Large Number Of Grid Points Research Articles

Vlasov on GPU (VOG project)

This work concerns the numerical simulation of the Vlasov-Poisson equation using semi-Lagrangian methods on Graphics Processing Units (GPU). To accomplish this goal, modifications to traditional methods had to be implemented. First and foremost, a reformulation of semi-Lagrangian methods is performed, which enables us to rewrite the governing equations as a circulant matrix operating on the vector of unknowns. This product calculation can be performed efficiently using FFT routines. Nowadays GPU is no more limited to single precision; however, single precision may still be preferred with respect to performance and available memory. So, in order to be able to deal with single precision, a δf type method is adopted which only needs refinement in specialized areas of phase space but not throughout. Thus, a GPU Vlasov-Poisson solver can indeed perform high precision simulations (since it uses very high order of reconstruction and a large number of grid points in phase space). We show results for more academic test cases and also for physically relevant phenomena such as the bump on tail instability and the simulation of Kinetic Electrostatic Electron Nonlinear (KEEN) waves.

A finite difference construction of the spheroidal wave functions

A finite difference construction of the spheroidal wave functions

Cartesian grid method for the compressible Euler equations using simplified ghost point treatments at embedded boundaries

We introduce two new approaches called the simplified and the modified simplified ghost point treatments for solving the 2D compressible Euler equations near embedded boundaries for the Cartesian grid method. These approaches are second order accurate for second order schemes near the embedded boundaries, if the wall boundary is in the middle between fluid and ghost points. We assign values to the ghost points near embedded solid boundaries from mirror points in the fluid to reflect the presence of the solid boundaries. In the simplified ghost point treatment, we consider the closest grid points on the grid lines through the ghost points in the x- and y-directions as the mirror points of the ghost points depending on which directions are closest to the directions normal to the embedded boundaries. In the modified simplified ghost point treatment, we choose mirror points not only on the grid lines through the ghost points in the x- or y-directions, but also on the diagonals through the ghost points. The primitive variables at the mirror points are mirrored to the ghost points using local symmetry boundary conditions. The simplified ghost point treatments at embedded boundaries are tested for supersonic flow over a circular arc airfoil and a circular cylinder. Applications to supersonic flow over multiple circular cylinders and a 2D model of the F-22 fighter aircraft demonstrate the flexibility of the ghost point treatments. Another advantage of these new approaches is that they are easily extendable to higher order methods and to 3D. The Cartesian grid method requires a larger number of grid points than the standard body-fitted grid method. We found a good agreement between the results obtained with the simplified and the modified simplified ghost point treatments and the reference solutions in the literature.

An Application of the Level Set Method to Underwater Acoustic Propagation

An algorithm for computing wavefronts, based on the high frequency approximation to the wave equation, is presented. This technique applies the level set method to underwater acoustic wavefront propagation in the time domain. The level set method allows for computation of the acoustic phase function using established numerical techniques to solve a first order transport equation to a desired order of accuracy. Traditional methods for solving the eikonal equation directly on a fixed grid limit one to only the first arrivals, so these approaches are not useful when multi-path propagation is present. Applying the level set model to the problem allows for the time domain computation of the phase function on a fixed grid, without having to restrict to first arrival times. The implementation presented has no restrictions on range dependence or direction of travel, and offers improved efficiency over solving the full wave equation which under the high frequency assumption requires a large number of grid points to resolve the highly oscillatory solutions. Boundary conditions are discussed, and an approach is suggested for producing good results in the presence of boundary reflections. An efficient method to compute the amplitude from the level set method solutions is also presented. Comparisons to analytical solutions are presented where available, and numerical results are validated by comparing results with exact solutions where available, a full wave equation solver, and with wavefronts extracted from ray tracing software.

