Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport.
Abstract not provided.
Gulian, Mamikon↗
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Results for “Diffusion problems”
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V&V for turbulent mixing in the intermediate asymptotic regime
Turbulent fluid flow interacts nonlinearly with a number of processes, including reactive flow (chemistry), radiation, and the transport of particles embedded in the flow. Subgrid scale modeling of these nonlinear processes is a major challenge to computational science. To this end, turbulent mixing of dissimilar fluids is a driver. Here we study validation issues for a class of turbulent mixing simulations. In previous work, we have obtained agreement between simulation and experiment. The present paper addresses a remaining issue, the role of meniscus related boundary effects for immiscible fluids in the validation process. Validation and even verification has been controversial for turbulent mixing flows, as commonly used numerical models are diffusive and the resulting numerical diffusion modifies the parameters of the problem and its solution. Additionally, numerical diffusion of concentration and temperature is well documented in the scientific literature for Eulerian simulations. Lagrangian simulations, which might appear to avoid this problem, are subject to mesh tangling, ensuing regridding and use of Arbitrary Lagrangian Eulerian codes to substitute for a pure Lagrangian algorithm. In practice, Lagrangian methods are potentially subject to the same numerical diffusion issues when used to study complex interface instabilities.
97 MATHEMATICS AND COMPUTING↗
Assimilating partial observation to enhance feedback control of stochastic dynamical systems
Here, in this paper, we present a novel methodology to tackle feedback optimal control problems in scenarios where the exact state of the controlled process is unknown. It integrates data assimilation techniques and optimal control solvers to manage partial observation of the state process, a common occurrence in practical scenarios. Traditional stochastic optimal control methods assume full state observation, which is often not feasible in real-world fluid dynamics control problems. Our approach underscores the significance of utilizing observational data to inform control policy design. Specifically, we introduce a kernel learning backward stochastic differential equation (SDE) filter to enhance data assimilation efficiency and propose a sample-wise stochastic optimization method within the stochastic maximum principle framework. We demonstrate the efficacy and accuracy of our method in the control of advection-diffusion-reaction flow problem and the Dubins airplane maneuvering problem with model uncertainty.
data driven↗
Direct discontinuous Galerkin method and its variations for second order elliptic equations
In this study, we study direct discontinuous Galerkin method (Liu and Yan in SIAM J Numer Anal 47(1):475–698, 2009) and its variations (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010; Vidden and Yan in J Comput Math 31(6):638–662, 2013; Yan in J Sci Comput 54(2–3):663–683, 2013) for 2nd order elliptic problems. A priori error estimate under energy norm is established for all four methods. Optimal error estimate under L 2 norm is obtained for DDG method with interface correction (Liu and Yan in Commun Comput Phys 8(3):541–564, 2010) and symmetric DDG method (Vidden and Yan in J Comput Math 31(6):638–662, 2013). A series of numerical examples are carried out to illustrate the accuracy and capability of the schemes. Numerically we obtain optimal (k+1)th order convergence for DDG method with interface correction and symmetric DDG method on nonuniform and unstructured triangular meshes. An interface problem with discontinuous diffusion coefficients is investigated and optimal (k+1)th order accuracy is obtained. Peak solutions with sharp transitions are captured well. Highly oscillatory wave solutions of Helmholz equation are well resolved.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Charging of a conducting sphere in a weakly ionized collisional plasma: Temporal dynamics and stationary state
