coordinate and time.

The least‐square design of minimum‐order allpass‐based infinite impulse response multi‐notch filters

SummaryThe paper investigates the least‐square design of minimum‐order infinite impulse response multi‐notch filters. Monotonically decreasing nature of the stable allpass filter's phase response, characterizing the allpass‐based multi‐notch filter, along with the simple relation between coefficients and the phase response of the allpass filter, allows the formulation of the multi‐notch filter's magnitude response specifications as the linear equality constraints regarding the positions of notch, left‐ and right‐hand cutoff frequencies. Since the number of such constraints equals triple the number of notch frequencies, while the number of unknown filter coefficients equals double the number of notch frequencies, variable elimination is first performed to ensure that specifications regarding the notch frequencies positions are strictly satisfied. The remaining overdetermined system of linear equations is then solved in the least‐square sense, leading to the approximate satisfaction of left‐ and right‐hand cutoff frequencies positions. Proposed design method is also compared with some of the existing infinite impulse response multi‐notch filter design methods.

Complex associated to some systems of PDE

In this paper, we construct the complex associated to an overdetermined system of PDE of order one with constant coefficients using the theory of exterior differential system of Cartan (see [1]). We show that if the tableau associated to the system of PDE is in involution then the complex contains only operators of order one given by the successive torsions of the system (see section 2 for the details). Moreover in section 3, we give a simple direct proof of “the involution of tableau” by prolongation process in this particular case (theorem of Cartan-Kuranishi in this context). Finally in the last section, we construct the complex associated to the Cauchy-Fueter operator (in fact, more general system of PDE containing Cauchy-Fueter equations) using the previous results. We can note that the system of PDE associated to the Cauchy-Fueter operator is not in involution contrarily of the De Rham or Dolbeault operators (this complex was first constructed by [5] by using computer algebra methods, and by [6] and [7] with the help of theory of Leray's spectral sequence). We can add that the key point to obtain many of the above results is a useful characterization of the involution of a tableau given in section 2.

On fourth order accuracy stable difference scheme for a multi-point overdetermined elliptic problem

In this paper fourth order of accuracy difference scheme for approximate solution of a multi-point elliptic overdetermined problem in a Hilbert space is proposed. The existence and uniqueness of the solution of the difference scheme are obtained by using the functional operator approach. Stability, almost coercive stability, and coercive stability estimates for the solution of difference scheme are established. These theoretical results can be applied to construct a stable highly accurate difference scheme for approximate solution of multi-point overdetermined boundary value problem for multidimensional elliptic partial differential equations.

Novel Geometries for Stereotactic Localizers

Introduction: The N-localizer is generally utilized in a 3-panel or, more rarely, a 4-panel system for computing stereotactic positions. However, a stereotactic frame that incorporates a 2-panel (bipanel) N-localizer system with panels affixed to only the left and right sides of the frame offers several advantages: improved ergonomics to attach the panels, reduced claustrophobia for the patient, mitigation of posterior panel contact with imaging systems, and reduced complexity. A bipanel system that comprises two standard N-localizer panels yields only two three-dimensional (3D) coordinates, which are insufficient to solve for the stereotactic matrix without further information. While additional information to determine the stereotactic positions could include scalar distances from Digital Imaging and Communications in Medicine (DICOM) metadata or 3D regression across the imaging volume, both have risks related to noise and error propagation. Therefore, we sought to develop new stereotactic localizers that comprise only lateral fiducials (bipanel) that leave the front and back regions of the patient accessible but that contain enough information to solve for the stereotactic matrix using each image independently. Methods: To solve the stereotactic matrix, we assumed the need to compute three or more 3D points from a single image. Several localizer options were studied using Monte Carlo simulations to understand the effect of errors on the computed target location. The simulations included millions of possible combinations for computing the stereotactic matrix in the presence of random errors of 1mm magnitude. The matrix then transformed coordinates for a target that was placed 50mm anterior, 50mm posterior, 50mm lateral, or 50mm anterior and 50mm lateral to the centre of the image. Simulated cross-sectional axial images of the novel localizer systems were created and converted into DICOM images representing computed tomography (CT) images. Results: Three novel models include the M-localizer, F-localizer, and Z-localizer. For each of these localizer systems, optimized results were obtained using an overdetermined system of equations made possible by more than three diagonal bars. In each case, the diagonal bar position was computed using standard N-localizer mathematics. Additionally, the M-localizer allowed adding a computation using the Sturm-Pastyr method. Monte Carlo simulation demonstrated that the Z-localizer provided optimal results.Conclusion: The three proposed novel models meet our design objectives. Of the three, the Z-localizer produced the least propagation of error. The M-localizer was simpler and had slightly more error than the Z-localizer. The F-localizer produced more error than either the Z-localizer or M-localizer. Further study is needed to determine optimizations using these novel models.

