Non-unique Solutions Research Articles

An overview of the method of fundamental solutions—Solvability, uniqueness, convergence, and stability

In this paper we give an overview of the Method of Fundamental Solutions (MFS) as a heuristic numerical method. It is truly meshless. Its concept and numerical implementation are simple. It has the flexibility of using various forms of fundamental solutions, singular, hypersingular, or nonsingular, and mixing with general solutions and particular solutions, for different purposes. The collocation matrix, however, is not guaranteed to be invertible. There are other issues. For example, in using the logarithmic fundamental solution, a degenerate scale can exist, causing the nonuniqueness of solution. For metaharmonic operators, such as the Helmholtz equation, the zeros of the fundamental solutions can create spurious resonance frequencies. These and other issues are discussed, and remedies are offered.The traditional error analysis for the MFS shows that when the boundary condition is prescribed by a harmonic function, the convergence is exponential either by increasing the number of terms in the approximation, or by increasing the radius of the fictitious boundary. In practical problems, however, a harmonic boundary condition hardly exists. Numerical experiments show that the sources need to stay close to the boundary, and there is an optimal distance. Based on the maximum principle, a posteriori error can be monitored on the boundary to seek the optimal fictitious boundary location. Other topics discussed include the origin of the MFS, the equivalence between the MFS and the Trefftz collocation method, effective condition number, nonsingular MFS, and solving ill-posed and inverse problems.

Artificial neural network application in an implemented lightning locating system

Time difference of arrival (TDOA) technique is one of many bases to determine lightning strike location employed in a lightning locating system (LLS). In this technique, at least four measurement sensors are required to correctly locate a lightning strike. Usage of fewer number of sensors will result in non-unique solutions to the generated hyperbolas, and hence wrong lightning strike point. This research aims to correctly determine the strike point even if only three measuring sensors are utilized. An artificial neural network (ANN) based algorithm was developed for a 400 km2 coverage area in Southern Malaysia using time of arrival data collected at the three measuring stations over a certain period. The Levenberg–Marquardt algorithm is demonstrated to correctly identify the lightning strike coordinates with an average error of 350 m. The algorithm has helped the three-station TDOA-based LLS to successfully locate the lightning strike point with a remarkable accuracy comparable to that of commercial systems.

Hybrid Nanofluid Flow Past a Shrinking Cylinder with Prescribed Surface Heat Flux

This numerical study was devoted to examining the occurrence of non-unique solutions in boundary layer flow due to deformable surfaces (cylinder and flat plate) with the imposition of prescribed surface heat flux. The hybrid Al2O3-Cu/water nanofluid was formulated using the single phase model with respective correlations of hybrid nanofluids. The governing model was simplified by adopting a similarity transformation. The transformed differential equations were then numerically computed using the efficient bvp4c solver with the ranges of the control parameters 0.5%≤ϕ1,ϕ2≤1.5% (Al2O3 and Cu volumetric concentration), 0≤K≤0.2 (curvature parameter), 2.6<S≤3.2 (suction parameter) and −2.5<λ≤0.5 (stretching/shrinking parameter). Dual steady solutions are presentable for both a cylinder (K>0) and a flat plate (K=0) with the inclusion of only the suction (transpiration) parameter. The real and stable solutions were mathematically validated through the stability analysis. The Al2O3-Cu/water nanofluid with ϕ1=0.5% (alumina) and ϕ2=1.5% (copper) has the highest skin friction coefficient and heat transfer rate, followed by the hybrid nanofluids with volumetric concentrations (ϕ1=1%,ϕ2=1%) and (ϕ1=1.5%,ϕ2=0.5%), respectively. Surprisingly, the flat plate surface abates the separation of boundary layer while it enhances the heat transfer process.

Non-uniqueness for the ab-family of equations

We study the cubic ab-family of equations, which includes both the Fokas-Olver-Rosenau-Qiao (FORQ) and the Novikov (NE) equations. For a≠0, it is proved that there exist initial data in the Sobolev space Hs, s<3/2, with nonunique solutions. Multiple solutions are constructed by studying the collision of 2-peakon solutions. Furthermore, we prove the novel phenomenon that for some members of the family, collision between 2-peakons can occur even if the “faster” peakon is in front of the “slower” peakon.