We consider the interaction of a isolated conducting sphere with a collisional weakly ionized plasma in an external field. We assume that the plasma consists of two species of ions neglecting of electrons. We take into account charging of the sphere due to sedimentation of plasma ions on it, the field of the sphere charge and the space charge, as well as recombination and molecular diffusion. The nonstationary problem of interaction of the sphere with the surrounding plasma is solved numerically. The temporal dynamics of the sphere charge and plasma perturbations is analyzed, as well as the properties of the stationary state. It is shown that the duration of transient period is determined by the recombination time and by the reverse conductivity of ions. The temporal dynamics of the sphere charge and plasma perturbations is determined by the intensity of recombination processes relative to the influence of the space charge field and diffusion. The stationary absolute value of the sphere charge increases linearly with the external electric field, decreases with the relative intensity of recombination processes and increases in the presence of substantial diffusion. The scales of the perturbed region in the plasma are determined by the radius of the sphere, the external field, the effect of diffusion, and the relative intensity of recombination processes. In the limiting case of the absence of molecular diffusion and a strong external field, the properties of the stationary state coincide with those obtained earlier as a result of approximate solution.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
3D ADAPTIVE MESH REFINEMENT SIMULATIONS OF THE GAS CLOUD G2 BORN WITHIN THE DISKS OF YOUNG STARS IN THE GALACTIC CENTER
The dusty, ionized gas cloud G2 is currently passing the massive black hole in the Galactic Center at a distance of roughly 2400 Schwarzschild radii. We explore the possibility of a starting point of the cloud within the disks of young stars. We make use of the large amount of new observations in order to put constraints on G2's origin. Interpreting the observations as a diffuse cloud of gas, we employ three-dimensional hydrodynamical adaptive mesh refinement (AMR) simulations with the PLUTO code and do a detailed comparison with observational data. The simulations presented in this work update our previously obtained results in multiple ways: (1) high resolution three-dimensional hydrodynamical AMR simulations are used, (2) the cloud follows the updated orbit based on the Brackett-γ data, (3) a detailed comparison to the observed high-quality position–velocity (PV) diagrams and the evolution of the total Brackett-γ luminosity is done. We concentrate on two unsolved problems of the diffuse cloud scenario: the unphysical formation epoch only shortly before the first detection and the too steep Brackett-γ light curve obtained in simulations, whereas the observations indicate a constant Brackett-γ luminosity between 2004 and 2013. For a given atmosphere and cloud mass, we find a consistent model that can explain both, the observed Brackett-γ light curve and the PV diagrams of all epochs. Assuming initial pressure equilibrium with the atmosphere, this can be reached for a starting date earlier than roughly 1900, which is close to apo-center and well within the disks of young stars.
79 ASTRONOMY AND ASTROPHYSICS↗
Matrix-Free High-Performance Saddle-Point Solvers for High-Order Problems in \(\boldsymbol{H}(\operatorname{\textbf{div}})\)
Here, this work describes the development of matrix-free GPU-accelerated solvers for high-order finite element problems in H(div). The solvers are applicable to grad-div and Darcy problems in saddle-point formulation, and have applications in radiation diffusion and porous media flow problems, among others. Using the interpolation–histopolation basis, efficient matrix-free preconditioners can be constructed for the (1, 1)-block and Schur complement of the block system. With these approximations, block-preconditioned MINRES converges in a number of iterations that is independent of the mesh size and polynomial degree. The approximate Schur complement takes the form of an M-matrix graph Laplacian and therefore can be well-preconditioned by highly scalable algebraic multigrid methods. High-performance GPU-accelerated algorithms for all components of the solution algorithm are developed, discussed, and benchmarked. Numerical results are presented on a number of challenging test cases, including the “crooked pipe” grad-div problem, the SPE10 reservoir modeling benchmark problem, and a nonlinear radiation diffusion test case.
97 MATHEMATICS AND COMPUTING↗
An Efficient Parallel-in-Time Method for Optimization with Parabolic PDEs
To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equation and a backward-in-time adjoint equation to evaluate the reduced gradient in each iteration of the optimization method. In this study, we investigate the use of the parallel-in-time method PFASST in the setting of PDE-constrained optimization. In order to develop an efficient fully time-parallel algorithm, we discuss different options for applying PFASST to adjoint gradient computation, including the possibility of doing PFASST iterations on both the state and the adjoint equations simultaneously. Here, we also explore the additional gains in efficiency from reusing information from previous optimization iterations when solving each equation. Numerical results for both a linear and a nonlinear reaction-diffusion optimal control problem demonstrate the parallel speedup and efficiency of different approaches.