Contour integral method for European options with jumps

We develop an efficient method for pricing European options with jump on a single asset. Our approach is based on the combination of two powerful numerical methods, the spectral domain decomposition method and the Laplace transform method. The domain decomposition method divides the original domain into sub-domains where the solution is approximated by using piecewise high order rational interpolants on a Chebyshev grid points. This set of points are suitable for the approximation of the convolution integral using Gauss–Legendre quadrature method. The resulting discrete problem is solved by the numerical inverse Laplace transform using the Bromwich contour integral approach. Through rigorous error analysis, we determine the optimal contour on which the integral is evaluated. The numerical results obtained are compared with those obtained from conventional methods such as Crank–Nicholson and finite difference. The new approach exhibits spectrally accurate results for the evaluation of options and associated Greeks. The proposed method is very efficient in the sense that we can achieve higher order accuracy on a coarse grid, whereas traditional methods would required significantly more time-steps and large number of grid points.

Nonlinear observer with adaptive grid allocation for a fixed-bed adsorption process

Real-time knowledge of concentration profiles inside the adsorption bed is essential for the feedback control of a pressure swing adsorption (PSA) process. A difficulty encountered when developing an observer for this purpose is that the fixed-bed adsorption model is very stiff; thus, either a large number of grid points or an adaptive grid allocation is required to solve the model accurately through discretization. The objective of this study is to present a nonlinear observer with an adaptive grid allocation for an adsorption bed to estimate a spatially distributed state. Experimental verification revealed that the proposed technique can accurately and rapidly predict the breakthrough curve in addition to other process variables despite uncertainties in the model parameters and in the feed composition.

Extended Kalman filter with adaptive grid allocation for a fixed-bed adsorption process

Real-time knowledge of concentration profiles inside the adsorption bed is essential for feedback control of a pressure swing adsorption process. A difficulty encountered in developing an observer for this purpose is that the fixed-bed adsorption model is highly stiff and either a large number of grid points or an adaptive grid allocation is indispensable to accurately solve the model through discretization. The objective of this study is to present a nonlinear observer with adaptive grid allocation for an adsorption bed to estimate spatially distributed state. Experimental verification revealed that the proposed technique can accurately and rapidly predict the breakthrough curve in addition to other process variables despite the uncertainties in model parameters as well as the feed composition.

An Application of the Level Set Method to Underwater Acoustic Propagation

An algorithm for computing wavefronts, based on the high frequency approximation to the wave equation, is presented. This technique applies the level set method to underwater acoustic wavefront propagation in the time domain. The level set method allows for computation of the acoustic phase function using established numerical techniques to solve a first order transport equation to a desired order of accuracy. Traditional methods for solving the eikonal equation directly on a fixed grid limit one to only the first arrivals, so these approaches are not useful when multi-path propagation is present. Applying the level set model to the problem allows for the time domain computation of the phase function on a fixed grid, without having to restrict to first arrival times. The implementation presented has no restrictions on range dependence or direction of travel, and offers improved efficiency over solving the full wave equation which under the high frequency assumption requires a large number of grid points to resolve the highly oscillatory solutions. Boundary conditions are discussed, and an approach is suggested for producing good results in the presence of boundary reflections. An efficient method to compute the amplitude from the level set method solutions is also presented. Comparisons to analytical solutions are presented where available, and numerical results are validated by comparing results with exact solutions where available, a full wave equation solver, and with wavefronts extracted from ray tracing software.

Fast reconstruction and prediction of frozen flow turbulence based on structured Kalman filtering

Efficient and optimal prediction of frozen flow turbulence using the complete observation history of the wavefront sensor is an important issue in adaptive optics for large ground-based telescopes. At least for the sake of error budgeting and algorithm performance, the evaluation of an accurate estimate of the optimal performance of a particular adaptive optics configuration is important. However, due to the large number of grid points, high sampling rates, and the non-rationality of the turbulence power spectral density, the computational complexity of the optimal predictor is huge. This paper shows how a structure in the frozen flow propagation can be exploited to obtain a state-space innovation model with a particular sparsity structure. This sparsity structure enables one to efficiently compute a structured Kalman filter. By simulation it is shown that the performance can be improved and the computational complexity can be reduced in comparison with auto-regressive predictors of low order.