Multiple UAV-Mounted Base Station Placement and User Association With Joint Fronthaul and Backhaul Optimization

In this paper, we study a joint placement, resource allocation, and user association problem for UAV-assisted wireless networks with constrained backhaul links, where multiple UAV-mounted base stations (UBSs) are deployed to provide wireless services for ground users. We propose a novel framework to maximize the user throughput within the flight-time of UBSs and provides fairness among the users. We first obtain the optimal resource allocation schemes based on different fronthaul and backhaul conditions, and an efficient iterative algorithm is then developed to jointly optimize user association and UBS placement. The optimal UBS placement can be achieved by solving an unconstrained optimization problem which is a simplification of the initial constrained optimization problem based on the optimal resource allocation. We develop a dual-domain coordinated descent and bipartite graph matching based sub-process to identify an optimal user association that prefers the nearby UBSs, as the user association under constrained backhaul links have non-unique optimal solutions. Extensive simulations are conducted to verify the effectiveness of the proposed algorithm, and results show that our proposed method under constrained backhaul can improve both the average throughput by 49% and the fairness among the users by 47% in comparison with the method under ideal backhaul.

Optimal Estimation Retrievals and Their Uncertainties: What Every Atmospheric Scientist Should Know

AbstractRemote sensing instruments are heavily used to provide observations for both the operational and research communities. These sensors do not provide direct observations of the desired atmospheric variables, but instead, retrieval algorithms are necessary to convert the indirect observations into the variable of interest. It is critical to be aware of the underlying assumptions made by many retrieval algorithms, including that the retrieval problem is often ill posed and that there are various sources of uncertainty that need to be treated properly. In short, the retrieval challenge is to invert a set of noisy observations to obtain estimates of atmospheric quantities. The problem is often complicated by imperfect forward models, by imperfect prior knowledge, and by the existence of nonunique solutions. Optimal estimation (OE) is a widely used physical retrieval method that combines measurements, prior information, and the corresponding uncertainties based on Bayes’s theorem to find an optimal solution for the atmospheric state. Furthermore, OE also allows the relative contributions of the different sources of error to the uncertainty in the final retrieved atmospheric state to be understood. Here, we provide a novel Python library to illustrate the use of OE for inverse problems in the atmospheric sciences. We introduce two example problems: how to retrieve drop size distribution parameters from radar observations and how to retrieve the temperature profile from ground-based microwave sensors. Using these examples, we discuss common pitfalls, how the various error sources impact the retrieval, and how the quality of the retrieval results can be quantified.

A new multi-scale signed direct envelope inversion with more accurate amplitude polarity information

Subsurface parameter reconstruction using full waveform inversion is highly nonlinear with nonunique solutions. When the initial model with high precision is missing, the cycle skipping problem will be encountered, especially for the strong scattering media like salt structures. In order to adapt to the inversion of strong scattering media, direct envelope inversion (DEI) using directly the envelope Frechet derivatives was proposed to avoid the weakly nonlinear assumption for the model updating. In the original DEI method, the envelope polarity is not considered. To further improve the effectiveness of DEI, signed envelope inversion with polarity detection can be used. In this work, we develop a new polarity detection technique on the basis of seismic attributes, which can provide more accurate polarity information for the seismic envelope. Based on the above technique, the new multi-scale signed DEI method is proposed. The new technique is applied to the SEG/EAGE salt model. The numerical examples verified the effectiveness of this method.

Non-uniqueness solutions for the thin Carreau film flow and heat transfer over an unsteady stretching sheet

The present paper probed the problem of thin liquid film flow and heat transfer in the Carreau fluid past a permeable stretching sheet. The similarity transformation reduced the partial differential equations into a system of ordinary differential equations, which was then solved numerically by a collocation method. The influence of the governing parameters towards the model was discussed and presented graphically. The non-uniqueness solutions are observable as the governing parameters vary.