97 MATHEMATICS AND COMPUTING↗
An asymptotic-preserving stochastic Galerkin method for the radiative heat transfer equations with random inputs and diffusive scalings
In this paper, we develop an Asymptotic-Preserving (AP) stochastic Galerkin scheme for the radiative heat transfer equations with random inputs and diffusive scalings. In this problem the random inputs arise due to uncertainties in cross section, initial data or boundary data. We use the generalized polynomial chaos based stochastic Galerkin (gPC-SG) method, which is combined with the micro–macro decomposition based deterministic AP framework in order to handle efficiently the diffusive regime. For linearized problem we prove the regularity of the solution in the random space and consequently the spectral accuracy of the gPC-SG method. We also prove the uniform (in the mean free path) linear stability for the space-time discretizations. Several numerical tests are presented to show the efficiency and accuracy of proposed scheme, especially in the diffusive regime.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Singular value decomposition of adjoint flux distributions for Monte Carlo variance reduction
Monte Carlo (MC) shielding calculations often use weight windows (WWs) and biased sources formed from a deterministic estimate of the adjoint flux to improve the convergence rate of tallies. This requires a significant amount of computer memory, which can limit the memory available for high-resolution tally output. A new method is proposed for reducing these memory requirements by using singular value decomposition (SVD) in linear or logarithmic space to approximate the adjoint flux. This method’s performance is evaluated using the Shift and Denovo codes for streaming and diffusion base case problems, followed by problems using the Westinghouse AP1000 and the Joint European Torus. The log SVD reduced WW memory requirements by an order of magnitude in all cases without a significant performance penalty. Additionally, the linear SVD reduced biased source memory requirements by an order of magnitude, but further investigation is needed to account for observed limitations.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Fast permeability measurement for tight reservoir cores using only initial data of the one chamber pressure pulse decay test
Here, in this study, a mathematical model for fast determination of the permeabilities of tight rocks using measurements taken from the initial period of the One Chamber Pressure Pulse Decay (OC-PPD) test is presented. The model applies to measurements taken both before and after the pressure pulse front has reached the downstream end of the specimen. The analytical solutions for the pressure decay in the upstream chamber are derived based on a parabolic arc approximation of pore pressure distribution along the test specimen. This approximation allows converting the initial–boundary value problem of fluid diffusion in the specimen, governed by partial differential equations, to a system of ordinary differential equations that can be easily solved by explicit formulae. Thus, an explicit formula for the pressure decay rate is obtained, which enables inverse analysis of the initial experimental data to estimate the rock permeability. The proposed method expedites the pulse decay test as it does not require the system to reach equilibrium. The method is validated with three sets of experimental data of the OC-PPD test using helium as the diffusing fluid, for which the relative error of the permeability is found to be less than 6%. This method is particularly useful if the equilibrium time of the pulse decay test for rock specimens with permeabilities in the range of nano-Darcy takes hours or days.
early-time solution↗
Unifying Advective and Diffusive Descriptions of Bedform Pumping in the Benthic Biolayer of Streams
Abstract Many water quality and ecosystem functions performed by streams occur in the benthic biolayer, the biologically active upper (~5 cm) layer of the streambed. Solute transport through the benthic biolayer is facilitated by bedform pumping, a physical process in which dynamic and static pressure variations over the surface of stationary bedforms (e.g., ripples and dunes) drive flow across the sediment‐water interface. In this paper we derive two predictive modeling frameworks, one advective and the other diffusive, for solute transport through the benthic biolayer by bedform pumping. Both frameworks closely reproduce patterns and rates of bedform pumping previously measured in the laboratory, provided that the diffusion model's dispersion coefficient declines exponentially with depth. They are also functionally equivalent, such that parameter sets inferred from the 2D advective model can be applied to the 1D diffusive model, and vice versa. The functional equivalence and complementary strengths of these two models expand the range of questions that can be answered, for example, by adopting the 2D advective model to study the effects of geomorphic processes (such as bedform adjustments to land use change) on flow‐dependent processes and the 1D diffusive model to study problems where multiple transport mechanisms combine (such as bedform pumping and turbulent diffusion). By unifying 2D advective and 1D diffusive descriptions of bedform pumping, our analytical results provide a straightforward and computationally efficient approach for predicting, and better understanding, solute transport in the benthic biolayer of streams and coastal sediments.
Grant, Stanley B.↗
Data-driven linear time advance operators for the acceleration of plasma physics simulation
In this study, we demonstrate the application of data-driven linear operator construction for time advance with a goal of accelerating plasma physics simulation. We apply dynamic mode decomposition (DMD) to data produced by the nonlinear SOLPS-ITER (Scrape-off Layer Plasma Simulator - International Thermonuclear Experimental Reactor) plasma boundary code suite in order to estimate a series of linear operators and monitor their predictive accuracy via online error analysis. We find that this approach defines when these dynamics can be represented by a sequence of approximate linear operators and is essential for providing consistent projections when compared to an unconstrained application. For linear diffusion and advection–diffusion fluid test problems, we construct and apply operators within explicit and implicit time advance schemes, demonstrating that stability can be robustly guaranteed in each case. We further investigate the use of the linear time advance operators within several integration methods including forward Euler, backward Euler, and the matrix exponential. The application of this method to simulation data from SOLPS-ITER, with varying levels of Markov chain Monte Carlo numerical noise, shows that constrained DMD operators yield a capability to identify, extract, and integrate a (slow) subset of the present timescales. Example applications show that for projected speedup factors of [Formula: see text], and [Formula: see text], a mean relative error of 3%, 5%, and 8% and maximum relative error less than 20% are achievable, which appears acceptable for typical SOLPS-ITER steady-state simulations.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
HBMax: Optimizing Memory Efficiency for Parallel Influence Maximization on Multicore Architectures
The goal of influence maximization is to select k most-influential vertices or seeds in a network, where influence is defined by a given diffusion process. The problem has a number of important applications such as viral marketing, information spread, and epidemic control. Although computing optimal seed set is NP-Hard, due to the submodular nature of the problem efficient approximation algorithms exist. However, even state-of-the-art parallel implementations are limited by a sampling step that incurs large memory footprints. This in turn limits the problem size reach and approximation quality. In this work, we study the memory footprint of the sampling process collecting reverse reachability information in the IMM algorithm over large real-world social networks. We present an adaptive and memory-efficient optimization approach for a state-of-the-art multi-threaded parallel influence maximization algorithm. Our approach,HuffMax, uses a portion of the reverse reachable (RR) sets collected by the algorithm to learn the characteristics of the graph. Then, it compresses the intermediate reverse reachability information with Huffman coding, and queries directly on the compressed data to preserve the memory savings obtained through compression. We also propose an efficient sampling strategy based on the distribution of RR sets, which can further reduce the computation time for typical social networks with long-tail distributions. Considering a NUMA architecture, we scale up our solution on 128-core CPUs and reduce the memory footprint by up to 45.7% with negligible time overhead (or even faster) and without perceivable loss of accuracy.