Quantitative properties of sovereign default models: Solution methods matter

We study the sovereign default model that has been used to account for the cyclical behavior of interest rates in emerging market economies. This model is often solved using the discrete state space technique with evenly spaced grid points. We show that this method necessitates a large number of grid points to avoid generating spurious interest rate movements. This makes the discrete state technique significantly more inefficient than using Chebyshev polynomials or cubic spline interpolation to approximate the value functions. We show that the inefficiency of the discrete state space technique is more severe for parameterizations that feature a high sensitivity of the bond price to the borrowing level for the borrowing levels that are observed more frequently in the simulations. In addition, we find that the efficiency of the discrete state space technique can be greatly improved by (i) finding the equilibrium as the limit of the equilibrium of the finite-horizon version of the model, instead of iterating separately on the value and bond price functions and (ii) concentrating grid points in asset levels at which the bond price is more sensitive to the borrowing level and in levels that are observed more often in the model simulations. Our analysis is also relevant for the study of other credit markets.

Quantitative Properties of Sovereign Default Models: Solution Methods Matter

We study the sovereign default model that has been used to account for the cyclical behavior of interest rates in emerging market economies. This model is often solved using the discrete state space technique with evenly spaced grid points. We show that this method necessitates a large number of grid points to avoid generating spurious interestrate movements. This makes the discrete state technique significantly more inefficient than using Chebyshev polynomials or cubic spline interpolation to approximate the value functions. We show that the inefficiency of the discrete state space technique is more severe for parameterizations that feature a high sensitivity of the bond price to the borrowing level for the borrowing levels that are observed more frequently in the simulations. In addition, we find that the efficiency of the discrete state space technique can be greatly improved by (i) finding the equilibrium as the limit of the equilibrium of the finite-horizon version of the model, instead of iterating separately on the value and bond price functions and (ii) concentrating grid points in asset levels at which the bond price is more sensitive to the borrowing level and in levels that are observed more often in the model simulations. Our analysis is also relevant for the study of other credit markets.

Quantitative Properties of Sovereign Default Models: Solution Methods Matter

We study the sovereign default model that has been used to account for the cyclical behavior of interest rates in emerging market economies. This model is often solved using the discrete state space technique with evenly spaced grid points. We show that this method necessitates a large number of grid points to avoid generating spurious interest rate movements. This makes the discrete state technique significantly more inefficient than using Chebyshev polynomials or cubic spline interpolation to approximate the value functions. We show that the inefficiency of the discrete state space technique is more severe for parameterizations that feature a high sensitivity of the bond price to the borrowing level for the borrowing levels that are observed more frequently in the simulations. In addition, we find that the efficiency of the discrete state space technique can be greatly improved by (i) finding the equilibrium as the limit of the equilibrium of the finite-horizon version of the model, instead of iterating separately on the value and bond price functions and (ii) concentrating grid points in asset levels at which the bond price is more sensitive to the borrowing level and in levels that are observed more often in the model simulations. Our analysis questions the robustness of results in the sovereign default literature and is also relevant for the study of other credit markets.

Quantitative Properties of Sovereign Default Models: Solution Methods Matter

We study the sovereign default model that has been used to account for the cyclical behavior of interest rates in emerging market economies. This model is often solved using the discrete state space technique with evenly spaced grid points. We show that this method necessitates a large number of grid points to avoid generating spurious interestrate movements. This makes the discrete state technique significantly more inefficient than using Chebyshev polynomials or cubic spline interpolation to approximate the value functions. We show that the inefficiency of the discrete state space technique is more severe for parameterizations that feature a high sensitivity of the bond price to the borrowing level for the borrowing levels that are observed more frequently in the simulations. In addition, we find that the efficiency of the discrete state space technique can be greatly improved by (i) finding the equilibrium as the limit of the equilibrium of the finite-horizon version of the model, instead of iterating separately on the value and bond price functions and (ii) concentrating grid points in asset levels at which the bond price is more sensitive to the borrowing level and in levels that are observed more often in the model simulations. Our analysis is also relevant for the study of other credit markets. ; WP 10-04 replaces earlier versions listed as WP 09-13 and WP 06-11

