Implicit Approximation Research Articles

A combined mixed finite element method and discontinuous Galerkin method for hybrid-dimensional fracture models of two-phase flow

In this paper, a combined numerical method consisting of the mixed finite element method (MFE) for the pressure equation and the discontinuous Galerkin (DG) method for the saturation equation is proposed to solve hybrid-dimensional fracture models of incompressible two-phase flow in porous media. The hybrid-dimensional fracture models treat fractures as (d−1)-dimensional interfaces immersed in d-dimensional matrix domains and take fluid exchange between fractures and surrounding matrix into account. Fully implicit approximation schemes combining the MFE-DG method with the backward Euler time discretization for the models with both a single fracture and an intersecting fractures network are all formulated successfully. The stability of the discrete solution is analyzed, and optimal error estimates in H(div)-norm for the velocity and in L2-norm for the pressure are derived, as well as in the discrete H1-norm for the saturation. Numerical experiments with a single fracture and a T-junction intersecting fractures network are conducted to verify the accuracy of our theoretical analysis.

Accounting for intensity variation within pixels of Shack-Hartmann wavefront sensors

The accuracy of centroiding algorithms in Shack-Hartmann wavefront sensing is limited by the implicit approximation of uniform pixel illumination. Iterative centroiding algorithms are further limited by the consideration of full pixels to define the image domain for centroiding. Here, we demonstrate two practical and complementary approaches to mitigate both these sources of error. First, we consider partial or ‘fractional’ pixels to maintain centroiding area symmetry around the center of mass. Secondly, we propose methods to perform piece-wise polynomial interpolation to calculate intensity distribution within pixels, which is then used to estimate the centroid within each pixel area. This approach that accounts for intensity non-uniformity across pixels notably reduces centroid errors up to a factor of 5 across lenslet image widths ranging from 1.33 to 3.10 pixels full-width-half-maximum (FWHM). Consequently, wavefront sensing errors decrease from 14 % to 4 %, on average, for FWHM = 1.35 pixels, demonstrating a substantial benefit when the number of pixels per lenslet is minimized to enhance the signal-to-noise ratio or increase frame rate.

Multimodal and Multifactor Branching Time Active Inference.

Active inference is a state-of-the-art framework for modeling the brain that explains a wide range of mechanisms. Recently, two versions of branching time active inference (BTAI) have been developed to handle the exponential (space and time) complexity class that occurs when computing the prior over all possible policies up to the time horizon. However, those two versions of BTAI still suffer from an exponential complexity class with regard to the number of observed and latent variables being modeled. We resolve this limitation by allowing each observation to have its own likelihood mapping and each latent variable to have its own transition mapping. The implicit mean field approximation was tested in terms of its efficiency and computational cost using a dSprites environment in which the metadata of the dSprites data set was used as input to the model. In this setting, earlier implementations of branching time active inference (namely, BTAIVMP and BTAIBF) underperformed in relation to the mean field approximation (BTAI3MF) in terms of performance and computational efficiency. Specifically, BTAIVMP was able to solve 96.9% of the task in 5.1 seconds, and BTAIBF was able to solve 98.6% of the task in 17.5 seconds. Our new approach outperformed both of its predecessors by solving the task completely (100%) in only 2.559 seconds.

Tight conic approximation of testing regions for quantum statistical models and measurements

Quantum statistical models (i.e., families of normalized density matrices) and quantum measurements (i.e., positive operator-valued measures) can be regarded as linear maps: the former, mapping the space of effects to the space of probability distributions; the latter, mapping the space of states to the space of probability distributions. The images of such linear maps are called the testing regions of the corresponding model or measurement. Testing regions are notoriously impractical to treat analytically in the quantum case. Our first result is to provide an implicit outer approximation of the testing region of any given quantum statistical model or measurement in any finite dimension: namely, a region in probability space that contains the desired image, but is defined implicitly, using a formula that depends only on the given model or measurement. The outer approximation that we construct is minimal among all such outer approximations, and close, in the sense that it becomes the maximal inner approximation up to a constant scaling factor. Finally, we apply our approximation to provide sufficient conditions, that can be tested in a semi-device-independent way, for the ability to transform one quantum statistical model or measurement into another.

