Random variables - Necessary ingredient to simulation and Monte Carlo method.
Natural random number generator for simulation, Monte Carlo and other stochastic methods
Murry, H. F.↗
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Inverse Subspace Iteration for Spectral Stochastic Finite Element Methods
We study random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric positive-definite matrix operator, we explore the performance of algorithms for computing its eigenvalues and eigenvectors represented using polynomial chaos expansions. We formulate a version of stochastic inverse subspace iteration, which is based on the stochastic Galerkin finite element method, and we compare its accuracy with that of Monte Carlo and stochastic collocation methods. The coefficients of the eigenvalue expansions are computed from a stochastic Rayleigh quotient. Our approach allows the computation of interior eigenvalues by deflation methods, and we can also compute the coefficients of multiple eigenvectors using a stochastic variant of the modified Gram-Schmidt process. The effectiveness of the methods is illustrated by numerical experiments on benchmark problems arising from vibration analysis.
97 MATHEMATICS AND COMPUTING↗
Stochastic average model methods
We consider the solution of finite-sum minimization problems, such as those appearing in nonlinear least-squares or general empirical risk minimization problems. We are motivated by problems in which the summand functions are computationally expensive and evaluating all summands on every iteration of an optimization method may be undesirable. Here we present the idea of stochastic average model (SAM) methods, inspired by stochastic average gradient methods. SAM methods sample component functions on each iteration of a trust-region method according to a discrete probability distribution on component functions; the distribution is designed to minimize an upper bound on the variance of the resulting stochastic model. We present promising numerical results concerning an implemented variant extending the derivative-free model-based trust-region solver POUNDERS, which we name SAM-POUNDERS.
97 MATHEMATICS AND COMPUTING↗
An improved stochastic weighted particle method for boundary driven flows
Here, the stochastic weighted particle method (SWPM) is a generalization of the Direct Simulation Monte Carlo (DSMC) method where particle weights are variable and dynamic. SWPM is backed by a strong theoretical foundation but has not been critically evaluated for problems of practical interest. A thorough assessment of SWPM for boundary-driven flows reveals significant numerical artifacts near the boundary, notably a diverging heat flux. To correct the boundary heat flux, two modifications to SWPM are proposed: separated grouping and a spatially-dependent weight transfer function. To gauge the relative efficiency of SWPM in comparison to DSMC, a high-Mach-number wheel flow which forms a strong density gradient is also simulated.
97 MATHEMATICS AND COMPUTING↗
Two-nucleon 𝑆-wave interactions at the SU(3) flavor-symmetric point with 𝑚 𝑢𝑑 ≈ 𝑚$^{phys}_𝑠$: A first lattice QCD calculation with the stochastic Laplacian Heaviside method
We report on the first application of the stochastic Laplacian Heaviside method for computing multiparticle interactions with lattice QCD to the two-nucleon system. Like the Laplacian Heaviside method, this method allows for the construction of interpolating operators which can be used to construct a set of positive-definite two-nucleon correlation functions, unlike nearly all other applications of lattice QCD to two nucleons in the literature. It also allows for a variational analysis in which optimal linear combinations of the interpolating operators are formed that couple predominantly to the eigenstates of the system. Utilizing such methods has become of paramount importance to help resolve the discrepancy in the literature on whether two nucleons in either isospin channel form a bound state at pion masses heavier than physical, with the discrepancy persisting even in the SU(3)-flavor-symmetric point with all quark masses near the physical strange quark mass. This is the first in a series of papers aimed at resolving this discrepancy. In the present work, we employ the stochastic Laplacian Heaviside method without a hexaquark operator in the basis at a lattice spacing of 𝑎 ≈0.086 fm, lattice volume of 𝐿 = 48𝑎 ≈ 4.1 fm and pion mass 𝑚 𝜋 ≈ 714 MeV. With this setup, the observed spectrum of two-nucleon energy levels strongly disfavors the presence of a bound state in either the deuteron or dineutron channel.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Finite element structural analysis and optimization of space frames and trusses in a massively parallel environment
A set of massively parallel computational procedures are formulated to make finite element method based space frame and truss structural response analyses and stochastic method based structural optimization solution superefficient. Several example frame structural analysis and truss optimization problems are presented for demonstration purposes. All calculations are made on the massively parallel environment of CM-2 computer.