等角写像を用いて任意の2次元単連結領域に直交曲線格子を生成する一数値計算法(熱工学,内燃機関,動力など)

One easy way to numerically generate an orthogonal grid in 2D domain is presented and discussed. An orthogonal grid is generated by conformal mapping, i.e., this method finds at the same time both the regular complex function which determines mapping and the rectangular domain which is mapped to a given 2D domain. The grid lines are equal to isotherms of 2D steady heat conduction problems under certain boundary conditions, so a generated grid shape can be imagined well before computation. The formulation is really simple only using Cauchy's integral formula, a rudimentary theorem of complex analysis. Because the process is non-iterative, and the boundary integral is partly performed analytically in the formulation, computation time is short even though generating a large number of grid points. This method is not very flexible about grid spacing control, but is useful with respect to the simplicity and the potential of speedy calculation. Test calculations show that this method is successfully applied to various geometries.

Resolution Dependence of the Tropopause Inversion Layer in an Idealized Model for Upper-Tropospheric Anticyclones

Abstract This note investigates the dependence of the extratropical tropopause inversion layer (TIL) on numerical resolution in an idealized modeling framework. Axisymmetric upper-tropospheric anticyclones are constructed by specifying potential vorticity (PV) and solving the nonlinear PV-inversion problem. The PV distribution has a smooth but near-discontinuous change of PV across the tropopause in a transition zone with vertical depth δ. For fixed δ the strength of the TIL changes with changing resolution until the transition zone is resolved by a fairly large number of grid points. The quality-controlled numerical solutions are used to study the behavior for δ → 0. This limit can lead to very strong TILs, but no indications for divergent behavior were found.

Stochastic ground motion simulation of the 12 October 1992 Dahshour earthquake

The stochastic method for finite faults is applied to simulate the ground motion of the 12 October 1992, mb = 5.9, Dahshour earthquake. The method includes discritization of the fault plane into certain number of subfaults, and a ω-squared spectrum is assigned to each of them. Contributions from all subfaults are then empirically attenuated to the observation sites, where they are summed to produce the synthetic acceleration time-history. The method is first tested against its ability of reproducing the recording at Kottamya station. The calibrated model is then applied to calculate the synthetics at a large number of grid points covering the area around the fault plane. Simulated peak values are subsequently used to produce the synthetic peak horizontal acceleration map for the area.

Enhanced spectral resolution by high-dimensional NMR using the filter diagonalization method and “hidden” dimensions

High-dimensional (HD) NMR spectra have poorer digital resolution than low-dimensional (LD) spectra, for a fixed amount of experiment time. This has led to “reduced-dimensionality” strategies, in which several LD projections of the HD NMR spectrum are acquired, each with higher digital resolution; an approximate HD spectrum is then inferred by some means. We propose a strategy that moves in the opposite direction, by adding more time dimensions to increase the information content of the data set, even if only a very sparse time grid is used in each dimension. The full HD time-domain data can be analyzed by the filter diagonalization method (FDM), yielding very narrow resonances along all of the frequency axes, even those with sparse sampling. Integrating over the added dimensions of HD FDM NMR spectra reconstitutes LD spectra with enhanced resolution, often more quickly than direct acquisition of the LD spectrum with a larger number of grid points in each of the fewer dimensions. If the extra-dimensions do not appear in the final spectrum, and are used solely to boost information content, we propose the moniker hidden-dimension NMR. This work shows that HD peaks have unmistakable frequency signatures that can be detected as single HD objects by an appropriate algorithm, even though their patterns would be tricky for a human operator to visualize or recognize, and even if digital resolution in an HD FT spectrum is very coarse compared with natural line widths.

