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Physics-Informed Neural Networks (PINNs) have been widely used for solving partial differential equations (PDEs) of different types, including fractional PDEs (fPDES) (Pang et al., 2019). Herein, we propose a new general (quasi) Monte Carlo PINN (GMC-PINNs) for solving fPDEs on irregular domains. Specifically, instead of approximating fractional derivatives by Monte Carlo approximations of integrals as was done previously in Guo et al. (2022), we use a more general Monte Carlo approximation method to solve different fPDEs, which is valid for fractional differentiation under any definition. Moreover, based on the ensemble probability density function, the generated nodes are all located in denser regions near the target point where we perform the differentiation. This has an unexpected connection with known finite difference methods on non-equidistant or nested grids, and hence our method inherits their advantages. At the same time, the generated nodes exhibit a block-like dense distribution, leading to a good computational efficiency of this approach. We present the framework for using this algorithm and apply it to several examples. Our results demonstrate the effectiveness of GMC-PINNs in dealing with irregular domain problems and show a higher computational efficiency compared to the original fPINN method. We also include comparisons with the Monte Carlo fPINN (Guo et al., 2022). Finally, we use examples to demonstrate the effectiveness of the method in dealing with fuzzy boundary location problems, and then use the method to solve the coupled 3D fractional Bloch–Torrey equation defined in the ventricular domain of the human brain, and compare the results with classical numerical methods. [ABSTRACT FROM AUTHOR]

We study the dynamic evolution of COVID-19 caused by the Omicron variant via a fractional susceptible–exposed–infected–removed (SEIR) model. Preliminary data suggest that the symptoms of Omicron infection are not prominent and the transmission is, therefore, more concealed, which causes a relatively slow increase in the detected cases of the newly infected at the beginning of the pandemic. To characterize the specific dynamics, the Caputo–Hadamard fractional derivative is adopted to refine the classical SEIR model. Based on the reported data, we infer the fractional order and time-dependent parameters as well as unobserved dynamics of the fractional SEIR model via fractional physics-informed neural networks. Then, we make short-time predictions using the learned fractional SEIR model. [ABSTRACT FROM AUTHOR]

Patients with type 2 diabetes mellitus (T2DM) who have severe hypoglycemia (SH) poses a considerable risk of long-term death, especially among the elderly, demanding urgent medical attention. Accurate prediction of SH remains challenging due to its multifaced nature, contributed from factors such as medications, lifestyle choices, and metabolic measurements. In this study, we propose a systematic approach to improve the robustness and accuracy of SH predictions using machine learning models, guided by clinical feature selection. Our focus is on developing long-term SH prediction models using both semi-supervised learning and supervised learning algorithms. Using the action to control cardiovascular risk in diabetes trial, which includes electronic health records for over 10,000 individuals, we focus on studying adults with T2DM. Our results indicate that the application of a multi-view co-training method, incorporating the random forest algorithm, improves the specificity of SH prediction, while the same setup with Naive Bayes replacing random forest demonstrates better sensitivity. Our framework also provides interpretability of machine learning models by identifying key predictors for hypoglycemia, including fasting plasma glucose, hemoglobin A1c, general diabetes education, and NPH or L insulins. The integration of data routinely available in electronic health records significantly enhances our model’s capability to predict SH events, showcasing its potential to transform clinical practice by facilitating early interventions and optimizing patient management. By enhancing prediction accuracy and identifying crucial predictive features, our study contributes to advancing the understanding and management of hypoglycemia in this population. [ABSTRACT FROM AUTHOR]

Electrostatic correlation effects near charged planar surfaces immersed in a symmetric electrolytes solution are systematically studied by numerically solving the nonlinear six-dimensional electrostatic self-consistent equations. We compare our numerical results with widely accepted mean-field (MF) theory results, and find that the MF theory remains quantitatively accurate only in weakly charged regimes, whereas in strongly charged regimes, the MF predictions deviate drastically due to the electrostatic correlation effects. We also observe a first-order like phase-transition corresponding to the counterion condensation phenomenon in strongly charged regimes, and compute the phase diagram numerically within a wide parameter range. Finally, we investigate the interactions between two likely-charged planar surfaces, which repulse each other as MF theory predicts in weakly charged regimes. However, our results show that they attract each other above a certain distance in strongly charged regimes due to significant electrostatic correlations. [ABSTRACT FROM AUTHOR]

Polymer fluids are modeled with dissipative particle dynamics (DPD) as undiluted bead-spring chains and their solutions. The models are assessed by investigating their steady shear-rate properties. Non-Newtonian viscosity and normal stress coefficients, for shear rates from the lower to the upper Newtonian regimes, are calculated from both plane Couette and plane Poiseuille flows. The latter is realized as reverse Poiseuille flow (RPF) generated from two Poiseuille flows driven by uniform body forces in opposite directions along two-halves of a computational domain. Periodic boundary conditions ensure the RPF wall velocity to be zero without density fluctuations. In overlapping shear-rate regimes the RPF properties are confirmed to be in good agreement with those calculated from plane Couette flow with Lees–Edwards periodic boundary conditions (LECs), the standard virtual rheometer for steady shear-rate properties. The concentration and the temperature dependence of the properties of the model fluids are shown to satisfy the principles of concentration and temperature superposition commonly employed in the empirical correlation of real polymer-fluid properties. The thermodynamic validity of the equation of state is found to be a crucial factor for the achievement of time-temperature superposition. With these models, RPF is demonstrated to be an accurate and convenient virtual rheometer for the acquisition of steady shear-rate rheological properties. It complements, confirms, and extends the results obtained with the standard LEC configuration, and it can be used with the output from other particle-based methods, including molecular dynamics, Brownian dynamics, smooth particle hydrodynamics, and the lattice Boltzmann method. [ABSTRACT FROM AUTHOR]