Nonlinear Systems of Volterra Equations with Piecewise Smooth Kernels: Numerical Solution and Application for Power Systems Operation

The evolutionary integral dynamical models of storage systems are addressed. Such models are based on systems of weakly regular nonlinear Volterra integral equations with piecewise smooth kernels. These equations can have non-unique solutions that depend on free parameters. The objective of this paper was two-fold. First, the iterative numerical method based on the modified Newton–Kantorovich iterative process is proposed for a solution of the nonlinear systems of such weakly regular Volterra equations. Second, the proposed numerical method was tested both on synthetic examples and real world problems related to the dynamic analysis of microgrids with energy storage systems.

Incomplete analytic hierarchy process with minimum weighted ordinal violations

ABSTRACT Incomplete pairwise comparison matrices offer a natural way of expressing preferences in decision-making processes. Although ordinal information is crucial, there is a bias in the literature: cardinal models dominate. Ordinal models usually yield nonunique solutions; therefore, an approach blending ordinal and cardinal information is needed. In this work, we consider two cascading problems: first, we compute ordinal preferences, maximizing an index that combines ordinal and cardinal information; then, we obtain a cardinal ranking by enforcing ordinal constraints. Notably, we provide a sufficient condition (that is likely to be satisfied in practical cases) for the first problem to admit a unique solution and we develop a provably polynomial-time algorithm to compute it. The effectiveness of the proposed method is analyzed and compared with respect to other approaches and criteria at the state of the art.

Qualitative investigation of cytolytic and noncytolytic immune response in an HBV model

We investigate a mathematical model of infection by the hepatitis B virus (HBV) that includes cytolytic and noncytolytic immune response. The model exhibits a variety of steady-state solutions depending on parameter values, including nonunique and unique equilibrium solutions and periodic behavior. The disease-free equilibrium Ef and the positive-disease equilibrium Ed are examined. The basic reproduction ratio R0 is computed in order to examine the uniqueness and local asymptotic stability of equilibria and to understand the model’s biological implications for HBV dynamics.

Stability of hydromagnetic boundary layer flow of non-Newtonian power-law fluid flow over a moving wedge

We consider the two-dimensional laminar boundary-layer flow of power-law fluid over a moving wedge in which the uniform magnetic field is applied normally to the flow. The motion of the mainstream and wedge is approximated in terms of the power of the distance from leading boundary-layer edge which helps to reduce the governing partial differential equations to an ordinary differential equation. No analytic solution is feasible due to high nonlinearity; therefore, the numerical simulations are sought by Chebyshev collocation and shooting techniques. These techniques provide a means to assess physical insights into the hydrodynamics of complex flows and particularly the Chebyshev collocation method enables the construction of efficient tools suitable for nonlinear problems. This method often leads to a matrix-based analysis and an iterative technique needs to be employed to solve the problem. The various results on the physical parameters show that the shear-thinning and shear-thickening solutions are demarcated by the Newtonian fluid. It is also noticed that the system yielded a non-unique solution structure for a given set of parameters. However, the non-uniqueness of the solutions disappears for increasing the magnetic field. To assess the nature of the non-unique solutions, it is important to perform the linear stability using large-time asymptotics. This analysis gives as to which of these solutions is practically feasible and models the flow. Eigensolution-based analysis reveals that the first solution is always stable while the other one leading to unstable mode. The results further show that an increase in the magnetic field stabilizes the fluid flow over the moving wedge and promotes the unique solution to the problem. Further, the obtained numerical results are compared by examining the large λ asymptotics. It is observed that there is a qualitative comparison between solutions. The hydrodynamics behind the magnetic field and non-Newtonian fluid are discussed in detail.

Seismic inversion constrained by stress value changes

ABSTRACT Seismic inversion is an effective way to get stress distribution of rock in coal mine, but its usage is still limited by multi-solution problem in practice. As seismometers can only be deployed in limited areas in underground coal mine, it is impossible to solve multi-solution problem by adding more seismometers, thus, we propose a new seismic inversion method which introduces stress value changes as constraint to solve this problem. Numerical experiments show that the new method can reduce the mean distance between inversion results and true wave velocity distributions by about 11.5%, reduce the variance of the distances by about 91%, reduce the mean distance between inversion results and frequency-domain-truncated true wave velocity distributions by about 30.6% and reduce the variance of the distances by about 84.1%. These results indicate that the new method proposed in this paper can reduce the non-uniqueness of solutions in seismic inversion and improve the reliability and stability of seismic inversion.

