Stochastic Boundary Research Articles

Edge localized modes control by stochastic magnetic fields

This paper describes a new approach for modelling the pedestal energy transport in the presence of a small radial magnetic perturbation. The cases of a ballooning instability leading to Type I edge localized modes (ELMs) and a magnetic perturbation generated by external coils are treated. The model for Type I ELMs is based on the linear ideal MHD code MISHKA coupled with the non-linear energy transport code TELM in a realistic tokamak geometry. The main mechanism of the increased transport through the external transport barrier in this model of ELMs is due to the appearance of a radial velocity and a radial magnetic field perturbation due to the MHD mode. Both lead to additional transport perpendicular to the magnetic surface and hence to a relaxation of the pressure profile in the unstable zone. The typical Type I ELM time-cycle was reproduced numerically including the destabilization of the ballooning modes leading to the fast (250 μ s) collapse of the pedestal pressure followed by the edge pressure profile re-building on a diffusive time scale. A possible mechanism for the control of Type I ELMs using a stochastic plasma boundary created by external coils is modelled in this paper using data on ELM suppression by I-coils from the DIII-D experiment. In the stochastic layer the transverse transport is effectively increased by diffusion of the magnetic field lines. The modelling results demonstrate the possibility of decreasing the edge pressure gradient to a value that is just below the ideal ballooning limit, leading to a high confinement regime without Type I ELMs.

Reliable treatment of electrostatics in combined QM/MM simulation of macromolecules

A robust approach for dealing with electrostatic interactions for spherical boundary conditions has been implemented in the QM/MM framework. The development was based on the generalized solvent boundary potential (GSBP) method proposed by Im et al. [J. Chem. Phys. 114, 2924 (2001)], and the specific implementation was applied to the self-consistent-charge density-functional tight-binding approach as the quantum mechanics (QM) level, although extension to other QM methods is straightforward. Compared to the popular stochastic boundary-condition scheme, the new protocol offers a balanced treatment between quantum mechanics/molecular mechanics (QM/MM) and MM/MM interactions; it also includes the effect of the bulk solvent and macromolecule atoms outside of the microscopic region at the Poisson-Boltzmann level. The new method was illustrated with application to the enzyme human carbonic anhydrase II and compared to stochastic boundary-condition simulations using different electrostatic treatments. The GSBP-based QM/MM simulations were most consistent with available experimental data, while conventional stochastic boundary simulations yielded various artifacts depending on different electrostatic models. The results highlight the importance of carefully treating electrostatics in QM/MM simulations of biomolecules and suggest that the commonly used truncation schemes should be avoided in QM/MM simulations, especially in simulations that involve extensive conformational samplings. The development of the GSBP-based QM/MM protocol has opened up the exciting possibility of studying chemical events in very complex biomolecular systems in a multiscale framework.

Computational Study of IAG-Nucleoside Hydrolase: Determination of the Preferred Ground State Conformation and the Role of Active Site Residues

The mechanism of action of inosine-adenosine-guanosine nucleoside hydrolase (IAG-NH) has been investigated by long-term molecular dynamics (MD) simulation in TIP3P water using stochastic boundary conditions. Special attention has been given to the role of leaving group pocket residues and conformation of the bound substrate at the active site of IAG-NH. We also describe the positioning of the residues of an important flexible loop at the active site, which was previously unobservable by X-ray crystallography due to high B-factors. Five MD simulations have been performed with the Enzyme x Substrate complexes: Enzyme x anti-Adenosine with Asp40-COOH [E(40H) x Ade(a)], Enzyme x anti-Adenosine with Asp40-COO- [E(40) x Ade(a)], Enzyme x syn-Adenosine with Asp40-COOH [E(40H) x Ade(s)], Enzyme x syn-Adenosine with Asp40-COO- [E(40) x Ade(s)], and Enzyme x anti-Inosine with Asp40-COO- [E(40) x Ino(a)]. Overall, the structures generated from the MD simulation of E(40H) x Ade(s) preserve the catalytically important hydrogen bonds as well as electrostatic and hydrophobic interactions to provide a plausible catalytic structure. When deprotonated Asp40 (Asp4-COO-) is present, the active site is open to water solvent which interferes with the base stacking between Trp83 and nucleobase. A calculation using Poisson-Boltzmann equation module supports that Asp40 indeed has an elevated pK(app). Solvent accessible surface area (SASA) calculations on all the five MD structures shows that systems with protonated Asp40, namely, E(40H) x Ade(a) and E(40H) x Ade(s), have zero SASA. It is found that a water molecule is hydrogen-bonded to the N7 of the nucleobase and is probably the essential general acid to protonate the departing nucleobase anion. The N7-bonded water is in turn hydrogen-bonded to waters in a channel, held in place by the residues of the flexible loop, Tyr257, His247, and Cys245. Using normal-mode analysis with elastic network model, we find that the flexible loop explores a conformational space much larger than in the MD trajectory, leading to a "gating"-like motion with respect to the active site.