The flows of dilute polymer solutions in micro- and nanoscale channels are of both fundamental and practical importance in variety of applications in which the channel gap is of the same order as the size of the suspended particles or macromolecules. In such systems depletion layers are observed near solid-fluid interfaces, even in equilibrium, and the imposition of flow results in further cross-stream migration of the particles. In this work we employ dissipative particle dynamics to study depletion and migration in dilute polymer solutions in channels several times larger than the radius of gyration (Rg) of bead-spring chains. We compare depletion layers for different chain models and levels of chain representation, solvent quality, and relative wall-solvent-polymer interactions. By suitable scaling the simulated depletion layers compare well with the asymptotic lattice theory solution of depletion near a repulsive wall. In Poiseuille flow, polymer migration across the streamlines increases with the Peclet and the Reynolds number until the center-of-mass distribution develops two symmetric off-center peaks which identify the preferred chain positions across the channel. These appear to be governed by the balance of wall-chain repulsive interactions and an off-center driving force of the type known as the Segre–Silberberg effect. [ABSTRACT FROM AUTHOR]

Simulation studies for dilute polymeric systems are presented using the dissipative particle dynamics method. By employing two different thermostats, the velocity-Verlet and Lowe’s scheme, we show that the Schmidt number (Sc) of the solvent strongly affects nonequilibrium polymeric quantities. The fractional extension of wormlike chains subjected to steady shear is obtained as a function of Sc. Poiseuille flow in microchannels for fixed polymer concentration and varying number of repeated units within a chain is simulated. The nonuniform concentration profiles and their dependence on Sc are computed. We show the effect of the bounce-forward wall boundary condition on the depletion layer thickness. A power law fit of the velocity profile in stratified Poiseuille flow in a microchannel yields wall viscosities different from bulk values derived from uniform, steady plane Couette flow. The form of the velocity profiles indicates that the slip flow model is not useful for the conditions of these calculations. [ABSTRACT FROM AUTHOR]

Coarse graining of dense liquid-state systems can potentially lead to fast simulation times, thus providing an effective bridge between atomistic and continuum descriptions. Dissipative particle dynamics (DPD) is a stochastic Lagrangian method that provides a simple formal procedure for coarse graining. Here we analyze some of the fundamental modeling ideas of DPD and identify three factors that limit its application at high coarse-graining levels: interparticle force magnitude, compressibility, and geometric confinement. These artifacts lead to erroneous transport properties of highly coarse-grained DPD systems and thus incorrect dynamics in simulating complex fluids, e.g., colloids and polymers. [ABSTRACT FROM AUTHOR]

Highlights • We present the first ever three discretization schemes for solving the fractional magneto-hydrodynamic (FMHD) equations in bounded domains. • We use the lifting method for extending our methods to solve the FMHD equations with inhomogeneous boundary conditions. • The divergence-free constraint of the magnetic field is preserved for all the three schemes. • We applied these schemes for solving the lid-driven cavity flow with and without magnetic field, and observing that new flow patterns emerge as the fractional orders vary. Abstract We present two discretization methods for solving the fractional magneto-hydrodynamic (FMHD) equations with the fractional Laplacian defined in bounded domains. In the first method, we add a pseudo-pressure in the magnetic field equation to enforce that the magnetic field is divergence-free. In the second method, the magnetic field satisfies the divergence-free condition automatically with the no-slip boundary condition for the velocity and the perfectly conducting boundary condition for the magnetic field equation. In addition, we propose a discretization method for solving the modified fractional magneto-hydrodynamic (M-FMHD) equation. The divergence-free condition is preserved by employing the stream function in the M-FMHD system. Numerical experiments with fabricated solutions show that all three methods exhibit second-order accuracy in time for the velocity and magnetic fields. The pseudo-pressure and pressure also exhibit second-order convergence in time for small values of time step, however the pressure exhibits 1.5-order for convergence for large values of time step size when the fractional order α = 1. We use the spectral decomposition method for spatial discretization, and we demonstrate that exponential convergence is achieved for all fields and for any fractional order. We also applied these methods to solve the lid-driven cavity flow with and without magnetic field, and we observe new flow patterns emerging as the fractional order varies. [ABSTRACT FROM AUTHOR]

Diagnostic tests play a crucial role in establishing the presence of a specific disease in an individual. Receiver Operating Characteristic (ROC) curve analyses are essential tools that provide performance metrics for diagnostic tests. Accurate determination of the cutoff point in ROC curve analyses is the most critical aspect of the process. A variety of methods have been developed to find the optimal cutoffs. Although the R programming language provides a variety of package programs for conducting ROC curve analysis and determining the appropriate cutoffs, it typically needs coding skills and a substantial investment of time. Specifically, the necessity for data preprocessing and analysis can present a significant challenge, especially for individuals without coding experience. We have developed the CERA (ChatGPT-Enhanced ROC Analysis) tool, a user-friendly ROC curve analysis web tool using the shiny interface for faster and more effective analyses to solve this problem. CERA is not only user-friendly, but it also interacts with ChatGPT, which interprets the outputs. This allows for an interpreted report generated by R-Markdown to be presented to the user, enhancing the accessibility and understanding of the analysis results. [ABSTRACT FROM AUTHOR]

While there is currently a lot of enthusiasm about “big data”, useful data is usually “small” and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from small data. In particular, we introduce hidden physics models , which are essentially data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and nonlinear partial differential equations, to extract patterns from high-dimensional data generated from experiments. The proposed methodology may be applied to the problem of learning, system identification, or data-driven discovery of partial differential equations. Our framework relies on Gaussian processes, a powerful tool for probabilistic inference over functions, that enables us to strike a balance between model complexity and data fitting. The effectiveness of the proposed approach is demonstrated through a variety of canonical problems, spanning a number of scientific domains, including the Navier–Stokes, Schrödinger, Kuramoto–Sivashinsky, and time dependent linear fractional equations. The methodology provides a promising new direction for harnessing the long-standing developments of classical methods in applied mathematics and mathematical physics to design learning machines with the ability to operate in complex domains without requiring large quantities of data. [ABSTRACT FROM AUTHOR]