Shieh, Rong C.↗
A moment projection method for population balance dynamics with a shrinkage term
A new method of moments for solving the population balance equation is developed and presented. The moment projection method (MPM) is numerically simple and easy to implement and attempts to address the challenge of particle shrinkage due to processes such as oxidation, evaporation or dissolution. It directly solves the moment transport equation for the moments and tracks the number of the smallest particles using the algorithm by Blumstein and Wheeler (1973) . The performance of the new method is measured against the method of moments (MOM) and the hybrid method of moments (HMOM). The results suggest that MPM performs much better than MOM and HMOM where shrinkage is dominant. The new method predicts mean quantities which are almost as accurate as a high-precision stochastic method calculated using the established direct simulation algorithm (DSA).
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
An examination of the relative merits of stochastic and nonstatistical methods of computing primary ionization doses
Relative merits of stochastic and nonstatistical methods of computing primary ionization doses - radiation dose calculations
IONIZATION↗
A Stochastic Quasi-Newton Method in the Absence of Common Random Numbers
We present Q-SASS, a quasi-Newton method for unconstrained stochastic optimization that does not rely on common random numbers. Most existing quasi-Newton approaches leverage common random numbers to construct second-order updates. However, motivated by challenges in variational quantum algorithms—where such coordination is not possible—we consider the setting in which function values and gradients are accessible only through noisy probabilistic zeroth- and first-order oracles, and no common random numbers can be exploited. We derive high-probability tail bounds on the iteration complexity of our algorithm for nonconvex, convex, and strongly convex (more generally, those satisfying the PL condition) objective functions. Finally, we demonstrate the empirical benefits of our quasi-Newton updating scheme on both synthetic and quantum chemistry problems.
Complexity bound↗
Alpha Shape Analysis (ASA) Framework for Post- Clustering Property Determination in Atom Probe Tomographic Data
While application of clustering algorithms to atom probe tomography data have enabled quantification of solute clusters in terms of number density, size, and subcomposition there exist other properties (e.g., volume, surface area, and composition) that are better determined by defining an interface between the cluster and the surrounding matrix. The limitation in composition results from an ion selection step where the expected matrix ion types are omitted from the cluster search algorithm to enhance the contrast between the matrix and cluster and to reduce the complexity of the search. Previously, composition determination within solute clusters has utilized a secondary envelopment and erosion step on top of conventional methods such as maximum separation. In this work, we present a novel stochastic method that combines the particle identification fidelity of a conventional clustering algorithm with the analytical flexibility of mesh-based approaches through the generation of alpha shapes for each identified cluster. Furthermore, the corresponding mesh accounts for concave components of the clusters and determines the volume and surface area of the clusters; additionally, the mesh boundary is utilized to update the total composition according to the internal ions.
42 ENGINEERING↗
Probability and Cumulative Density Function Methods for the Stochastic Advection-Reaction Equation
We present a cumulative density function (CDF) method for the probabilistic analysis of $d$-dimensional advection-dominated reactive transport in heterogeneous media. We employ a probabilistic approach in which epistemic uncertainty on the spatial heterogeneity of Darcy-scale transport coefficients is modeled in terms of random fields with given correlation structures. Our proposed CDF method employs a modified Large-Eddy-Diffusivity (LED) approach to close and localize the nonlocal equations governing the one-point PDF and CDF of the concentration field, resulting in a $(d + 1)$ dimensional PDE. Compared to the classsical LED localization, the proposed modified LED localization explicitly accounts for the mean-field advective dynamics over the phase space of the PDF and CDF. To illustrate the accuracy of the proposed closure, we apply our CDF method to one-dimensional single-species reactive transport with uncertain, heterogeneous advection velocities and reaction rates modeled as random fields.
