Non-unique Solutions Research Articles

Algebraic Systems with Positive Coefficients and Positive Solutions

The paper is devoted to the existence, uniqueness and nonuniqueness of positive solutions to nonlinear algebraic systems of equations with positive coefficients. Such systems appear in large numbers of applications, such as steady-state equations in continuous and discrete dynamical models, Dirichlet problems, difference equations, boundary value problems, periodic solutions and numerical solutions for differential equations. We apply Brouwer’s fixed point theorem, Krasnoselskii’s fixed point theorem and monotone iterative methods in order to extend some known results and to obtain new results. We relax some hypotheses used in the literature concerning the strict monotonicity of the involved functions. We show that, in some cases, the unique positive solution can be obtained by a monotone increasing iterative method or by a monotone decreasing iterative method. As a consequence of one of our results, we recover the existence of a non-negative solution of the Leontief system and describe a monotone iterative method to find it.

Nonuniqueness for the ab-family of equations with peakon travelling waves on the circle

Peakon traveling wave solutions on the circle are derived for the cubic ab-family of equations, which includes both the Fokas–Olver–Rosenau–Qiao (FORQ) and Novikov (NE) equations. For a≠0, it is proved that there exists an initial data in the Sobolev space Hs, s<32, with nonunique solutions on the circle. We construct a two-peakon solution with an asymmetric peakon–antipeakon initial profile that collides at a finite time. At the time of collision, the two-peakon solution reduces to a single peakon solution, which will complete the proof of nonuniqueness.

Flame stabilisation by a highly conductive body: multiple steady-state solutions and time-dependent dynamics

We present an investigation of the stabilisation of premixed laminar flames by means of an isolated highly conductive bluff-body, a circular cylinder, placed in a uniform flow of a combustible mixture. It is shown that the problem has non-unique steady-state solutions for certain values of the parameters. Moreover, we solve the time-dependent equations to check the stability of the solutions and demonstrate the possibility of controlling the convergence to a certain steady-state solution.

Parameter identifiability analysis: Mitigating the non-uniqueness issue in the inverse identification of an anisotropic yield function

A large number of conventional mechanical tests are required to calibrate advanced anisotropic yield functions. A well-designed non-conventional mechanical test generating information-rich inhomogeneous strain fields enables to reduce the experimental effort via inverse identification methods. Gradient-based inverse identification methods have repeatedly been reported to yield non-unique solutions for advanced anisotropic yield functions. Although the generic design of non-conventional mechanical tests progressively improves by maximizing the information content using indicators for describing the inhomogeneity of the strain field, it remains questionable whether the uniqueness problem can be avoided for advanced anisotropic yield functions. The latter problem is studied in this paper through the concept of parameter identifiability. A general identifiability analysis framework is proposed enabling to efficiently quantify the parameters sensitivity strength as well as parameter interactions. The feasibility of the framework is assessed through the inverse identification of the Yld2000-2d yield function using a virtual experiment, i.e. a complex notched tensile specimen designed via shape optimization to maximize the strain field inhomogeneity. The advanced synthetic experimentation approach is crucial to avoid errors due to the choice of an inappropriate constitutive model, yet allows accounting for the intrinsic Digital Image Correlation (DIC) filter. For the non-conventional mechanical test under investigation, it is shown that the individual parameter sensitivity and parameter interaction jointly determine the inverse identification quality. It is observed that the detrimental effect of low individual parameter sensitivity on the identification quality can be mitigated by proper parameter subset selection. In addition, it is shown that the orientation of the principal material axes enables to significantly improve the identifiability of the sought model parameters of the anisotropic yield function. The virtual experiment shows that the studied notched tensile test with a tensile direction of 45° from the rolling direction enables to reliably identify all anisotropy parameters of the Yld2000-2d yield function. Finally, actual experiments confirm the importance of identifiability analysis and material orientation for the inverse identification of an anisotropic yield function, yet further research is required to arrive at a sufficiently accurate inverse identification quality.