Multiscale model reduction for the time fractional thermoporoelasticity problem in fractured and heterogeneous media

In this paper, we consider the time fractional thermoporoelasticity problem in fractured and heterogeneous media. The mathematical model with a time memory formalism is described by a coupled system of equations for pressure, temperature and displacements. We use an implicit finite difference approximation for temporal discretization. We present a fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) for two-dimensional model problems. Further, we use the Generalized Multiscale Finite Element Method (GMsFEM) for coarse grid approximation. The primary concept behind the proposed method is to streamline the complexity inherent in the thermoporoelasticity problem. Given that our model equation incorporates multiple fractional powers, leading to multiple unknowns with memory effects, we aim to address this intricacy by optimizing the problem’s dimensionality. As a result, the solution is sought on a coarse grid, a strategic choice that not only simplifies the computational cost but also contributes to significant time savings. We present numerical results for the two-dimensional model problems in heterogeneous fractured porous media. We derive relative errors between the reference fine grid solution and the multiscale solution for different numbers of multiscale basis functions. The results confirm that the proposed method is able to achieve good accuracy with a few degrees of freedoms on the coarse grid.

Evaluation of a New Approach for Entrainment and Detrainment Rate Estimation

AbstractEntrainment and detrainment rates (ε and δ) constitute the most critical free parameters in mass flux schemes commonly employed for cumulus parameterizations. Recently, Zhu et al. (2021) introduced a new approach that utilizes aircraft observations to simultaneously estimate ε and δ for cumulus clouds, overcoming the limitation of other observation‐based approaches that solely yield ε without offering insights into δ. This study aims to comprehensively evaluate the reliability of this new approach. First, evaluation using an Explicit Mixing Parcel Model demonstrates the capability of the new approach to back‐calculate predetermined ε and δ based on the physical properties before and after the entrainment mixing. Second, evaluation using large‐eddy simulations illustrates that the new approach yields consistent ε and δ profiles compared to the traditional approach. Sensitivity tests indicate a weak sensitivity of the estimated δ with the new approach to the entrained air source. A decrease in the proportion of cloudy air in the assumed detrained air leads to a reduction in the estimated δ, while ε remains unaffected. Finally, the most appropriate assumptions for entrained and detrained air are discussed. Estimating ε for cumulus parameterizations involves acquiring ambient air more than 500 m away from the cloud edge as entrained air. Due to implicit mean field approximations in the traditional approach, determining the optimal assumption for detrained air properties proves challenging. This study confirms the reliability of the new approach in estimating ε and δ, providing confidence in its application to extensive observational data and advancement in parameterization.

Implicit and Fully Discrete Approximation of the Supercooled Stefan Problem in the Presence of Blow-Ups

Implicit and Fully Discrete Approximation of the Supercooled Stefan Problem in the Presence of Blow-Ups

Convergence of a TPFA finite volume scheme for nonisothermal immiscible compressible two-phase flow in porous media

This paper presents and analyzes a finite volume scheme for a fully degenerate parabolic system modeling the compressible two-phase flow, such as water-gas, problem in porous media considering the effects of temperature. The problem is written in terms of the phase formulation, i.e. the saturation of one phase, the pressure of the second phase and the temperature are primary unknowns. We use a first-order time implicit Euler scheme. The spatial discretization uses a fully coupled fully implicit cell-centered two-point flux approximation scheme where upwinding of the flux across an interface between two control volumes is performed for each fluid phase. Under physically relevant assumptions on the data, we prove the maximum principle for both saturation and temperature. We obtain a priori estimates and we show that the discrete problem is well posed. Also, we exploit various a priori estimates as well as compactness arguments to establish the convergence of the numerical scheme toward the weak solution. The open-source platform DuMuX was employed for the implementation of the developed algorithm based on the above scheme. Numerical result are presented to demonstrate the efficiency of this scheme. The test addresses the evolution in 2D of CO2 injection into a layered aquifer. The algorithm's convergence with first order is validated by our numerical results, which support the theoretical results. To our best knowledge, this is the first convergence result of a finite volume scheme in the case of nonisothermal immiscible compressible two-phase flow in porous media.

Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients

This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings.