Stochastic BEM with spectral approach in elastostatic and elastodynamic problems with geometrical uncertainty

The present paper proposes a method for stochastic problems that have uncertainty in the boundary geometry. The method is developed by applying the spectral stochastic approach to the boundary element method and is called the spectral stochastic boundary element method (SSBEM). In the SSBEM, the uncertainty in the boundary geometry is represented by the Karhunen–Loève expansion. It is shown that, by utilizing material derivative, variation of boundary element matrices associated with the geometrical fluctuation of the boundary can be approximated by the Taylor expansion. The solution is represented by a stochastic process expressed in the form of polynomial chaos expansion. The stochastic equation is then projected on a homogeneous chaos space. This procedure reduces a stochastic equation to an ordinary linear matrix equation that can be solved by conventional schemes. The SSBEM can estimate not only mean values and variances of the solutions but also their probability density functions. In order to examine the performance, the SSBEM is applied to two-dimensional elastostatic and elastodynamic problems with geometrical boundary uncertainty. Computation results of the SSBEM exhibit good agreement with those obtained by Monte Carlo simulation. The efficiency of the SSBEM is verified by comparison of their computation times.

Stochastic boundary molecular dynamics simulation of l-ribose in the active site of Actinoplanes missouriensis xylose isomerase and its Val135Asn mutant with improved reaction rate

We used molecular dynamics simulations to study how a non-natural substrate, l-ribose, interacts with the active site of Actinoplanes missouriensis xylose isomerase. The simulations showed that l-ribose does not stay liganded in the active site in the same way as d-xylose, in which the oxygens O2 and O4 are liganded to the metal M1. The oxygen O4 of l-ribose moved away from the metal M1 to an upside down position. Furthermore, the distances of the carbons C1 and C2 of l-ribose to the catalytic metal M2 were higher than in the case of d-xylose. These findings explain the extremely low reaction rate of xylose isomerase with l-ribose. The mutation V135N close to the C5–OH of the substrate increased the reaction efficiency 2- to 4-fold with l-ribose. V135N did not affect the reaction with d-xylose and l-arabinose, whereas the reaction with d-glucose was impaired, probably due to a hydrogen bond between Asn-135 and the substrate. When l-ribose was the substrate, Asn-135 formed a hydrogen bond to Glu-181. As a consequence, O4 of l-ribose stayed liganded to the metal M1 in the V135N mutant in molecular dynamics simulations. This explains the decreased K m of the V135N mutant with l-ribose.

SOL and pedestal density profile evolution during DIII-D ELMy and ELM-suppressed H-mode operation

A fast profile reflectometer has been used to study the detailed characteristics of scrape-off-layer (SOL) and pedestal density profile evolution in DIII-D ELMy H-mode plasmas with high temporal (⩾10μs) and spatial (∼4mm) resolution. Results show a significant enhancement in radial particle transport to the outer vessel wall during Type-I ELMs. The radial particle transport time (∼140μs) becomes comparable to the parallel transport time, as a result of a high radial velocity (∼620m/s) of SOL profile expansion. The dynamics can vary from ELM-to-ELM. Scalings with pedestal density have been obtained, indicating that ELM is less affected as the pedestal density increases. Finally, during ELM-suppressed H-mode operation with a stochastic magnetic boundary, the radial extent of observed SOL profile modifications do not extend to the vessel wall, and profiles are steeper, substantially different from the SOL changes associated with Type-I ELMs.