A New Stochastic Process of Prestack Inversion for Rock Property Estimation

In order to enrich the current prestack stochastic inversion theory, we propose a prestack stochastic inversion method based on adaptive particle swarm optimization combined with Markov chain Monte Carlo (MCMC). The MCMC could provide a stochastic optimization approach, and, with the APSO, have a better performance in global optimization methods. This method uses logging data to define a preprocessed model space. It also uses Bayesian statistics and Markov chains with a state transition matrix to update and evolve each generation population in the data domain, then adaptive particle swarm optimization is used to find the global optimal value in the finite model space. The method overcomes the problem of over-fitting deterministic inversion and improves the efficiency of stochastic inversion. Meanwhile, the fusion of multiple sources of information can reduce the non-uniqueness of solutions and improve the inversion accuracy. We derive the APSO algorithm in detail, give the specific workflow of prestack stochastic inversion, and verify the validity of the inversion theory through the inversion test of two-dimensional prestack data in real areas.

A Sampling Algorithm to Compute the Set of Feasible Solutions for NonNegative Matrix Factorization with an Arbitrary Rank

Nonnegative matrix factorization (NMF) is a useful method to extract features from multivariate data, but an important and sometimes neglected concern is that NMF can result in nonunique solutions. Often, there exist a set of feasible solutions (SFS), which makes it more difficult to interpret the factorization. This problem is especially ignored in cancer genomics, where NMF is used to infer information about the mutational processes present in the evolution of cancer. In this paper the extent of nonuniqueness is investigated for two mutational counts data, and a new sampling algorithm that can find the SFS is introduced. Our sampling algorithm is easy to implement and applies to an arbitrary rank of NMF. This is in contrast to state of the art, where the NMF rank must be smaller than or equal to four. For lower ranks we show that our algorithm performs similar to the polygon inflation algorithm that is developed in relation to chemometrics. Furthermore, we show how the size of the SFS can have a high influence on the appearing variability of a solution. Our sampling algorithm is implemented in the R package SFS (https://github.com/ragnhildlaursen/SFS).

An intelligent traceability method of water pollution based on dynamic multi-mode optimization.

Drinking water safety is a safety issue that the whole society attaches great importance to currently. For sudden water pollution accidents, it is necessary to trace the water pollution source in real time to determine the pollution source’s characteristic information and provide technical support to emergency management departments for decision making. The problems of water pollution’s real-time traceability are as follows: non-uniqueness and dynamic real time of pollution sources. Aiming at these two difficulties, an intelligent traceability algorithm based on dynamic multi-mode optimization was designed and proposed in the work. As a multi-mode optimization problem, pollution traceability could have multiple similar optimal solutions. Firstly, the new algorithm divided the population reasonably through the optimal subpopulation division strategy, which made the nodes’ distribution in a single subpopulation more similar and conducive to local optimization. Then, a similar peak penalty strategy was used to eliminate similar solutions and reduce the non-unique solutions’ number, since real-time traceability required higher algorithm convergence than traditional offline traceability and dynamic problems with parameter changes, historical information preservation, and adaptive initialization strategies could make reasonable use of the algorithm’s historical knowledge to improve the population space and increase the population convergence rate when the problem changed. The experimental results showed the proposed new algorithm’s effectiveness in solving problems—accurately tracing the source of pollution, and obtain corresponding characteristic information in a short time.

Estimating reservoir properties using downhole temperature and pressure data

Predicting the behavior of an underground reservoir requires precise reservoir properties determination. One of the most effective tools for this purpose is well test analysis. Conventional well test analysis employed bottomhole pressure data to understand the reservoir performance better. As the heterogeneity of the system increases, the complexity of well test analysis increases and nonunique solutions can be obtained. Using another source of data, like bottomhole temperature and wellbore temperature profile, can significantly minimize the uncertainty in reservoir model identification, particularly in more heterogeneous reservoirs like multi-layered systems.In the current study, the permeability of various reservoir models has been estimated via optimization approaches. For this purpose, uncertain reservoir parameters have been considered as optimization variables. The particle swarm optimization (PSO) and differential evolution (DE) algorithms have been utilized to find the optimal value of these parameters. The temperature and pressure data have been assumed as the input parameters and were simultaneously analyzed to determine the properties of reservoir layers more accurately. Some statistical parameters were employed to verify the accuracy of the developed approaches. Using simultaneous analysis of the bottomhole data and wellbore temperature profile, the average absolute percent relative error (AAPRE) values for the multi-layered heterogeneous reservoir model was 1.57%. This approach, compared to the bottomhole pressure analysis technique, in which AAPRE of 19.91% was obtained, led to significantly more reliable results.Furthermore, the best history-matched models created through various analysis approaches were selected to perform the prediction scenarios. Absolute relative error (ARE) values for the multi-layered heterogeneous reservoir model for bottomhole pressure analysis and simultaneous analysis were 12.00% and 0.26%, respectively. The results demonstrated that employing the simultaneous bottomhole temperature and pressure analysis approach has the lowest estimations error, reducing the uncertainties in the reservoir model predictions.