Refined Schemes for Computing the Dynamics of Elastoviscoplastic Media

For a stable numerical solution of the system of equations governing an elastoviscoplastic continuous medium model, a second-order explicit-implicit scheme is proposed. An explicit approximation is used for the equations of motion, and an implicit approximation, for the constitutive relations containing a small relaxation time parameter in the denominator of the nonlinear free terms. A second-order implicit approximation for isotropic and anisotropic elastoviscoplastic models is constructed to match the orders of approximation of the explicit elastic and implicit adjustment steps. Refined formulas for correcting the stress deviators after the elastic step are derived for various viscosity function representations. The resulting solutions of the second-order implicit approximation for the stress deviators of the elastoviscoplastic equations admit passage to the limit as the relaxation time tends to zero. The correcting formulas obtained via this passage to the limit can be treated as regularizers of the numerical solutions to the elastoplastic systems.

Bioconvective periodic MHD Eyring-Powell fluid flow around a rotating cone: Influence of multiple diffusions and oxytactic microorganisms

Investigating the effects of periodic magnetic field and triple diffusion on bioconvection flow through a rotating cone populated by oxytactic bacteria and an Eyring-Powell fluid is innovative and significant. The magneto-bioconvection flow of an Eyring-Powell multi-diffusive fluid across a spinning cone is investigated in this paper, considering the effect of oxytactic microorganisms. The non-similar technique is used in the mathematical analysis of the flow over a spinning cone. The flow under consideration contains two diffusive species: liquid hydrogen and liquid oxygen. In light of the periodic magnetic field, the surface gradients, notably skin friction, exhibit wavy effects in the boundary layer domain. The governing equations for the fluid flow in the current flow problem, accompanied by heat diffusion, species diffusion, rotation, bioconvection, and periodic magnetic, are highly coupled nonlinear PDEs dependent on the proper initial and boundary conditions. Mangler's transformations convert them into non-dimensional forms, and numerical non-similar solutions are produced using implicit finite difference approximation and quasi-linearisation. The enriching values of bioconvective Rayleigh number Rb decline the velocity F, as a result, lessen the friction between the surrounding fluid and the cone's surface. The improving Peclet number Pe and microbial density difference σ reduced the microorganism density profile and heightened the microorganism density number Re-1/2Nn.

Multiscale Model Reduction with Local Online Correction for Polymer Flooding Process in Heterogeneous Porous Media

In this work, we consider a polymer flooding process in heterogeneous media. A system of equations for pressure, water saturation, and polymer concentration describes a mathematical model. For the construction of the fine grid approximation, we use a finite volume method with an explicit time approximation for the transports and implicit time approximation for the flow processes. We employ a loose coupling approach where we first perform an implicit pressure solve using a coarser time step. Subsequently, we execute the transport solution with a minor time step, taking into consideration the constraints imposed by the stability of the explicit approximation. We propose a coupled and splitted multiscale method with an online local correction step to construct a coarse grid approximation of the flow equation. We construct multiscale basis functions on the offline stage for a given heterogeneous field; then, we use it to define the projection/prolongation matrix and construct a coarse grid approximation. For an accurate approximation of the nonlinear pressure equation, we propose an online step with calculations of the local corrections based on the current residual. The splitted multiscale approach is presented to decoupled equations into two parts related to the first basis and all other basis functions. The presented technique provides an accurate solution for the nonlinear velocity field, leading to accurate, explicit calculations of the saturation and concentration equations. Numerical results are presented for two-dimensional model problems with different polymer injection regimes for two heterogeneity fields.

Impulsive motion causes time-dependent nonlinear convective Williamson nanofluid flow across a rough conical surface: Semi-similar analysis

The time-dependent nonlinear convective Williamson nanofluid flow over a rough conical surface in the presence of numerous diffusions and Arrhenius activation energy effects are the main focus of the investigation in this paper. The unsteady flow is essentially due to the impulsive mainstream flow. Two diffusive species, namely, liquid Hydrogen and liquid Nitrogen, occur in the flow under investigation. A sinusoidal waveform mathematically models the rough conical surface with small amplitude and high frequency. Thus, the surface gradients, namely skin friction, exhibit wavy effects in the boundary layer regime. In the current flow problem, the governing equations for the fluid flow are highly coupled nonlinear PDEs that depend on the proper initial and boundary conditions. Mangler's transformations are used to convert them into non-dimensional forms, and implicit finite difference approximation and quasilinearization are employed to produce numerical semi-similar solutions. The numerical results representing the boundary layer effects are displayed graphically and analyzed for parametric values. The friction experienced by the rough surface is strongly fluctuating for higher values of Williamson parameter W. The heat transfer rate Rex-1/2Nu increases approximately by 61% and 63% at ξ=0.8 when Nt decreases from Nt=0.5 to Nt=0.1, for the cases of W = 0 (Newtonian fluid) and W = 1(Williamson fluid), respectively. Additionally, there is a strong correlation between the current findings and the related results previously published in the literature.