Influence of stochastic disturbances on the quasi-wavelet solution of the convection–diffusion equation

A quasi-wavelet numerical method (QWNM) is introduced for solving the convection–diffusion equation (CDE). The results manifest that the calculated bandwidth has an extremum. When the bandwidth takes the value of the extremum, the accuracy of the solution for the CDE by using the QWNM is relatively high and better than that by using the up-wind scheme. Under the condition of stochastic boundary disturbances of different amplitudes, the results of the QWNM are a little worse than those of the up-wind scheme when the integral time is longer. However, when stochastic boundary disturbances of equal amplitudes occur, the solutions of the equation by using the QWNM and the up-wind scheme can be identical if the bandwidth takes an integer greater than or equal to 20. When the parameter is stochastically disturbed, the root-mean-square error of the quasi-wavelet solution of the equation is smaller than that of the up-wind scheme solution if the bandwidth is 10. When the initial values are stochastically disturbed and the bandwidth equals 10, the accuracy of the quasi-wavelet solution is relatively high and better than that of the up-wind scheme solution.

Suppression of large edge localized modes in high confinement DIII-D plasmas with a stochastic magnetic boundary

Large 70Hz Type-I edge localized modes (ELMs) are converted into small 130Hz oscillations using edge resonant magnetic perturbations (RMPs) from a coil with currents ⩽0.4% Ip in double null DIII-D plasmas. When the RMP is properly phased with respect to the background field errors, all but a few isolated ELM-like events are suppressed. The impulsive pedestal energy loss ΔEELM/Δt1/2 to the scrape-of layer is reduced a factor of ⩾20 relative to the Type-I ELMs and the core confinement is unaffected by the perturbation field. Significant changes in the properties of the ELMs are also observed when edge RMPs are applied to lower single null plasmas but the nature of these changes are much more complex. Both lower single null and double null plasmas are being studied to determine how ELM control techniques based on the application of edge RMPs can be expected to scale to future devices such as ITER.

An Efficient Linear-Scaling Ewald Method for Long-Range Electrostatic Interactions in Combined QM/MM Calculations

A method is presented for the efficient evaluation of long-range electrostatic forces in combined quantum mechanical and molecular mechanical (QM/MM) calculations of periodic systems. The QM/MM-Ewald method is a linear-scaling electrostatic method that utilizes the particle mesh Ewald algorithm for calculation of point charge interactions of molecular mechanical atoms and a real-space multipolar expansion for the quantum mechanical electrostatic terms plus a pairwise periodic correction factor for the QM and QM/MM interactions that does not need to be re-evaluated during the self-consistent field procedure. The method is tested in a series of molecular dynamics simulations of the ion-ion association of ammonium chloride and ammonium metaphosphate and the dissociative phosphoryl transfer of methyl phosphate and acetyl phosphate. Results from periodic boundary molecular dynamics (PBMD) simulations employing the QM/MM-Ewald method are compared with corresponding PBMD simulations using electrostatic cutoffs and with results from nonperiodic stochastic boundary molecular dynamics (SBMD) simulations, with cutoffs and with full electrostatics (no cutoff). The present method allows extension of linear-scaling Ewald methods to molecular simulations of enzyme and ribozyme reactions that use combined QM/MM potentials.

Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation

This work studies a linear elliptic problem with uncertainty. The introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations. The major emphasis of this work is the probabilistic treatment of uncertainty, addressing the problem of solving linear elliptic boundary value problems with stochastic coefficients. If the stochastic coefficients are known functions of a random vector, then the stochastic elliptic boundary value problem is turned into a parametric deterministic one with solution u(y,x), y∈Γ, x∈D, where D⊂Rd, d=1,2,3, and Γ is a high-dimensional cube. In addition, the function u is specified as the solution of a deterministic variational problem over Γ×D. A tensor product finite element method, of h-version in D and k-, or, p-version in Γ, is proposed for the approximation of u. A priori error estimates are given and an adaptive algorithm is also proposed. Due to the high dimension of Γ, the Monte Carlo finite element method is also studied here. This work compares the asymptotic complexity of the numerical methods, and shows results from numerical experiments. Comments on the uncertainty in the probabilistic characterization of the coefficients in the stochastic formulation are included.