An open problem in the numerical solution of fractional partial differential equations (FPDEs) is how to obtain high-order accuracy for singular solutions; even for smooth right-hand sides solutions of FPDEs are singular. Here, we consider the one-dimensional diffusion equation with general two-sided fractional derivative characterized by a parameter p ∊ [0; 1]; for p = 1=2 we recover the Riesz fractional derivative, while for p = 1, 0 we obtain the one-sided fractional derivative. We employ a Petrov-Galerkin projection in a properly weighted Sobolev space with (two-sided) Jacobi polyfracnomials as basis and test functions. In particular, we derive these two-sided Jacobi polyfractonomials as eigenfunctions of a Sturm-Liouville problem with weights uniquely determined by the parameter p. We provide a rigorous analysis and obtain optimal error estimates that depend on the regularity of the forcing term, i.e., for smooth data (corresponding to singular solutions) we obtain exponential convergence, while for smooth solutions we obtain algebraic convergence. We demonstrate the sharpness of our error estimates with numerical examples, and we present comparisons with a competitive spectral collocation method of tunable accuracy. We also investigate numerically deviations from the theory for inhomogeneous Dirichlet boundary conditions as well as for a fractional diffusion-reaction equation. [ABSTRACT FROM AUTHOR]

We consider the viscous Burgers equation with a fractional nonlinear term as a model involving non-local nonlinearities in conservation laws, which, surprisingly, has an analytical solution obtained by a fractional extension of the Hopf–Cole transformation. We use this model and its inviscid limit to develop stable spectral and discontinuous Galerkin spectral element methods by employing the concept of spectral vanishing viscosity (SVV). For the global spectral method, SVV is very effective and the computational cost is O ( N 2 ) , which is essentially the same as for the standard Burgers equation. We also develop a local discontinuous Galerkin (LDG) spectral element method to improve the accuracy around discontinuities, and we again stabilize the LDG method with the SVV operator. Finally, we solve numerically the inviscid fractional Burgers equation both with the spectral and the spectral element LDG methods. We study systematically the stability and convergence of both methods and determine the effectiveness of each method for different parameters. [ABSTRACT FROM AUTHOR]

Highlights from the article: However, purely data-driven approaches for machine learning present difficulties when the data is I scarce i and of I variable fidelity i relative to the complexity of the system. Laurent Stainier, et al., proposed model-free data-driven methods in mechanics material data identification and solvers. Model-free data-driven methods in mechanics: material data identification and solvers.

While several high-order methods have been developed for fractional PDEs (FPDEs) with fixed order, there are no such methods for FPDEs with field-variable order. These equations allow multiphysics simulations seamlessly, e.g. from diffusion to sub-diffusion or from wave dynamics transitioning to diffusion, by simply varying the fractional order as a function of space or time. We develop an exponentially accurate fractional spectral collocation method for solving linear/nonlinear FPDEs with field-variable order. Following the spectral theory, developed in [1] for fractional Sturm–Liouville eigenproblems, we introduce a new family of interpolants, called left-/right-sided and central fractional Lagrange interpolants . We employ the fractional derivatives of (left-/right-sided) Riemann–Liouville and Riesz type and obtain the corresponding fractional differentiation matrices by collocating the field-variable fractional orders. We solve several FPDEs including time- and space-fractional advection-equation, time- and space-fractional advection–diffusion equation, and finally the space-fractional Burgers' equation to demonstrate the performance of the method. In addition, we develop a spectral penalty method for enforcing inhomogeneous initial conditions. Our numerical results confirm the exponential-like convergence of the proposed fractional collocation methods. [ABSTRACT FROM AUTHOR]

We develop new probabilistic and deterministic approaches for moment statistics of stochastic partial differential equations with pure jump tempered α-stable (TαS) Lévy processes. With the compound Poisson (CP) approximation or the series representation of the TαS process, we simulate the moment statistics of stochastic reaction-diffusion equations with additive TαS white noises by the probability collocation method (PCM) and the Monte Carlo (MC) method. PCM is shown to be more efficient and accurate than MC in relatively low dimensions. Then as an alternative approach, we solve the generalized Fokker-Planck equation that describes the evolution of the density for stochastic overdamped Langevin equations to obtain the density and the moment statistics for the solution following two different approaches. First, we solve an integral equation for the density by approximating the TαS processes as CP processes; second, we directly solve the tempered fractional PDE (TFPDE). We show that the numerical solution of TFPDE achieves higher accuracy than PCM at a lower cost and we also demonstrate agreement between the histogram from MC and the density from the TFPDE. [ABSTRACT FROM AUTHOR]

We study confined Brownian motion by investigating the memory function of a $$d$$ -dimensional hypercube ( $$d\ge 2$$ ), which is subject to a harmonic potential and suspended in an ideal gas confined by two parallel walls. For elastic walls and under the infinite-mass limit, we obtain analytic expressions for the force autocorrelation function and the memory function. The transverse-direction memory function possesses a nonnegative tail decaying like $$t^{-(d-1)}$$ , from which anomalous diffusion is expected for $$d=2$$ . For $$d=3$$ , the position-dependent friction coefficient becomes larger than the unconfined case and the increment is inversely proportional to the square of the distance from the wall. We also perform molecular dynamics simulations with thermal walls and/or a finite-mass hypercube. We observe faster decay due to the thermal wall ( $$t^{-3}$$ for $$d=2$$ and $$t^{-5}$$ for $$d=3$$ under the fully thermalizing wall) and convergence behaviors of the finite-mass memory function, which are different from the unconfined case. [ABSTRACT FROM AUTHOR]

The vortex-induced vibrations (VIV) of a flexible cylinder inclined at 60° are investigated by means of direct numerical simulation, at a Reynolds number equal to 500, based on the cylinder diameter and inflow velocity. The cylinder has a circular cross-section and a length to diameter aspect ratio equal to 50; it is modeled as a tension-dominated structure which is free to oscillate in the in-line and cross-flow directions. The behavior of the coupled fluid–structure system is examined for two values of the tension. Particular attention is paid to the validity of the independence principle (IP) which states that the inclined and normal-incidence body cases are comparable if the inflow velocity normal component is used to scale the physical quantities. The flexible cylinder exhibits regular VIV for both values of the tension. In the high-tension configuration, where the in-line bending of the structure remains small, the IP is shown to be valid for the prediction of the cylinder responses and the fluid forces. In contrast, in the lower-tension configuration, the behavior of the fluid–structure system deviates from the IP. It is shown that this deviation is connected to the larger in-line bending of the structure which leads to considerably different profiles of the flow velocity locally perpendicular to the body in the inclined and normal cylinder cases. Since the system behavior appears to be mainly driven by this component of the flow, the profile modification induced by the larger in-line bending results in distinct responses: multi-frequency vibrations are observed in the inclined cylinder case whereas mono-frequency oscillations of larger amplitudes develop at normal incidence. [ABSTRACT FROM AUTHOR]