Thermophoresis particle deposition of CoFe2O4-TiO2 hybrid nanoparticles on micropolar flow through a moving flat plate with viscous dissipation effects

PurposeThis study aims to investigate the micropolar fluid flow through a moving flat plate containing CoFe2O4-TiO2 hybrid nanoparticles with the substantial influence of thermophoresis particle deposition and viscous dissipation.Design/methodology/approachThe partial differential equations are converted to the similarity equations of a particular form through the similarity variables. Numerical outcomes are computed by applying the built-in program bvp4c in MATLAB. The process of flow, heat and mass transfers phenomena are examined for several physical aspects such as the hybrid nanoparticles, micropolar parameter, the thermophoresis particle deposition and the viscous dissipation.FindingsThe friction factor, heat and mass transfer rates are higher with an increment of 1.4%, 2.2% and 1.4%, respectively, in the presence of the hybrid nanoparticles (with 2% volume fraction). However, they are declined because of the rise of the micropolar parameter. The imposition of viscous dissipation reduces the heat transfer rate, significantly. Meanwhile, thermophoresis particle deposition boosts the mass transfer. Multiple solutions are developed for a certain range of physical parameters. Lastly, the first solution is shown to be stable and reliable physically.Originality/valueAs far as the authors have concerned, no work on thermophoresis particle deposition of hybrid nanoparticles on micropolar flow through a moving flat plate with viscous dissipation effect has been reported in the literature. Most importantly, this current study reported the stability analysis of the non-unique solutions and, therefore, fills the gap of the study and contributes to new outcomes in this particular problem.

Technology Focus: Well Testing (February 2022)

Bias is always a factor in decision making. To put it a catchier way, “if you have a brain, you are biased.” This was the opening to a recent soft-skills training course I took on the science of better decision making. I left pondering the implications of bias on pressure transient analysis. Well tests are conducted throughout the resource life cycle to establish well and reservoir properties and to optimize depletion planning. But this requires well-test interpretation, which, in turn, requires choosing among nonunique solutions. One key to reducing nonuniqueness is to integrate pressure data with information from other disciplines, including - Routine core analysis, core description, and well logs - Formation-tester mobilities and interval pressure transient tests - Production logging and distributed temperature surveys - Fieldwide production and injection history - Seismic data and geologic understanding This is where technology comes in. New tools have made the unmeasureable measurable and have provided data in larger quantities with improved precision and for longer periods of time. While it may be an engineer’s dream, more and better data do not necessarily lead to better decisions. As with any sound scientific or engineering study, one must think about bias. Is the data being unconsciously cherry-picked to mold the desired outcome? After all, the subsurface is complex enough that multiple assumptions and hypotheses remain plausible; not all information may fit the storyline, and some might be dismissed or underweighted before the full range of possibilities is considered. A critical component of interpretation includes acknowledging one’s own bias and carefully listening to and seriously considering alternative and opposing views. Cold-eyes reviews and devil’s advocates are part of a centuries-old toolkit proven to root out bias, ambiguity, and unnecessary complexity, leading to better-defined problems and, ultimately, better decision making. Interpreting data from new and well-established technologies under the framework of the Socratic method paves the way to a clearer view of the subsurface. This month’s papers demonstrate the continued relevance and diversity of our discipline. The ambitious permeability log, referred to as “elusive” in my column 2 years ago, now boasts a tangible pilot in a Middle Eastern carbonate field. Well tests and cleanup operations do not always require flaring. Alternative fluid-disposal solutions, tailored to project specifics, show potential to meet the timely topic of emissions. Machine learning offers a key to deciphering well connectivity in field data, overcoming noise and other suboptimal conditions. Recommended additional reading at OnePetro: www.onepetro.org. IPTC 21780 - Design and Implementation of a Water-Based Micronized Weighting Agent Fluid System for Deepwater Drillstem Testing Operation in Environmentally Restricted Location by Marcelo Dourado Motta, MI-SWACO, a Schlumberger Company, et al. SPE 205145 - Systematic Application of Pressure and Temperature Transient Analysis in an Oil Field: A Case Study by Khafiz Muradov, Heriot-Watt University, et al.