advection-reaction equation↗
Improved Modeling of Finite-Rate Turbulent Combustion Processes in Research Combustors
The objective of this thesis is to further develop and test a stochastic model of turbulent combustion in recirculating flows. There is a requirement to increase the accuracy of multi-dimensional combustion predictions. As turbulence affects reaction rates, this interaction must be more accurately evaluated. In this work a more physically correct way of handling the interaction of turbulence on combustion is further developed and tested. As turbulence involves randomness, stochastic modeling is used. Averaged values such as temperature and species concentration are found by integrating the probability density function (pdf) over the range of the scalar. The model in this work does not assume the pdf type, but solves for the evolution of the pdf using the Monte Carlo solution technique. The model is further developed by including a more robust reaction solver, by using accurate thermodynamics and by more accurate transport elements. The stochastic method is used with Semi-Implicit Method for Pressure-Linked Equations. The SIMPLE method is used to solve for velocity, pressure, turbulent kinetic energy and dissipation. The pdf solver solves for temperature and species concentration. Thus, the method is partially familiar to combustor engineers. The method is compared to benchmark experimental data and baseline calculations. The baseline method was tested on isothermal flows, evaporating sprays and combusting sprays. Pdf and baseline predictions were performed for three diffusion flames and one premixed flame. The pdf method predicted lower combustion rates than the baseline method in agreement with the data, except for the premixed flame. The baseline and stochastic predictions bounded the experimental data for the premixed flame. The use of a continuous mixing model or relax to mean mixing model had little effect on the prediction of average temperature. Two grids were used in a hydrogen diffusion flame simulation. Grid density did not effect the predictions except for peak temperature and tangential velocity. The hybrid pdf method did take longer and required more memory, but has a theoretical basis to extend to many reaction steps which cannot be said of current turbulent combustion models.
VanOverbeke, Thomas J.↗
Analog methods for on-line system identification using noisy measurements.
Nonlinear system on-line identification in presence of noise, using stochastic methods and analog equipment
Ricker, D. W.↗
Investigation of the Attila Deterministic Solver for Out-of-Field Organ Dose Calculations for Radiotherapy Patients
An undeniable paradox in the current practice of radiation oncology is that the therapeutic radiation often used to treat cancer is also a known carcinogen. Indeed, the potential for inducing second cancers in patients receiving radiotherapy has been well established. Retrospective epidemiological studies are an important way to assess the risks of late effects in radiotherapy patients. Such investigations attempt to correlate organ dose with the incidence of secondary cancers in a cohort of patients. Individualized patient dosimetry is a critical component of these studies and is challenging for two reasons. First, medical images describing the patient's anatomy are either non-existent or, if they exist, often only cover a portion of the body. This means that whole body computational human phantoms must be used as a surrogate for the patient's true anatomy in order to perform dose calculations far from the treatment site. Second, the gold standard dosimetry approach calls for the use of Monte Carlo radiation transport simulations which are far too slow for studies involving large-scale patient cohorts. This paper describes a preliminary effort at the National Cancer Institute aimed at addressing these dosimetry challenges. The Attila software (Varian Medical Systems, Palo Alto, CA) is investigated as an alternative to Monte Carlo for performing out-of-field organ dose calculations for radiotherapy patients. Whereas as Monte Carlo uses stochastic methods, Attila can deterministically solve the linear Boltzmann equation throughout the entire solution space. It is hypothesized that dose calculations with Attila will offer comparable accuracy to Monte Carlo, but with significant faster computation times. At least one study has demonstrated this for in-field dose calculations [3]. To our knowledge Attila has not yet been used to perform out-of-field organ dosimetry. (authors)
61 RADIATION PROTECTION AND DOSIMETRY↗
A Comparison of the FW-CADIS and MR-CADIS Variance Reduction Methods
The Consistent Adjoint Driven Importance Sampling (CADIS) method, which uses an adjoint deterministic estimate for making variance reduction parameters that optimize Monte Carlo (MC) shielding calculations, was introduced in 1998. This method was designed to optimize a single tally, typically an integrated response such as a detector count rate or a dose rate. Because both deterministic and stochastic methods are employed, CADIS is referred to as a hybrid method. In 2007, the forward-weighted CADIS (FW-CADIS) method was introduced to optimize several tallies at once, giving more uniform relative uncertainties among the tallies for both high-flux and low-flux areas of the problem. This method can be used for several small detectors, for semi-global problems (mesh tally over a large fraction of the problem), or for global problems (mesh tally over the entire problem). Optimization can be done for an integrated response such as a dose rate D(r) as a function of position, the total flux {phi}(r) as a function of position, or the space/energy flux {phi}(r,E). Both the CADIS and FW-CADIS methods have been in the MAVRIC sequence of SCALE for three versions (6.0 in 2009, 6.1 in 2011, and 6.2 in 2016). The ADVANTG package released in 2015 uses the CADIS and FW-CADIS methods to accelerate MCNP neutron/photon shielding calculations. In 2014, the multi-response CADIS (MR-CADIS) method was presented as a way to obtain even more uniform relative uncertainties for problems with multiple tallies than those obtained with FW-CADIS. The method does not use a forward deterministic calculation but instead uses one adjoint calculation for every tally to be optimized in the Monte Carlo calculation. This work compares the above methods.