Heterogeneity and Network Structure in the Dynamics of Diffusion: Comparing Agent-Based and Differential Equation Models

When is it better to use agent-based (AB) models, and when should differential equation (DE) models be used? Whereas DE models assume homogeneity and perfect mixing within compartments, AB models can capture heterogeneity across individuals and in the network of interactions among them. AB models relax aggregation assumptions, but entail computational and cognitive costs that may limit sensitivity analysis and model scope. Because resources are limited, the costs and benefits of such disaggregation should guide the choice of models for policy analysis. Using contagious disease as an example, we contrast the dynamics of a stochastic AB model with those of the analogous deterministic compartment DE model. We examine the impact of individual heterogeneity and different network topologies, including fully connected, random, Watts-Strogatz small world, scale-free, and lattice networks. Obviously, deterministic models yield a single trajectory for each parameter set, while stochastic models yield a distribution of outcomes. More interestingly, the DE and mean AB dynamics differ for several metrics relevant to public health, including diffusion speed, peak load on health services infrastructure, and total disease burden. The response of the models to policies can also differ even when their base case behavior is similar. In some conditions, however, these differences in means are small compared to variability caused by stochastic events, parameter uncertainty, and model boundary. We discuss implications for the choice among model types, focusing on policy design. The results apply beyond epidemiology: from innovation adoption to financial panics, many important social phenomena involve analogous processes of diffusion and social contagion.

Finite elements for elliptic problems with stochastic coefficients

We describe a deterministic finite element (FE) solution algorithm for a stochastic elliptic boundary value problem (sbvp), whose coefficients are assumed to be random fields with finite second moments and known, piecewise smooth two-point spatial correlation function. Separation of random and deterministic variables (parametrization of the uncertainty) is achieved via a Karhunen–Loève (KL) expansion. An O(NlogN) algorithm for the computation of the KL eigenvalues is presented, based on a kernel independent fast multipole method (FMM). Truncation of the KL expansion gives an (M,1) Wiener polynomial chaos (PC) expansion of the stochastic coefficient and is shown to lead to a high dimensional, deterministic boundary value problem (dbvp). Analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r=(r1,…,rM) in an (M,r) Wiener PC expansion for the approximate solution. Pointwise error bounds for the FEM approximations of KL eigenpairs, the truncation of the KL expansion and the FE solution to the dbvp are given. Numerical examples show that M depends on the spatial correlation length of the random diffusion coefficient. The variable polynomial degree r in PC-stochastic Galerkin FEM allows to handle KL expansions with M up to 30 and r1 up to 10 in moderate time.

Suppression of large edge-localized modes in high-confinement DIII-D plasmas with a stochastic magnetic boundary.

A stochastic magnetic boundary, produced by an applied edge resonant magnetic perturbation, is used to suppress most large edge-localized modes (ELMs) in high confinement (H-mode) plasmas. The resulting H mode displays rapid, small oscillations with a bursty character modulated by a coherent 130 Hz envelope. The H mode transport barrier and core confinement are unaffected by the stochastic boundary, despite a threefold drop in the toroidal rotation. These results demonstrate that stochastic boundaries are compatible with H modes and may be attractive for ELM control in next-step fusion tokamaks.

Solution of heat transfer problems for stochastic boundary conditions

Solutions are known for a great deal of heat transfer problems with stationary or analytically given transient boundary conditions. In this paper the well-known Duhamel theorem will be extended for boundary conditions not stated analytically. The fact that the energy supply to a system depends on time in a stochastic manner is a problem in many technical applications. It will be shown by comparison with well-known analytical calculations, that the extended method of Duhamel can be used for calculating the temperature distributions and heat fluxes in any case of transient variations of heat supply or temperatures forced upon the system. The method leads to a long series of error or trigonometrical functions. In order to compute these in real time, an EDP with high capacity and velocity is necessary. But that should be no problem nowadays.

Solution of heat transfer problems for stochastic boundary conditions

Solution of heat transfer problems for stochastic boundary conditions

Stochastic boundary conditions to the convection-diffusion equation including chemical reactions at solid surfaces.