An approach for automatic parameters evaluation in unconventional oil reservoirs with deep reinforcement learning

Accurate estimation of unconventional reservoir parameters is of great significance to improve the development effect and prolong the life cycle of production wells. Reservoir parameter estimation based on pressure transient analysis (PTA) is a mainstream method due to its ease of use. However, the non-unique solution and human bias make the reliability of the method less than ideal. Therefore, a robust automatic interpretation method is urgently needed to alleviate these problems. In this work, we propose an automatic parameter evaluation method of unconventional reservoirs based on a combination of deep reinforcement learning (DRL) and PTA. Our key insight is to treat the PTA process, namely the pressure derivative curve matching process, as Markov decision process (MDP) and solve the optimal matching policy through DRL algorithm. Based on this idea, we trained an agent to automatically adjust the parameters of the trilinear flow model, a classic PTA model, and finally complete the pressure derivative curve matching to evaluate the unconventional oil reservoirs parameters. To make the training converge, branch deep Q-network with independent rewards strategy (IR-BDQ) was proposed to train the agent. Results show that IR-BDQ algorithm can effectively improve the convergence speed and parameter evaluation accuracy compared with other DRL algorithms. The results of 1000 curve matching tests showed that the mean average relative errors of parameters is 13.1%. In addition, comparison with the supervised learning algorithm reveals that the proposed method has the smallest variance of parameter inversion errors, indicating that the method has good robustness. Finally, the case study shows that the proposed method can effectively alleviate the non-unique solution problem, which is of great significance to improve the repeatability of parameter evaluation results in unconventional oil reservoirs.

Online Acoustic System Identification Exploiting Kalman Filtering and an Adaptive Impulse Response Subspace Model

We introduce a novel algorithm for online estimation of Acoustic Impulse Responses (AIRs) which allows for fast convergence by exploiting prior knowledge about the fundamental structure of AIRs. The proposed method assumes that the variability of AIRs of an acoustic scene is confined to a low-dimensional manifold which is embedded in a high-dimensional space of possible AIR estimates. We discuss various approaches which exploit a training data set of AIRs, e.g., high-accuracy AIR estimates from the acoustic scene, to learn a local affine subspace approximation of the AIR manifold. The model is motivated by the idea of describing the generally nonlinear AIR manifold locally by tangential hyperplanes and its validity is verified for simulated data. Subsequently, we describe how the manifold assumption can be used to enhance online AIR estimates by projecting them onto an adaptively estimated subspace. Motivated by the assumption of manifolds being locally Euclidean, the parameters determining the adaptive subspace are learned from the nearest neighbor AIR training samples to the current AIR estimate. To assess the proximity of training data AIRs to the current AIR estimate, we introduce a probabilistic extension of the Euclidean distance which improves the performance for applications with non-white excitation signals. Furthermore, we describe how model imperfections can be tackled by a soft projection of the AIR estimates. The proposed algorithm exhibits significantly faster convergence properties in comparison to a high-performance state-of-the-art algorithm. Furthermore, we show an improved steady-state performance for speech-excited system identification scenarios suffering from high-level interfering noise and nonunique solutions.

A deep reinforcement learning (DRL) based approach for well-testing interpretation to evaluate reservoir parameters

Parameter inversions in oil/gas reservoirs based on well test interpretations are of great significance in oil/gas industry. Automatic well test interpretations based on artificial intelligence are the most promising to solve the problem of non-unique solution. In this work, a new deep reinforcement learning (DRL) based approach is proposed for automatic curve matching for well test interpretation, by using the double deep Q-network (DDQN). The DDQN algorithms are applied to train agents for automatic parameter tuning in three conventional well-testing models. In addition, to alleviate the dimensional disaster problem of parameter space, an asynchronous parameter adjustment strategy is used to train the agent. Finally, field applications are carried out by using the new DRL approaches. Results show that step number required for the DDQN to complete the curve matching is the least among, when comparing the naive deep Q-network (naive DQN) and deep Q-network (DQN). We also show that DDQN can improve the robustness of curve matching in comparison with supervised machine learning algorithms. Using DDQN algorithm to perform 100 curve matching tests on three traditional well test models, the results show that the mean relative error of the parameters is 7.58% for the homogeneous model, 10.66% for the radial composite model, and 12.79% for the dual porosity model. In the actual field application, it is found that a good curve fitting can be obtained with only 30 steps of parameter adjustment.