The primary objective of this study is to introduce a stochastic framework based on generalized polynomial chaos (gPC) for uncertainty quantification in numerical ocean wave simulations. The techniques we present can be easily extended to other numerical ocean simulation applications. We perform stochastic simulations using a relatively new numerical method to simulate the HISWA (Hindcasting Shallow Water Waves) laboratory experiment for directional near-shore wave propagation and induced currents in a shallow-water wave basin. We solve the phased-averaged equation with hybrid discretization based on discontinuous Galerkin projections, spectral elements, and Fourier expansions. We first validate the deterministic solver by comparing our simulation results against the HISWA experimental data as well as against the numerical model SWAN (Simulating Waves Nearshore). We then perform sensitivity analysis to assess the effects of the parametrized source terms, current field, and boundary conditions. We employ an efficient sparse-grid stochastic collocation method that can treat many uncertain parameters simultaneously. We find that the depth-induced wave-breaking coefficient is the most important parameter compared to other tunable parameters in the source terms. The current field is modeled as random process with large variation but it does not seem to have a significant effect. Uncertainty in the source terms does not influence significantly the region before the submerged breaker whereas uncertainty in the incoming boundary conditions does. Considering simultaneously the uncertainties from the source terms and boundary conditions, we obtain numerical error bars that contain almost all experimental data, hence identifying the proper range of parameters in the action balance equation. [ABSTRACT FROM AUTHOR]

Abstract: Current discretizations of fractional differential equations lead to numerical solutions of low order of accuracy. Here, we present different methods for fractional ODEs that lead to exponentially fast decay of the error. First, we develop a Petrov–Galerkin (PG) spectral method for Fractional Initial-Value Problems (FIVPs) of the form and Fractional Final-Value Problems (FFVPs) , where , subject to Dirichlet initial/final conditions. These schemes are developed based on a new spectral theory for fractional Sturm–Liouville problems (FSLPs), which has been recently developed in [1]. Specifically, we obtain solutions to FIVPs and FFVPs in terms of the new fractional (non-polynomial) basis functions, called Jacobi polyfractonomials, which are the eigenfunctions of the FSLP of first kind (FSLP-I). Correspondingly, we employ another space of test functions as the span of polyfractonomial eigenfunctions of the FSLP of second kind (FSLP-II). Subsequently, we develop a Discontinuous Spectral Method (DSM) of Petrov–Galerkin sense for the aforementioned FIVPs and FFVPs, where the basis functions do not satisfy the initial/final conditions. Finally, we extend the DSM scheme to a Discontinuous Spectral Element Method (DSEM) for efficient longer time-integration and adaptive refinement. In these discontinuous schemes, we employ the asymptotic eigensolutions to FSLP-I & -II, which are of Jacobi polynomial forms, as basis and test functions. Our numerical tests confirm the exponential/algebraic convergence, respectively, in p- and h-refinements, for various test cases with integer- and fractional-order solutions. [Copyright &y& Elsevier]

The exponentially decaying behavior of the momentum-momentum and momentumforce time correlation functions of Brownian motion at large times has been extensively used for the numerical evaluation of the friction coefficient γ from molecular dynamics simulations. We perform numerical analysis on these methods and address issues related to the appropriate choice of large time and the rate of convergence of these methods. To this end, we obtain asymptotic expansions of the time correlation functions with respect to the reduced mass μ of the Brownian particle. For two important limit procedures of achieving the Brownian limit, certain forms of the asymptotic expansion of the Mori memory function K(t) are introduced by physical arguments, and then the asymptotic expansions of the time correlation functions are expressed in terms of the limit of K(t), i.e., K0(t) = limμ→∝ K(t), and the next-order correction term K1(t). For the infinite mass limit case, we show that the numerical methods of estimating γ from the exponential decay of the time correlation functions produce γ + O(μ-1), where the first-order correction depends on the microscopic nature of K0(t) (i.e., deviation of K0(t) from the Dirac delta function γδ(t)) as well as the contribution of K1(t) (i.e., deviation of K(t) from K0(t)). We also analyze the ratio of the momentum-force correlation function to the momentum-momentum correlation function, which gives instantaneous exponential decay rate. For the thermodynamic limit case, we consider the Rayleigh gas system to investigate the finite-volume effect due to the boundary conditions and to demonstrate that the lowest-order terms of the asymptotic expansions may fail to describe some characteristic behavior of the time correlation functions. We perform systematic molecular dynamics simulations of the Rayleigh gas system to confirm the theoretical predictions. [ABSTRACT FROM AUTHOR]

We present a new dynamic elastic network model (DENM) that describes the unfolding process of a force-loaded protein. The protein interaction network and its potentials are constructed based on information of its native-state structure obtained from the Protein Data Bank, with network nodes positioned at the Cα coordinates of the protein backbone. Specifically, to mimic the unfolding process, i.e., to simulate the process of overcoming the local energy barrier on the free energy landscape with force loading, the noncovalent protein network bonds (i.e., hydrogen bonds, salt bridges, hydrophobic contacts, etc.) are broken one-by-one with a certain probability, while the strong covalent bonds along the backbone (i.e., peptide bonds, disulfide bonds, etc.) are kept intact. The jumping event from local energy minima (bonds breaking rate) are chosen according to Kramer's theory and the Bell model. Moreover, we exploit the self-similar structure of proteins at different scales to design an effective coarse-graining procedure for DENM with optimal parameter selection. The robustness of DENM is validated by coarse-grained molecular dynamics (MD) simulation against atomistic MD simulation of force-extension processes of the Fibrinogen and Titin Immunoglobulin proteins. We observe that the native structure of the proteins determines the total unfolding dynamics (including large deviations) and not just the fluctuations around the native state. [ABSTRACT FROM AUTHOR]