Simulations of heat and mass transport may require complex nonlinear boundary conditions to describe the flow of mass and energy across an interface. Although stochastic methods do not suffer from the numerical diffusion of grid-based methods, they typically lose accuracy in the vicinity of interfacial boundaries. In this work we introduce ideas and algorithms to account for mass (or energy) transfer at reactive interfaces, with accuracies comparable to the bulk phase. We show how to introduce particles into the system with the correct distribution near the interface, as well as the correct flux through the interface. The algorithms have been tested in a channel flow, for which accurate numerical solutions can be independently calculated.

Stochastic bifurcation in duffing system subject to harmonic excitation and in presence of random noise

A global analysis of stochastic bifurcation in a special kind of Duffing system, named as Ueda system, subject to a harmonic excitation and in presence of random noise disturbance is studied in detail by the generalized cell mapping method using digraph. It is found that for this dissipative system there exists a steady state random cell flow restricted within a pipe-like manifold, the section of which forms one or two stable sets on the Poincare cell map. These stable sets are called stochastic attractors (stochastic nodes), each of which owns its attractive basin. Attractive basins are separated by a stochastic boundary, on which a stochastic saddle is located. Hence, in topological sense stochastic bifurcation can be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value. Through numerical simulations the evolution of the Poincare cell maps of the random flow against the variation of noise intensity is explored systematically. Our study reveals that as a powerful tool for global analysis, the generalized cell mapping method using digraph is applicable not only to deterministic bifurcation, but also to stochastic bifurcation as well. By this global analysis the mechanism of development, occurrence, and evolution of stochastic bifurcation can be explored clearly and vividly.

Atomistic simulations of the adhesion hysteresis mechanism of atomic scale dissipation innon-contact atomic force microscopy

A possible mechanism of atomic scale dissipation in non-contact atomic force microscopy(NC-AFM) is investigated using a non-equilibrium classical molecular dynamicssimulation method with stochastic boundary conditions. We find that for simple flatcrystal surfaces, such as the MgO(001) which does not possess intrinsic surface softvibrational modes, dissipation effects can be explained by tip induced soft modesassociated with well-known atomic instabilities produced by the approaching tip.Extrapolating results of calculations performed for a succession of tip oscillationfrequencies, we suggest that dissipation energies can be obtained via this mechanismwhich are of the same order of magnitude of experimentally observed contrasts.

Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations

We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the com- putations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method gener- ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The sec- ond method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h -o rp-version, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.

A multi-spectral non-local method for retrieval of boundary layer cloud properties from optical remote sensing data

A multi-spectral non-local (MSN) method is developed for advanced retrieval of boundary layer cloud properties from remote sensing data, as an alternative to the independent pixel approximation (IPA) method. The non-local method uses data at both the target pixel and neighboring pixels to retrieve cloud properties such as pixel-averaged cloud optical thickness and effective droplet radius. Radiance data to be observed from space were simulated by a three-dimensional (3D) radiation model and a stochastic boundary layer cloud model with two-dimensional (horizontal and vertical) variability in cloud liquid water and effective radius. An adiabatic assumption is used for each cloud column to model the geometrical thickness and vertical profiles of cloud liquid water content and effective droplet radius, neglecting drizzle and cloud brokenness for simplicity. The dependence of radiative smoothing and roughening on horizontal scale, optical thickness and single scattering albedo are investigated. Then, retrieval methods using 250-m horizontal resolution data onboard new generation satellites are discussed. The regression model for the MSN method was trained based on datasets from numerical simulations. The training was performed with respect to various domain averages of optical thickness and effective radius, because smoothing and roughening effects are strongly dependent on the two variables. Retrieval accuracy is discussed here with datasets independent of those used in the training, towards assessing the generality of the technique. It is demonstrated that retrieval accuracy of cloud optical thickness, which is often retrieved from single-spectral visible-wavelength data, is improved the most using neighboring pixel data and secondly using multi-spectral data, and ideally with both. When the IPA retrieval method is applied to optical thickness and effective radius, the root-mean-square relative errors can be 15–90%, depending on solar and view directions. In contrast, the MSN method has errors of 4–10%, which is smaller than IPA by a factor of 2–10. It is also suggested that the accuracy of the MSN method is insensitive to some assumptions in the inhomogeneous cloud input data used to train the regression model.