Symmetry of positive solutions of elliptic equations with mixed boundary conditions in a sub-spherical sector

The paper is devoted to the qualitative properties of positive solutions to a semilinear elliptic equation in a planar sub-spherical sector. Under certain range of amplitudes, we prove some monotonicity properties via the method of moving planes. The symmetry properties follow from the uniqueness of the corresponding over-determined problem by Farina and Valdinoci (2013 Am. J. Math.).

ON A BOUNDARY VALUE PROBLEM FOR ONE OVERDETERMINED SYSTEM IN A HALF-PLANE

This paper considers a boundary value problem for an overdetermined system of equations in a half-plane. This problem arises in particular when solving a stationary system of the two-velocity hydrodynamics with one pressure and homogeneous divergent conditions and the Dirichlet boundary conditions for two phase velocities, as well as in problems of electrodynamics. The generalized solution to a stationary system of the two-velocity hydrodynamics in the case of two-dimensional unbounded domains, for instance, in a half-plane, has a significant difference from the three-dimensional case. Namely, in the two-dimensional case for the velocities it is impossible to satisfy the pre-set conditions at infinity and the condition of boundedness at infinity is imposed. In this case, the medium is considered to be homogeneous, and the energy dissipation occurs due to the shear viscosities of the phases of the subsystems, and other effects are not discussed in this paper. The mass transfer occurs due to the mass force. With an appropriate choice of functional spaces, the existence and uniqueness of a generalized solution with an appropriate stability estimate has been proved.

New computational methods for inverse wave scattering with a new filtering technique

This note is concerned with inverse wave scattering in one and two dimensional domains. It seeks to recover an unknown function based on measurements collected at the boundary of the domain. For one-dimensional problem, only one point of the domain is assumed to be accessible. For the two dimensional domain, the outer boundary is assumed to be accessible. It develops two iterative algorithms, in which an assumed initial guess for the unknown function is updated. The first method uses a set of sampling functions to formulate a moment problem for the correction to the assumed value. This method is applied to both one-dimensional and two dimensional domains. For two dimensional Helmholtz equation, it relies on a new effective filtering technique which is another contribution of the present work. The second method uses a direct formulation to recover the correction term. This method is only developed for the one-dimensional case. For all cases presented here, the correction to the assumed value is obtained by solving an over-determined linear system through the use of least-square minimization. Tikhonov regularization is also used to stabilize the least-square solution. A number of numerical examples are used to show their applicability and robustness to noise.

Compressive Covariance Sensing-Based Power Spectrum Estimation of Real-Valued Signals Subject to Sub-Nyquist Sampling

In this work, an estimate of the power spectrum of a real-valued wide-sense stationary autoregressive signal is computed from sub-Nyquist or compressed measurements in additive white Gaussian noise. The problem is formulated using the concepts of compressive covariance sensing and Blackman-Tukey nonparametric spectrum estimation. Only the second-order statistics of the original signal, rather than the signal itself, need to be recovered from the compressed signal. This is achieved by solving the resulting overdetermined system of equations by application of least squares, thereby circumventing the need for applying the complicated ℓ 1 -minimization otherwise required for the reconstruction of the original signal. Moreover, the signal need not be spectrally sparse. A study of the performance of the power spectral estimator is conducted taking into account the properties of the different bases of the covariance subspace needed for compressive covariance sensing, as well as different linear sparse rulers by which compression is achieved. A method is proposed to benefit from the possible computational efficiency resulting from the use of the Fourier basis of the covariance subspace without considerably affecting the spectrum estimation performance.

Near-Surface Geothermal Reservoir Imaging based on the Customized Dense Seismic Network

Exploration of geothermal resources has become one of the major drivers for urbanization and economic transformation in southeast China, and it calls for temporal and spatial resolution improvements on conventional geophysical techniques under complex near-surface conditions. A dense seismic network including 192 sensors, with its aperture around 4.8 km, was deployed to record ambient noise for local-scale geothermal reservoir imaging in central Zhejiang Province, southeast China. With the ultrashort (less than 5 days) observation, a subset of the total station pairs was extracted to conduct seismic interferometry and dispersion measurements. This customized subnet was qualified to be an overdetermined system to achieve similar performance of entire network for the eikonal tomography with higher efficiency. The recorded high-frequency noise sources turned out to be rather homogeneous distributions, except for the dominant direction of northern Jinhua urban area. Azimuthal effects and wavelength restriction of dispersion measurements were quantified by simulating source–receiver responses under non-equipartitions of energy. The azimuth–velocity variations of field dataset verified that phase velocity measurements using the tested wavelength restriction were robust under different source–receiver orientations. The obtained phase velocity maps were comparable with the geological map. Furthermore, the inverted body wave velocities coincide well with the borehole data. Shear wave velocity profiles revealed corresponding fracture and depression structures of the reservoir. Finally, multiple geophysical properties delineate the shallow geothermy to be karst fissure water heated at certain earth temperature gradient.