Anomalous Dissipation in Passive Scalar Transport

We study anomalous dissipation in hydrodynamic turbulence in the context of passive scalars. Our main result produces an incompressible \(C^\infty ([0,T)\times {\mathbb {T}}^d)\cap L^1([0,T]; C^{1-}({\mathbb {T}}^d))\) velocity field which explicitly exhibits anomalous dissipation. As a consequence, this example also shows the non-uniqueness of solutions to the transport equation with an incompressible \(L^1([0,T]; C^{1-}({\mathbb {T}}^d))\) drift, which is smooth except at one point in time. We also give a sufficient condition for anomalous dissipation based on solutions to the inviscid equation becoming singular in a controlled way. Finally, we discuss connections to the Obukhov-Corrsin monofractal theory of scalar turbulence along with other potential applications.

К ОБОСНОВАНИЮ ОДНОЗНАЧНОСТИ ПРОДОЛЖЕНИЯ РЕШЕНИЯ ЗАДАЧ О ДЕФОРМИРОВАНИИ МЯГКИХ ОБОЛОЧЕК МЕТОДОМ ДИФФЕРЕНЦИРОВАНИЯ ПО ПАРАМЕТРУ

Criterion of hyperelastic soft shell deforming nonlinear problem numerical solution continuation uniqueness assessment is suggested in the work. The criterion can be used when carrying out calculations. It is based on investigation of properties of Jacobi matrix of linear algebraic equations system which is formed when using parameter differentiation method. This method allows reducing nonlinear boundary value problem solution to a couple of quasilinear boundary value and nonlinear initial problems and applying initial parameters method of solving linear boundary value problems. For assessment of solution continuation uniqueness in each point of integration interval one needs controlling magnitudes of resolving differential equation system right-hand sides vector components, as well as calculating determinant and rank of Jacobi matrix of algebraic equations system formed as a result of initial parameters method using, and expanded Jacobi matrix rank calculation. For testing suggested criterion the problem of neohookean material hemisphere static inflation by uniformly applied pressure is considered. Solution of this problem as certain values of numerical algorithm parameters leads to various calculation difficulties – calculation stability loss, big errors of calculation results, non-uniqueness of solution which reason requires additional investigations. It is shown that in the points, where mentioned difficulties are met, determinateness conditions for the function of right-hand sides of differential equation system formulated within the framework of the suggested criterion are broken.

Double solutions and stability analysis of slip flow past a stretching/shrinking sheet in a carbon nanotube

A stagnation point flow past a stretching/shrinking surface in carbon nanotubes (CNTs) with slip effects is investigated in this paper. Applying transformations of similarity, the governing partial differential equations are modified to the nonlinear ordinary differential equations. Afterward, they are numerically solved in Matlab by the bvp4c solver. The single-wall CNTs and multi-wall CNTs are used, including water as a base fluid. The effects of the flow parameters are investigated, shown in the form of graphs, and physically evaluated for the dimensionless velocity, temperature, skin friction, and Nusselt numbers. According to our findings, the unique solution exists for stretching sheets, whereas non-unique solutions are obtainable for shrinking sheets. The stability analysis is utilized to discover which solution is stable.

Uncertain solutions and multiple steady states in the built ventilation flows

In past decades, several peers and not too many publications have reported that uncertain solutions existed in the building ventilation flows. In the present work, numerical, theoretical and experimental researches have been jointly applied for the non-unique flow solutions occurring in the built environment, covering from thermal driven flows, pressure driven flows and multi component flows. General multiple steady flows could be observed when different initial conditions were imposed. Independent flow branch could switch from each other during the critical point or some sudden transitions were imposed. Theoretical solutions could give rough routes for multiple flows, whereas numerical ones could give such detailed transitions delicately. Experimental model results well demonstrated those theoretical and numerical results. Future low carbon ventilation engineering applications could benefit from present fundamental researches.

Numerical Study of the Non-Uniqueness of Solutions to the Showalter-Sidorov Problem for a Mathematical Model of I-Beam Deformation

Numerical Study of the Non-Uniqueness of Solutions to the Showalter-Sidorov Problem for a Mathematical Model of I-Beam Deformation

Non-uniqueness of solution for initial value problem of impulsive fractional partial differential equations

Non-uniqueness of solution for initial value problem of impulsive fractional partial differential equations

Non-uniqueness of solution for initial value problem of impulsive fractional partial differential equations

Non-uniqueness of solution for initial value problem of impulsive fractional partial differential equations