Comparison of generalized differential quadrature and Galerkin methods for the analysis of micro-electro-mechanical coupled systems

This paper presents a comprehensive comparison study between the generalized differential quadrature (GDQ) and the well-known global Galerkin method for analysis of pull-in behavior of nonlinear micro-electro-mechanical coupled systems. The nonlinear governing integro-differential equation for double clamped MEMS devices which was derived using variational principle by the authors [Sadeghian H, Rezazadeh G, Osterberg PM. Application of the generalized differential quadrature method to the study of pull-in phenomena of MEMS switches. J Microelectromech Syst 2007;16(6):1334–40] is discretized by applying Galerkin and GDQ methods. The divergence instability or pull-in phenomenon is analyzed. Obtained results are compared with the results of the pervious works. The Galerkin method is implemented with effect of number of used shape functions. Different types of trail functions on calculated pull-in voltage are examined. Furthermore, compare to one term and two terms truncation Galerkin method, it is observed that the GDQ with small number of grid points (non-uniform) performs accurate results for nonlinear micro-electro-mechanical coupled behavior which requires a large number of grid points at high-order approximation.

A conservative and non-oscillatory scheme for Vlasov code simulations

Abstract A new numerical positive interpolation technique for conservation laws and its application to Vlasov code simulations are presented. In recent Vlasov simulation codes, the Vlasov equation is solved based on the numerical interpolation method because of its simplicity of algorithm and its ease of programming. However, a large number of grid points are needed in both configuration and velocity spaces to suppress numerical diffusion. In this paper we propose a new high-order interpolation scheme for Vlasov simulations. The current scheme is non-oscillatory and conservative and is well-designed for Vlasov simulations. This is compared with the latest interpolation schemes by performing one-dimensional electrostatic Vlasov simulations.

A simplified age-stage model for copepod population dynamics

MEPS Marine Ecology Progress Series Contact the journal Facebook Twitter RSS Mailing List Subscribe to our mailing list via Mailchimp HomeLatest VolumeAbout the JournalEditorsTheme Sections MEPS 360:179-187 (2008) - DOI: https://doi.org/10.3354/meps07314 A simplified age-stage model for copepod population dynamics Qiao Hu*, Cabell S. Davis, Colleen M. Petrik Biology Department, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543, USA *Email: qiao_hu@alum.mit.edu ABSTRACT: Complex 3D biological-physical models are becoming widely used in marine and freshwater ecology. These models are highly valued synthesizing tools because they provide insights into complex dynamics that are difficult to understand using purely empirical methods or theoretical analytical models. Of particular interest has been the incorporation of concentration-based copepod population dynamics into 3D physical transport models. These physical models typically have large numbers of grid points and therefore require a simplified biological model. However, concentration-based copepod models have used a fine resolution age-stage structure to prevent artificially short generation times, known as numerical ‘diffusion.’ This increased resolution has precluded use of age-stage structured copepod models in 3D physical models due to computational constraints. In this paper, we describe a new method, which tracks the mean age of each life stage instead of using age classes within each stage. We then compare this model to previous age-stage structured models. A probability model is developed with the molting rate derived from the mean age of the population and the probability density function (PDF) of molting. The effects of temperature and mortality on copepod population dynamics are also discussed. The mean-age method effectively removes the numerical diffusion problem and reproduces observed median development times (MDTs) without the need for a high-resolution age-stage structure. Thus, it is well-suited for finding solutions of concentration-based zooplankton models in complex biological-physical models. KEY WORDS: Plankton · Copepods · Modeling · Marine ecology · Oceanography · Limnology · Methodology · Mean age Full text in pdf format PreviousNextCite this article as: Hu Q, Davis CS, Petrik CM (2008) A simplified age-stage model for copepod population dynamics. Mar Ecol Prog Ser 360:179-187. https://doi.org/10.3354/meps07314 Export citation RSS - Facebook - Tweet - linkedIn Cited by Published in MEPS Vol. 360. Online publication date: May 22, 2008 Print ISSN: 0171-8630; Online ISSN: 1616-1599 Copyright © 2008 Inter-Research.