Abstract: We first consider a regular fractional Sturm–Liouville problem of two kinds RFSLP-I and RFSLP-II of order . The corresponding fractional differential operators in these problems are both of Riemann–Liouville and Caputo type, of the same fractional order . We obtain the analytical eigensolutions to RFSLP-I & -II as non-polynomial functions, which we define as Jacobi poly-fractonomials. These eigenfunctions are orthogonal with respect to the weight function associated with RFSLP-I & -II. Subsequently, we extend the fractional operators to a new family of singular fractional Sturm–Liouville problems of two kinds, SFSLP-I and SFSLP-II. We show that the primary regular boundary-value problems RFSLP-I & -II are indeed asymptotic cases for the singular counterparts SFSLP-I & -II. Furthermore, we prove that the eigenvalues of the singular problems are real-valued and the corresponding eigenfunctions are orthogonal. In addition, we obtain the eigen-solutions to SFSLP-I & -II analytically, also as non-polynomial functions, hence completing the whole family of the Jacobi poly-fractonomials. In numerical examples, we employ the new poly-fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results. [Copyright &y& Elsevier]

Abstract: We consider elliptic stochastic partial differential equations (SPDEs) with random coefficients and solve them by expanding the solution using generalized polynomial chaos (gPC). Under some mild conditions on the coefficients, the solution is “sparse” in the random space, i.e., only a small number of gPC basis makes considerable contribution to the solution. To exploit this sparsity, we employ reweighted minimization to recover the coefficients of the gPC expansion. We also combine this method with random sampling points based on the Chebyshev probability measure to further increase the accuracy of the recovery of the gPC coefficients. We first present a one-dimensional test to demonstrate the main idea, and then we consider 14 and 40 dimensional elliptic SPDEs to demonstrate the significant improvement of this method over the standard minimization method. For moderately high dimensional (∼10) problems, the combination of Chebyshev measure with reweighted minimization performs well while for higher dimensional problems, reweighted only is sufficient. The proposed approach is especially suitable for problems for which the deterministic solver is very expensive since it reuses the sampling results and exploits all the information available from limited sources. [Copyright &y& Elsevier]

Abstract: The multi-frequency vortex-induced vibrations of a cylindrical tensioned beam of aspect ratio 200, free to move in the in-line and cross-flow directions within first a linearly and then an exponentially sheared current are investigated by means of direct numerical simulation, at a Reynolds number equal to 330. The shape of the inflow profile impacts the spectral content of the mixed standing-traveling wave structural responses: narrowband vibrations are excited within the lock-in area, which is limited to a single region lying in the high flow velocity zone, for the linear shear case; in contrast, the lock-in condition occurs at several spanwise locations in the exponential shear case, resulting in broadband responses, containing a wide range of excited frequencies and spatial wavenumbers. The broadband in-line and cross-flow vibrations occurring for the exponential shear current have a phase difference that lies within a specific range along the entire span; this differs from the phase drift noted for narrowband responses in linear shear flow. Lower vibration amplitudes, time-averaged and fluctuating in-line force coefficients are observed for the exponential shear current. The cross-flow force coefficient has comparable magnitude for both inflow profiles along the span, except in zones where the broadband vibrations are under the lock-in condition but not the narrowband ones. As in the narrowband case, the fluid forces associated with the broadband responses are dominated by high frequencies related to high-wavenumber vibration components. Considerable variability of the effective added mass coefficients along the span is noted in both cases. [Copyright &y& Elsevier]

Vasoocclusion crisis is a key hallmark of sickle cell anemia. Although early studies suggest that this crisis is caused by blockage of a single elongated cell, recent experiments have revealed that vasoocclusion is a complex process triggered by adhesive interactions among different cell groups in multiple stages. However, the quantification of the biophysical characteristics of sickle cell anemia remains an open issue. Based on dissipative particle dynamics, we develop a multiscale model for the sickle red blood cells (SS-RBCs), accounting for diversity in both shapes and cell rigidities, to investigate the precise mechanism of vasoocclusion. First, we investigate the adhesive dynamics of a single SS-RBC in shear flow and static conditions, and find that the different cell groups (SS2: young-deformable SS-RBCs, ISCs: rigid-irreversible SS-RBCs) exhibit heterogeneous adhesive behavior due to the diverse cell morphologies and membrane rigidities. We quantify the observed adhesion behavior (in static conditions) in terms of a balance of free energies due to cell adhesion and deformation, and propose a power law that relates the free-energy increase as a function of the contact area. We further simulate postcapillary flow of SS-RBC suspensions with different cell fractions. The more adhesive SS2 cells interact with the vascular endothelium and trap ISC cells, resulting in vasoocclusion in vessels less than Graphic depending on the hematocrit. Under inflammation, adherent leukocytes may also trap ISC cells, resulting in vasoocclusion in even larger vessels. [ABSTRACT FROM AUTHOR]

Abstract: The mechanisms of phasing between the in-line and cross-flow vortex-induced vibrations of a cylindrical tensioned beam in non-uniform flow are studied by direct numerical simulation. Three types of responses are considered, mono-frequency, narrowband, and broadband multi-frequency vibrations; in all cases, in-line and cross-flow vibration components occurring with a frequency ratio of 2 are phase-locked within regions of wake-body synchronization. The in-line/cross-flow phase difference exhibits a persistent spanwise drift when vibration components present significant traveling-wave behavior; this drift depends linearly on the in-line/cross-flow wavenumber difference, controlled by the beam non-linear dispersion relation and also impacted by the effective added mass variability. [Copyright &y& Elsevier]