Radiative hybrid nanofluid flow past a rotating permeable stretching/shrinking sheet

Purpose This paper aims to discuss a stability analysis on Cu-Al2O3/water nanofluid having a radiation and suction impacts over a rotating stretching/shrinking sheet. Design/methodology/approach The partial differential equations are converted into nonlinear ordinary differential equations using similarity transformation and then being solved numerically using built in function in Matlab software (bvp4c). The effects of pertinent parameters on the temperature and velocity profiles together with local Nusselt number and skin friction are reported. Findings Compared to previously published studies, the current work is noticed to be in good deal. The analysis further shows that the non-unique solutions exist for certain shrinking parameter values. Hence, a stability analysis is executed using a linear temporal stability analysis and concluded that the second solution is unstable, while the first solution is stable. The effect of suction parameter is observed to be significant in obtaining the solutions. The improvement of the local skin friction and the decrease of the local Nusselt number on the shrinking surface are observed with the increment of the copper nanoparticle volume fractions. Originality/value The originality of current work is the numerical solutions and stability analysis of hybrid nanofluid in rotating flow. This work has also resulted in producing the non-unique solutions for the shrinking sheet, and a stability analysis has also been executed for this flow showing that the second solution is unstable, while the first solution is stable. This paper is therefore valuable for engineers and scientist to get acquainted with the properties of the flow, its behavior and the way to predict it. The authors admit that all the findings are original and were not published anywhere else.

Network flows that solve least squares for linear equations

This paper presents a first-order distributed continuous-time algorithm for computing the least-squares solution to a linear equation over networks. Given the uniqueness of the solution, with nonintegrable and diminishing step size, convergence results are provided for fixed graphs. The exact rate of convergence is also established for various types of step size choices falling into that category. For the case where non-unique solutions exist, convergence to one such solution is proved for constantly connected switching graphs with square integrable step size. Validation of the results and illustration of the impact of step size on the convergence speed are made using a few numerical examples.

The hydraulic efficiency of single fractures: correcting the cubic law parameterization for self-affine surface roughness and fracture closure

Abstract. Quantifying the hydraulic properties of single fractures is a fundamental requirement to understand fluid flow in fractured reservoirs. For an ideal planar fracture, the effective flow is proportional to the cube of the fracture aperture. In contrast, real fractures are rarely planar, and correcting the cubic law in terms of fracture roughness has therefore been a subject of numerous studies in the past. Several empirical relationships between hydraulic and mechanical aperture have been proposed based on statistical variations of the aperture field. However, often, they exhibit non-unique solutions, attributed to the geometrical variety of naturally occurring fractures. In this study, a non-dimensional fracture roughness quantification scheme is acquired, opposing effective surface area against relative fracture closure. This is used to capture deviations from the cubic law as a function of quantified fracture roughness, here termed hydraulic efficiencies. For that, we combine existing methods to generate synthetic 3-D fracture voxel models. Each fracture consists of two random, 25 cm2 wide self-affine surfaces with prescribed roughness amplitude, scaling exponent, and correlation length, which are separated by varying distances to form fracture configurations that are broadly spread in the newly formed two-parameter space (mean apertures in submillimeter range). First, we performed a percolation analysis on 600 000 synthetic fractures to narrow down the parameter space on which to conduct fluid flow simulations. This revealed that the fractional amount of contact and the percolation probability solely depend on the relative fracture closure. Next, Stokes flow calculations are performed, using a 3-D finite differences code on 6400 fracture models to compute directional permeabilities. The deviations from the cubic law prediction and their statistical variability for equal roughness configurations were quantified. The resulting 2-D solution fields reveal decreasing cubic law accordance down to 1 % for extreme roughness configurations. We show that the non-uniqueness of the results significantly reduces if the correlation length of the aperture field is much smaller than the spatial extent of the fracture. An equation was provided that predicts the average behavior of hydraulic efficiencies and respective fracture permeabilities as a function of their statistical properties. A model to capture fluctuations around that average behavior with respect to their correlation lengths has been proposed. Numerical inaccuracies were quantified with a resolution test, revealing an error of 7 %. By this, we propose a revised parameterization for the permeability of rough single fractures, which takes numerical inaccuracies of the flow calculations into account. We show that this approach is more accurate compared to existing formulations. It can be employed to estimate the permeability of fractures if a measure of fracture roughness is available, and it can readily be incorporated in discrete fracture network modeling approaches.