A New Viscosity Implicit Approximation Method for Solving Variational Inequalities over the Common Fixed Points of Nonexpansive Mappings in Symmetric Hilbert Space

In this paper, based on the viscosity approximation method and the hybrid steepest-descent iterative method, a new implicit iterative algorithm is presented for finding the common fixed points set of a finite family of nonexpansive mappings in a reflexive Hilbert space, which is called a symmetric space. We prove that the sequence generated by this new implicit rule strongly converges to the unique solution of a class of variational inequalities under certain appropriate conditions of the parameters. Moreover, we also study the applications to a broader family of strictly pseudo-contractive mappings and generalized equilibrium problems that involve several variational inequality problems, optimization problems, and fixed-point problems. Finally, numerical results are provided to clarify the stability and effectiveness of the algorithm and to compare with some existing iterative algorithms.

Entropy generation analysis from the time-dependent quadratic combined convective flow with multiple diffusions and nonlinear thermal radiation

Diffusions of multiple components have numerous applications such as underground water flow, pollutant movement, stratospheric warming, and food processing. Particularly, liquid hydrogen is used in the cooling process of the aeroplane. Further, liquid nitrogen can find applications in cooling equipment or electronic devices, i.e ., high temperature superconducting (HTS) cables. So, herein, we have analysed the entropy generation (EG), nonlinear thermal radiation and unsteady (time-dependent) nature of the flow on quadratic combined convective flow over a permeable slender cylinder with diffusions of liquid hydrogen and nitrogen. The governing equations for flow and heat transfer characteristics are expressed in terms of nonlinear coupled partial differential equations. The solutions of these equations are attempted numerically by employing the quasilinearization technique with the implicit finite difference approximation. It is found that EG is minimum for double diffusion (liquid hydrogen and heat diffusion) than triple diffusion (diffusion of liquid hydrogen, nitrogen and heat). The enhancing values of the radiation parameter R d and temperature ratio θ w augment the fluid temperature for steady and unsteady cases as well as the local Nusselt number. Because, the fluid absorbs the heat energy released due to radiation, and in turn releases the heat energy from the cylinder to the surrounding surface.

Quadratic combined convective flow about yawed cylinder in presence of time variations and magnetic effects: entropy analysis

The significance of time-dependent variations in a quadratic combined convective magnetohydrodynamic (MHD) flow around an infinite yawed cylinder with entropy analysis is explored in the present investigation. There are numerous real-world applications wherein the yawed-shaped bodies are used extensively, for example, overhead cables, bridge stay cables, chimney stacks etc. First, the dimensional governing equations are made dimensionless by applying the appropriate transformations of non-similar nature. After that, using the Quasilinearization technique, the resulting equations are linearised and discretized by employing implicit finite difference approximations. In unsteady flow, velocity distributions along chordwise and spanwise directions reduced when compared with the steady case. As the cylinder's yaw angle grows, fluid pressure inside it rises, increasing its velocity in all directions. The surface drag coefficients along chord & spanwise directions and the energy transfer rate increase approximately by 104%, 67% and 40%, respectively, as R i increases from −2 to 10, for accelerating flow ( α > 0 ) at θ = 45 0 and τ = 1 . An increase in the yaw angle results in more entropy generation (EG). It shows that yawed cylinder overcomes the EG difficulties faced in the case of a vertical cylinder. It is revealed that, by using a yawed cylinder with MHD, one can restrict the amount of energy loss.