We investigate three force autocorrelation functions 〈F(0) · F(t)〉, 〈F+(0) · F+(t)〉, and 〈F0(0) · F0(t)〉 and the friction coefficient ɣ for the Rayleigh model (a massive particle in an ideal gas) by analytic methods and molecular-dynamics (MD) simulations. Here, F and F+ are the total force and the Mori fluctuating force, respectively, whereas F0 is the force on the Brownian particle under the frozen dynamics, where the Brownian particle is held fixed and the solvent particles move under the external potential due to the presence of the Brownian particle. By using ensemble averaging and the ray representation approach, we obtain two expressions for 〈F0(0) · F0(t)〉 in terms of the one-particle trajectory and corresponding expressions for ɣ by the time integration of these expressions. Performing MD simulations of the near-Brownian-limit (NBL) regime, we investigate the convergence of 〈F(0) · F(t)〉 and #12296;F+(0) · F+(t)〉 and compare them with (F0(0) · F0(t)〉- We show that for a purely repulsive potential between the Brownian particle and a solvent particle, both expressions for 〈F0(0) · F0(t)〉 produce 〈F+(0) · F+(t)) in the NBL regime. On the other hand, for a potential containing an attractive component, the ray representation expression produces only the contribution of the nontrapped solvent particles. However, we show that the net contribution of the trapped particles to ɣ disappears, and hence we confirm that both the ensemble-averaged expression and the ray representation expression for ɣ are valid even if the potential contains an attractive component. We also obtain a closed-form expression of ɣ for the square-well potential. Finally, we discuss theoretical and practical aspects for the evaluation of 〈F0(0) · F0(t)〉 and ɣ. [ABSTRACT FROM AUTHOR]

We propose a class of novel fractional-order optimization algorithms. We define a fractional-order gradient via the Caputo fractional derivatives that generalizes integer-order gradient. We refer it to as the Caputo fractional-based gradient, and develop an efficient implementation to compute it. A general class of fractional-order optimization methods is then obtained by replacing integer-order gradients with the Caputo fractional-based gradients. To give concrete algorithms, we consider gradient descent (GD) and Adam, and extend them to the Caputo fractional GD (CfGD) and the Caputo fractional Adam (CfAdam). We demonstrate the superiority of CfGD and CfAdam on several large scale optimization problems that arise from scientific machine learning applications, such as ill-conditioned least squares problem on real-world data and the training of neural networks involving non-convex objective functions. Numerical examples show that both CfGD and CfAdam result in acceleration over GD and Adam, respectively. We also derive error bounds of CfGD for quadratic functions, which further indicate that CfGD could mitigate the dependence on the condition number in the rate of convergence and results in significant acceleration over GD. [ABSTRACT FROM AUTHOR]

Abstract: The neurovascular unit is a coordinated and interactional system of neurons, astrocytes, and microvessels in the brain. A central autoregulation mechanism observed in this unit is functional hyperemia, in which the microvasculature dilates in response to local neural activity in order to meet the increased demand for blood flow and oxygen. We have developed the first interactional model of bidirectional signaling in the neurovascular unit. The vascular model includes a description of vasomotion, the vascular oscillatory response to transmural pressure, observed in vivo. The communication mechanisms in the model include neural synaptic glutamate and potassium signaling to the astrocytes, potassium signaling from the astrocyte to the microvasculature, and astrocytic mechanosensation of vascular changes. The model response of the astrocyte to arteriolar dilation is validated with recent in vivo and in vitro experimental results. The model reproduces for the first time the in vitro observed phenomenon in which arteriole radius and Ca2+ oscillations, “vasomotion,” are damped due to neural induced astrocytic signaling. [Copyright &y& Elsevier]

Abstract: We develop a new high-order hybrid discretization of the phased-averaged (action balance) equation to simulate ocean waves. We employ discontinuous Galerkin (DG) discretization on an unstructured grid in geophysical space and Fourier-collocation along the directional and frequency coordinates. The original action balance equation is modified to facilitate absorbing boundary conditions along the frequency direction; this modification enforces periodicity at the frequency boundaries so that the fast convergence of Fourier-collocation holds. In addition, a mapping along the directional coordinate is introduced to cluster the collocation points around steep directional spectra. Time-discretization is accomplished by the TVD Runge–Kutta scheme. The overall convergence of the scheme is exponential (spectral). We successfully verified and validated the method against several analytical solutions, observational data, and experimental results. [Copyright &y& Elsevier]

Abstract: We develop, analyze and validate a new method for simulating fluid–structure interactions (FSIs), which is based on fictitious mass and fictitious damping in the structure equation. We employ a partitioned method for the fluid and structure motions in conjunction with sub-iteration and Aitken relaxation. In particular, the use of such fictitious parameters requires sub-iterations in order to reduce the induced error in addition to the local temporal truncation error. To this end, proper levels of tolerance for terminating the sub-iteration procedure have been obtained in order to recover the formal order of temporal accuracy. For the coupled FSI problem, these fictitious terms have a significant effect, leading to better convergence rate and hence substantially smaller number of sub-iterations. Through analysis we identify the proper range of these parameters, which we then verify by corresponding numerical tests. We implement the method in the context of spectral element discretization, which is more sensitive than low-order methods to numerical instabilities arising in the explicit FSI coupling. However, the method we present here is simple and general and hence applicable to FSI based on any other discretization. We demonstrate the effectiveness of the method in applications involving 2D vortex-induced vibrations (VIV) and in 3D flexible arteries with structural density close to blood density. We also present 3D results for a patient-specific aneurysmal flow under pulsatile flow conditions examining, in particular, the sensitivity of the results on different values of the fictitious parameters. [Copyright &y& Elsevier]

Abstract: The occurrence of lock-in, defined as the local synchronization between the vortex shedding frequency and the cross-flow structural vibration frequency, is investigated in the case of a tensioned beam of length to diameter ratio 200, free to move in both the in-line and cross-flow directions, and immersed in a linear shear current. Direct numerical simulation is employed at three Reynolds numbers, from 110 to 1100, so as to include the transition to turbulence in the wake. The Reynolds number influences the response amplitudes, but in all cases we observed similar fluid–structure interaction mechanisms, resulting in high-wavenumber vortex-induced vibrations consisting of a mixture of standing and traveling wave patterns. Lock-in occurs in the high oncoming velocity region, over at least 30% of the cylinder length. In the case of multi-frequency response, at any given spanwise location lock-in is principally established at one of the excited vibration frequencies, usually the locally predominant one. The spanwise patterns of the force and added mass coefficients exhibit different behaviors within the lock-in versus the non-lock-in region. The spanwise zones where the flow provides energy to excite the structural vibrations are located mainly within the lock-in region, while the flow damps the structural vibrations in the non-lock-in region. [Copyright &y& Elsevier]