The computation of turbulent phenomena

We address multiple aspects of the numerical modeling of turbulence and mixing. Fundamental issues of thermodynamics related to nonuniqueness of solutions of the fluid Euler are addressed here. Recent developments of Front Tracking, a part of this analysis, is reviewed as well. The work presented is the result of multiple collaboration teams, and includes in some cases announcements of results yet to be published. The present paper provides a unique opportunity to view the entire range as a unified contribution to science.

Frame Potential in CPn Some Numerical and Analytical Results

In this paper we discuss minimizing a generalized frame potential - F(p,m,n) - for vectors in CPn. We discuss the minimal frame potential as well as the space of solutions for the minimization problem. We investigate this question both numerically and analytically. We find that for some values of p - Equiangular Tight Frames (ETF) are unique solutions, in contrast to the case of p = 2 where they are non-unique solutions.

Improved lithology prediction in channelized reservoirs by integrating stratigraphic forward modelling: Towards improved model calibration in a case study of the Holocene Rhine-Meuse fluvio-deltaic system

Stratigraphic forward modelling (SFM) provides the means to produce geologically coherent and realistic models. In this paper, we demonstrate the possibility of matching lithological variability simulated with a basin-scale advection-diffusion SFM to a data-rich real-world setting, i.e. the Holocene Rhine-Meuse fluvio-deltaic system in the Netherlands. SFM model calibration to real-world data in general has proven non-trivial. This study focuses on a novel inversion process constrained by the top surface and the sand proportion observed at specific pseudo-wells in the study area. Goodness-of-fit expressed by a new fitness function gives the error calculated as the average of two calibration constraints. Computational efficiency was increased significantly by implementing a new optimization process in two hierarchical steps: a) optimization in terms of sediment load and discharge, which are the most influential parameters having the largest uncertainty and b) optimization with respect to the remaining uncertain parameters, these being sediment transport parameters. The calibration process described allows for the most optimal combination of achieving acceptable levels of goodness-of-fit, feasible runtimes and multiple (non-unique) solutions to obtain synthetic stratigraphic output best matching real-world datasets.By removing model realizations which are geologically unrealistic, calibrated SFM models provide a multiscale stratigraphic framework for reconstructing static models of reservoirs which are consistent with the palaeogeographic layout, basin-fill history and external drivers (e.g. sea level, sediment supply). The static reservoir models that are matched with highest certainty therefore contain the highest geological realism and may be used to improve deep subsurface reservoir or aquifer property prediction.The new methodology was applied to the well-established Holocene Rhine-Meuse dataset, which allows a rigorous testing of the optimization; the calibrated SFM allows investigation of controls of the Holocene development on the sedimentary system.

An application of Phragm\xe9n\u2013Lindel\xf6f theorem to the existence of ground state solutions for the generalized Schr\xf6dinger equation with optimal control

In this paper, we develop optimal Phragmén–Lindelöf methods, based on the use of maximum modulus of optimal value of a parameter in a Schrödinger functional, by applying the Phragmén–Lindelöf theorem for a second-order boundary value problems with respect to the Schrödinger operator. Using it, it is possible to find the existence of ground state solutions of the generalized Schrödinger equation with optimal control. In spite of the fact that the equation of this type can exhibit non-uniqueness of weak solutions, we prove that the corresponding Phragmén–Lindelöf method, under suitable assumptions on control conditions of the nonlinear term, is well-posed and admits a nonempty set of solutions.