42 ENGINEERING↗
Solar flux forecasting using mutual information with an optimal delay
Solar flux F(sub 10.7) directly affects the atmospheric density, thereby changing the lifetime and prediction of satellite orbits. For this reason, accurate forecasting of F(sub 10.7) is crucial for orbit determination of spacecraft. Our attempts to model and forecast F(sub 10.7) uncovered highly entangled dynamics. We concluded that the general lack of predictability in solar activity arises from its nonlinear nature. Nonlinear dynamics allow us to predict F(sub 10.7) more accurately than is possible using stochastic methods for time scales shorter than a characteristic horizon, and with about the same accuracy as using stochastic techniques when the forecasted data exceed this horizon. The forecast horizon is a function of two dynamical invariants: the attractor dimension and the Lyapunov exponent. In recent years, estimation of the attractor dimension reconstructed from a time series has become an important tool in data analysis. In calculating the invariants of the system, the first necessary step is the reconstruction of the attractor for the system from the time-delayed values of the time series. The choice of the time delay is critical for this reconstruction. For an infinite amount of noise-free data, the time delay can, in principle, be chosen almost arbitrarily. However, the quality of the phase portraits produced using the time-delay technique is determined by the value chosen for the delay time. Fraser and Swinney have shown that a good choice for this time delay is the one suggested by Shaw, which uses the first local minimum of the mutual information rather than the autocorrelation function to determine the time delay. This paper presents a refinement of this criterion and applies the refined technique to solar flux data to produce a forecast of the solar activity.
Ashrafi, S.↗
Stochastic Robustness: Towards a Comprehensive Robustness Tool
Stochastic robustness is a simple technique to determine the robustness of linear, time-invariant systems by Monte Carlo methods. Stochastic stability robustness has been described previously. Those results are extended here to provide insight into control system design for performance. Together, stochastic stability and performance robustness concepts constitute a comprehensive tool that can be used to analyze control system robustness properties. As well, they offer control system design insight that can set the stage for stochastic robustness synthesis. The concept of stochastic stability robustness is reviewed, stochastic performance robustness is introduced, and stochastic robustness synthesis is described qualitatively. Confidence intervals necessary for comparing control laws statistically are presented.
Ray, Laurie Ryan↗
Influence of pseudopotentials on excitation energies from selected configuration interaction and diffusion Monte Carlo
Due to their diverse nature, the faithful description of excited states within electronic structure theory methods remains one of the grand challenges of modern theoretical chemistry. Quantum Monte Carlo (QMC) methods have been applied very successfully to ground state properties but still remain generally less effective than other non-stochastic methods for electronically excited states. Nonetheless, we have recently reported accurate excitation energies for small organic molecules at the fixed-node diffusion Monte Carlo (FN-DMC) within a Jastrow-free QMC protocol relying on a deterministic and systematic construction of nodal surfaces using the selected configuration interaction (sCI) algorithm known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). Albeit highly accurate, these all-electron calculations are computationally expensive due to the presence of core electrons. One very popular approach to remove these chemically-inert electrons from the QMC simulation is to introduce pseudopotentials (also known as effective core potentials). Taking the water molecule as an example, we investigate the influence of Burkatzki-Filippi-Dolg (BFD) pseudopotentials and their associated basis sets on vertical excitation energies obtained with sCI and FN-DMC methods. Although these pseudopotentials are known to be relatively safe for ground state properties, we evidence that special care may be required if one strives for highly accurate vertical transition energies. Indeed, comparing all-electron and valence-only calculations, we show that using pseudopotentials with the associated basis sets can induce differences of the order of 0.05 eV on the excitation energies. Fortunately, a reasonable estimate of this shift can be estimated at the sCI level.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