Abstract: We present an iterative semi-implicit scheme for the incompressible Navier–Stokes equations, which is stable at CFL numbers well above the nominal limit. We have implemented this scheme in conjunction with spectral discretizations, which suffer from serious time step limitations at very high resolution. However, the approach we present is general and can be adopted with finite element and finite difference discretizations as well. Specifically, at each time level, the nonlinear convective term and the pressure boundary condition – both of which are treated explicitly in time – are updated using fixed-point iteration and Aitken relaxation. Eigenvalue analysis shows that this scheme is unconditionally stable for Stokes flows while numerical results suggest that the same is true for steady Navier–Stokes flows as well. This finding is also supported by error analysis that leads to the proper value of the relaxation parameter as a function of the flow parameters. In unsteady flows, second- and third-order temporal accuracy is obtained for the velocity field at CFL number 5–14 using analytical solutions. Systematic accuracy, stability, and cost comparisons are presented against the standard semi-implicit method and a recently proposed fully-implicit scheme that does not require Newton’s iterations. In addition to its enhanced accuracy and stability, the proposed method requires the solution of symmetric only linear systems for which very effective preconditioners exist unlike the fully-implicit schemes. [Copyright &y& Elsevier]

Abstract: We combine multi-element polynomial chaos with analysis of variance (ANOVA) functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity. Specifically, we employ the multi-element probabilistic collocation method MEPCM and so we refer to the new method as MEPCM-A. We investigate the dependence of the convergence of MEPCM-A on two decomposition parameters, the polynomial order μ and the effective dimension ν, with , and N the nominal dimension. Numerical tests for multi-dimensional integration and for stochastic elliptic problems suggest that for monotonic convergence of the method. We also employ MEPCM-A to obtain error bars for the piezometric head at the Hanford nuclear waste site under stochastic hydraulic conductivity conditions. Finally, we compare the cost of MEPCM-A against Monte Carlo in several hundred dimensions, and we find MEPCM-A to be more efficient for up to 600 dimensions for a specific multi-dimensional integration problem involving a discontinuous function. [Copyright &y& Elsevier]

Abstract: We propose a simple and effective numerical procedure for solving elliptic problems with non-Gaussian random coefficients, assuming that samples of the non-Gaussian random inputs are available from a statistical model. Given a correlation function, the Karhunen–Loève (K–L) expansion is employed to reduce the dimensionality of random inputs. Using the kernel density estimation technique, we obtain the marginal probability density functions (PDFs) of the random variables in the K–L expansion, based on which we define an auxiliary joint PDF. We then implement the generalized polynomial chaos (gPC) method via a collocation projection according to the auxiliary joint PDF. Based on the observation that the solution has an analytic extension in the parametric space, we ensure that the polynomial interpolation achieves point-wise convergence in the parametric space regardless of the PDF, where the energy norm is employed in the physical space. Hence, we can sample the gPC solution using the joint PDF instead of the auxiliary one to obtain the correct statistics. We also implement Monte Carlo methods to further refine the statistics using the gPC solution for variance reduction. Numerical results are presented to demonstrate the efficiency of the proposed approach. [Copyright &y& Elsevier]

Abstract: Multiscale flow phenomena in microfluidic and biomedical applications require the use of heterogeneous modeling approaches. In this paper we present a hybrid method based on coupling the Molecular Dynamics (MD) method, the Dissipative Particle Dynamics (DPD) method, and the incompressible Navier–Stokes (NS) equations. MD, DPD, and NS are formulated in separate subdomains and are coupled via an overlapping region by communicating state information at the subdomain boundaries. Imposition of boundary conditions in the MD and DPD systems involves particle insertion and deletion, specular wall reflection and body force terms. The latter includes a boundary pressure force in order to minimize near-boundary density fluctuations, and an adaptive shear force which enforces the tangential velocity component of boundary conditions. The triple-decker algorithm is verified for prototype flows, including simple and multi-layer fluids (Couette, Poiseuille, and lid-driven cavity), using highly accurate reference solutions. A zero-thickness interface is also possible if it is aligned with the flow streamlines. [Copyright &y& Elsevier]

Abstract: We report an error estimate of the multi-dimensional fast Gauss transform (FGT), which is much sharper than that previously reported in the literature. An application to the Karhunen–Loeve decomposition in the three-dimensional physical space is also presented that shows savings of three orders of magnitude in time and memory compared to a direct solver. [Copyright &y& Elsevier]

We develop a multi-element generalized polynomial chaos (ME-gPC) method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs. Given a stochastic input with an arbitrary probability measure, its random space is decomposed into smaller elements. Subsequently, in each element a new random variable with respect to a conditional probability density function (PDF) is defined, and a set of orthogonal polynomials in terms of this random variable is constructed numerically. Then, the generalized polynomial chaos (gPC) method is implemented element-by-element. Numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost. Efficiency and convergence of ME-gPC are studied numerically by considering some commonly used random variables. ME-gPC provides an efficient and flexible approach to solving differential equations with random inputs, especially for problems related to long-term integration, large perturbation, and stochastic discontinuities. [ABSTRACT FROM AUTHOR]

Abstract: We propose new schemes for integrating the stochastic differential equations of dissipative particle dynamics (DPD) in simulations of dilute polymer solutions. The hybrid DPD models consist of hard potentials that describe the microscopic dynamics of polymers and soft potentials that describe the mesoscopic dynamics of the solvent. In particular, we develop extensions to the velocity-Verlet and Lowe’s approaches – two representative DPD time-integrators – following a subcycling procedure whereby the solvent is advanced with a timestep much larger than the one employed in the polymer time-integration. The introduction of relaxation parameters allows optimization studies for accuracy while maintaining the low computational complexity of standard DPD algorithms. We demonstrate through equilibrium simulations that a 10-fold gain in efficiency can be obtained with the time-staggered algorithms without loss of accuracy compared to the non-staggered schemes. We then apply the new approach to investigate the scaling response of polymers in equilibrium as well as the dynamics of λ-phage DNA molecules subjected to shear. [Copyright &y& Elsevier]