Chen, Xinyu↗
A Semi-Algebraic Two Level Solver
We develop a simple semi-algebraic 2-level solver built on traditional multigrid ideas. It is designed to be easily incorporated into existing simulation software. It exhibits good convergence for many classes of challenging problems including discontinuous diffusion, convection- diffusion, and Helmholtz equations. It has built-in structure that makes it simple to generalize in several interesting directions.
97 MATHEMATICS AND COMPUTING↗
Monte Carlo based investigation of Berry phase for depth resolved characterization of biomedical scattering samples
The propagation of light in turbid media is an active area of research with relevance to numerous investigational fields, e.g., biomedical diagnostics and therapeutics. The statistical random-walk nature of photon propagation through turbid media is ideal for computational based modeling and simulation. Ready access to super computing resources provide a means for attaining brute force solutions to stochastic light-matter interactions entailing scattering by facilitating timely propagation of sufficient (>10million) photons while tracking characteristic parameters based on the incorporated physics of the problem. One such model that works well for isotropic but fails for anisotropic scatter, which is the case for many biomedical sample scattering problems, is the diffusion approximation. In this report, we address this by utilizing Berry phase (BP) evolution as a means for capturing anisotropic scattering characteristics of samples in the preceding depth where the diffusion approximation fails. We extend the polarization sensitive Monte Carlo method of Ramella-Roman, et al.,1 to include the computationally intensive tracking of photon trajectory in addition to polarization state at every scattering event. To speed-up the computations, which entail the appropriate rotations of reference frames, the code was parallelized using OpenMP. The results presented reveal that BP is strongly correlated to the photon penetration depth, thus potentiating the possibility of polarimetric depth resolved characterization of highly scattering samples, e.g., biological tissues.
back scattered Mueller matrix↗
Resolvent-Techniques for Multiple Exercise Problems
We study optimal multiple stopping of strong Markov processes with random refraction periods. The refraction periods are assumed to be exponentially distributed with a common rate and independent of the underlying dynamics. Our main tool is using the resolvent operator. In the first part, we reduce infinite stopping problems to ordinary ones in a general strong Markov setting. This leads to explicit solutions for wide classes of such problems. Starting from this result, we analyze problems with finitely many exercise rights and explain solution methods for some classes of problems with underlying Lévy and diffusion processes, where the optimal characteristics of the problems can be identified more explicitly. We illustrate the main results with explicit examples.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Space-energy separated representations for multigroup neutron diffusion using proper generalized decompositions
Proper generalized decomposition (PGD) can be described as a numerical extension of separation of variables and thus be employed as a solution technique for multi-dimensional problems, where dimensions should be understood in the phase-space context of the governing law at hand. This paper presents a PGD approach on efficiently solving problems involving multigroup neutron diffusion. Two PGD approaches are described: space-only and space-energy decompositions. Numerical results include 2-D and 3-D examples with two-, seven-, and 145-group structures. The PGD solutions are compared against traditional multi-dimensional finite element discretization. For few-group problems, both PGD approaches prove effective for mildly heterogeneous geometries, but showed reduced performance with increasing heterogeneity. The space-energy representation was found to be slower than the space-only approach for two-group problems, but proved more effective for seven-group problems. For even larger numbers of groups, the space-energy PGD decomposition was very effective at reducing the computational time.
42 ENGINEERING↗