Error quantification of the Arrhenius blending rule for viscosity of hydrocarbon mixtures

Six hundred and seventy-five measurements of dynamic viscosity and density have been used to assess the prediction error of the Arrhenius blending rule for kinematic viscosity of hydrocarbon mixtures. Major trends within the data show that mixture complexity–binary to hundreds of components—and temperature are more important determinants of prediction error than differences in molecular size or hydrogen saturation between the components of the mixtures. Over the range evaluated, no correlation between prediction error and mole fractions was observed, suggesting the log of viscosity truly is linear in mole fraction, as indicated by the Arrhenius blending rule. Mixture complexity and temperature also impact molar volume and its prediction. However, a linear regression between the two model errors explains less than 20% of the observed variation, indicating that mixture viscosity and/or molar volume are not linear with respect to temperature and/or mixture complexity. Extensive discussion of the intermolecular forces and the geometric arrangement of molecules and vacancies in liquids, which ultimately determines its viscosity, is brought into context with the implicit approximations behind the Arrhenius blending rule. The complexity of this physics is not compatible with a simple algebraic correction to the model. However, sufficient data is now available to determine confidence intervals around the prediction of fuel viscosity based on its component mole fractions and viscosities. At −40°C, when all identified components are pure molecules the modeling error is 13.2% of the predicted (nominal) viscosity times the root mean square of the component mole fractions.

SGD-Based Cascade Scheme for Higher Degrees Wiener Polynomial Approximation of Large Biomedical Datasets

The modern development of the biomedical engineering area is accompanied by the availability of large volumes of data with a non-linear response surface. The effective analysis of such data requires the development of new, more productive machine learning methods. This paper proposes a cascade ensemble that combines the advantages of using a high-order Wiener polynomial and Stochastic Gradient Descent algorithm while eliminating their disadvantages to ensure a high accuracy of the approximation of such data with a satisfactory training time. The work presents flow charts of the learning algorithms and the application of the developed ensemble scheme, and all the steps are described in detail. The simulation was carried out based on a real-world dataset. Procedures for the proposed model tuning have been performed. The high accuracy of the approximation based on the developed ensemble scheme was established experimentally. The possibility of an implicit approximation by high orders of the Wiener polynomial with a slight increase in the number of its members is shown. It ensures a low training time for the proposed method during the analysis of large datasets, which provides the possibility of its practical use in the biomedical engineering area.

Implicit Randomized Progressive-Iterative Approximation for Curve and Surface Reconstruction

In this paper, we propose the implicit randomized progressive-iterative approximation (IR-PIA) method for curve and surface reconstruction. By introducing the effective probability criterion, the IR-PIA method selects the working elements of the collocation matrix to adjust the control coefficients. It is proved that the sequence of curves and surfaces generated by the control coefficients converge to the least-norm results in expectation. The IR-PIA method reduces the computation complexity and speeds up the curve and surface reconstruction compared with the implicit progressive-iterative approximation (I-PIA) method (Hamza et al., 2020). Numerical examples show that the IR-PIA method can be more efficient than the I-PIA method (Hamza et al., 2020). • The IR-PIA method converge to the least-norm solution in expectation. • The IR-PIA method eliminates the extra zero-level sheets effectively. • The working elements are selected to update the control coefficients. • The larger entries of the difference vectors can be annihilated as far as possible.

A computational macroscale model for the time fractional poroelasticity problem in fractured and heterogeneous media

In this paper, we consider the poroelasticity problem with a time memory formalism that couples the pressure and displacement, and we assume this multiphysics process occurs in multicontinuum media. A coupled system of equations for pressures in each continuum and elasticity equations for displacements of the medium are included in the mathematical model.Based on the Caputo’s time fractional derivative, we derive an implicit finite difference approximation for time discretization. Also, a Discrete Fracture Model (DFM) is used to model fluid flow through fractures and treat the complex network of fractures. Further, we develop a coarse grid approximation based on the Generalized Multiscale Finite Element Method (GMsFEM), where we solve local spectral problems for construction of the multiscale basis functions. The main idea of the proposed method is to reduce the dimensionality of the problem because our model equation has multiple fractional powers, there multiple unknowns with memory effects. Consequently, the solution is on a coarse grid, which saves some computational time.We present numerical results for the two-dimensional model problems in fractured heterogeneous porous media. After, we investigate error analysis between reference (fine-scale) solution and multiscale solution with different numbers of multiscale basis functions. The results show that on a coarse grid, the proposed approach can achieve good accuracy.