Abstract: In this paper we focus on the long-term behavior of generalized polynomial chaos (gPC) and multi-element generalized polynomial chaos (ME-gPC) for partial differential equations with stochastic coefficients. First, we consider the one-dimensional advection equation with a uniform random transport velocity and derive error estimates for gPC and ME-gPC discretizations. Subsequently, we extend these results to other random distributions and high-dimensional random inputs with numerical verification using the algebraic convergence rate of ME-gPC. Finally, we apply our results to noisy flow past a stationary circular cylinder. Simulation results demonstrate that ME-gPC is effective in improving the accuracy of gPC for a long-term integration whereas high-order gPC cannot capture the correct asymptotic behavior. [Copyright &y& Elsevier]

Abstract: We develop a formulation for the single-fluid/two-temperature equations for simulating two-species, compressible, non-equilibrium plasma flows. The divergence-free condition of the magnetic field is enforced via the characteristic decomposition of an extended nine-wave system. The source terms are modified appropriately to improve energy and momentum conservation accuracy. A spectral/hp element algorithm is employed in the discretization combined with a discontinuous Galerkin formulation for the advective and diffusive contributions. The formulation is conservative, and monotonicity is enforced by appropriately lowering the spectral order around discontinuities. A new MHD flux introduced here is the MHD-HLLC (Harten–Lax–van Leer Contact wave) flux that preserves monotonicity and resolves contact discontinuities better. Exponential convergence is demonstrated for a magneto-hydrostatic problem. Two tests are presented using the new MHD-HLLC flux. Also, the differences between the single-temperature and the two-temperature models are presented for two-dimensional plasma flows around bluff bodies. [Copyright &y& Elsevier]

Abstract: We formulate a Multi-Element generalized Polynomial Chaos (ME-gPC) method to deal with long-term integration and discontinuities in stochastic differential equations. We first present this method for Legendre-chaos corresponding to uniform random inputs, and subsequently we generalize it to other random inputs. The main idea of ME-gPC is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value. In each subdomain or random element, we then employ a generalized polynomial chaos expansion. We develop a criterion to perform such a decomposition adaptively, and demonstrate its effectiveness for ODEs, including the Kraichnan–Orszag three-mode problem, as well as advection–diffusion problems. The new method is similar to spectral element method for deterministic problems but with h–p discretization of the random space. [Copyright &y& Elsevier]

Abstract: We present a Galerkin-free, proper orthogonal decomposition (POD)-assisted computational methodology for numerical simulations of the long-term dynamics of the incompressible Navier–Stokes equations. The approach is based on the “equation-free” framework: we use short, appropriate initialized bursts of full direct numerical simulations (DNS) of the Navier–Stokes equations to observe, estimate, and accelerate, through “projective integration”, the evolution of the flow dynamics. The main assumption is that the long-term dynamics of the flow lie on a low-dimensional, attracting, and invariant manifold, which can be parametrized, not necessarily spanned, by a few POD basis functions. We start with a discussion of the consistency and accuracy of the approach, and then illustrate it through numerical examples: two-dimensional periodic and quasi-periodic flows past a circular cylinder. We demonstrate that the approach can successfully resolve complex flow dynamics at a reduced computational cost and that it can capture the long-term asymptotic state of the flow in cases where traditional Galerkin-POD models fail. The approach trades the overhead involved in developing POD-Galerkin and POD-nonlinear Galerkin codes, for the repeated (yet short, and on demand) use of an existing full DNS simulator. Moreover, since in this approach the POD modes are used to observe rather than span the true system dynamics, the computation is much less sensitive than POD-Galerkin to values of the system parameters (e.g., the Reynolds number) and the particular simulation data ensemble used to obtain the POD basis functions. [Copyright &y& Elsevier]

Abstract: Dissipative particle dynamics (DPD) is a potentially very effective approach in simulating mesoscale hydrodynamics. However, because of the soft potentials employed, the simple no-slip boundary conditions are difficult to impose. In this work, we first identify some of these difficulties and subsequently we propose a new method, based on an equivalent force between wall- and DPD-particles, to impose boundary conditions. We demonstrate the validity of this approach for steady problems (Poiseuille flow, lid-driven cavity) as well as for the unsteady oscillating flow over a flat plate. [Copyright &y& Elsevier]

We study the intrinsic nature of the finite system-size effect in estimating shear viscosity of dilute and dense fluids within the framework of the Green–Kubo approach. From extensive molecular dynamics simulations, we observe that the size effect on shear viscosity is characterized by an oscillatory behavior with respect to system size L at high density and by a scaling behavior with an L−1 correction term at low density. Analysis of the potential contribution in the shear-stress autocorrelation function reveals that the former is configurational and is attributed to the inaccurate description of the long-range spatial correlations in finite systems. Observation of the long-time inverse-power decay in the kinetic contribution confirms its hydrodynamic nature. The L−1 correction term of shear viscosity is explained by the sensitive change in the long-time tail obtained from a finite system. [ABSTRACT FROM AUTHOR]

Direct numerical simulation advances in the field of flow–structure interactions are reviewed both from a deterministic and stochastic point of view. First, results of complex wake flows resulting from vibrating cylindrical bluff bodies in linear and exponential sheared flows are presented. On the structural side, non-linear modeling of cable structures with variable tension is derived and applied to the problem of a catenary riser of complex shape. Finally, a direct approach using Polynomial Chaos to modeling uncertainty associated with flow–structure interaction is also described. The method is applied to the two-dimensional flow–structure interaction case of an elastically mounted cylinder with random structural parameters subject to vortex-induced vibrations. [Copyright &y& Elsevier]