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PubMed2002

Analysis of Monte Carlo methods applied to blackbody and lower emissivity cavities.

Monte Carlo methods are often applied to the calculation of the apparent emissivities of blackbody cavities. However, for cavities with complex as well as some commonly encountered geometries, the emission Monte Carlo method experiences problems of convergence. The emission and absorption Monte Carlo methods are compared on the basis of ease of implementation and convergence speed when applied to blackbody sources. A new method to determine solution convergence compatible with both methods is developed, and the convergence speeds of the two methods are compared through the application of both methods to a right-circular cylinder cavity. It is shown that the absorption method converges faster and is easier to implement than the emission method when applied to most blackbody and lower emissivity cavities.

Journal Article↗

PubMed1995

Building proteins from C alpha coordinates using the dihedral probability grid Monte Carlo method.

Dihedral probability grid Monte Carlo (DPG-MC) is a general-purpose method of conformational sampling that can be applied to many problems in peptide and protein modeling. Here we present the DPG-MC method and apply it to predicting complete protein structures from C alpha coordinates. This is useful in such endeavors as homology modeling, protein structure prediction from lattice simulations, or fitting protein structures to X-ray crystallographic data. It also serves as an example of how DPG-MC can be applied to systems with geometric constraints. The conformational propensities for individual residues are used to guide conformational searches as the protein is built from the amino-terminus to the carboxyl-terminus. Results for a number of proteins show that both the backbone and side chain can be accurately modeled using DPG-MC. Backbone atoms are generally predicted with RMS errors of about 0.5 A (compared to X-ray crystal structure coordinates) and all atoms are predicted to an RMS error of 1.7 A or better.

Amino Acids↗

PubMed1997

Monte Carlo methods in clinical research: applications in multivariable analysis.

BACKGROUND: Monte Carlo methods use "simulated" analyses with random numbers for solving problems, particularly those that defy solutions using mathematical theory alone. Research using Monte Carlo simulations is very popular in many branches of science and is sometimes done in clinical investigation. The origins and basic strategy of the technique, however, may not be well known to clinical researchers. The purpose of this paper is to describe the history and general principles of Monte Carlo methods and to demonstrate how Monte Carlo simulations were recently applied to examine a phenomenon in multivariable statistical analysis called the number of outcome events per independent variable (EPV). For example, in a cohort of 200 people, with 50 deaths and 5 independent (predictor) variables, EPV = 50/5 = 10. METHODS: The "real-world" data came from a clinical trial of 673 patients in which 7 variables were cogent predictors of 252 deaths, so that EPV = 252/7 = 36. For the Monte Carlo simulations, special models were used while allowing simulations of proportional hazards and logistic regression to maintain the basic relationship of variables and the same size of the original population, at EPV values of 2, 5, 10, 15, 20, and 25. RESULTS: The Monte Carlo simulations confirmed a previously undocumented "rule of thumb" stating that when the EPV is less than 10-20, the algebraic models used in logistic regression and proportional hazards regression may be unreliable, leading to imprecise or spurious results. CONCLUSION: Monte Carlo techniques offer attractive methods for clinical investigators to use in solving problems that are not amenable to customary mathematical approaches.

Humans↗

PubMed2002

Calculation of the average solid angle subtended by a detector to source in a parallel plane by a Monte Carlo method.

A short computer program is described for a PC which uses a Monte Carlo method to calculate the average solid angle subtended by a rectangular or circular detector window to a coaxial or non-coaxial rectangular, circular or point source. The advantage of the Monte Carlo method is that it allows the calculation of average solid angle for source-detector geometries that are difficult to analyse by analytical methods. The values of solid angle are calculated to accuracies of typically better than 0.1%. The calculated values from the Monte Carlo method agree closely with those produced by polygon approximation and numerical integration by Gardner and Verghese, and others.

Algorithms↗

PubMed2001

[Foundations of the Monte Carlo method for dose calculation in radiotherapy].

Monte Carlo (MC) methods applied in dose calculation are based on fundamental principles of radiation interaction with matter. In contrast to other methods, the accuracy of dose calculation achievable with MC depends only on the determination of the beam quality and the interaction coefficients. Using MC techniques it is possible to predict the dose for clinical photon and electron beams with an accuracy of > +/- 2%. Especially for inhomogeneous regions like head, neck, and lung, the MC technique can significantly improve the accuracy compared to conventional algorithms. Therefore, in the present paper the basic features of the MC method are reviewed in the context of treatment planning in radiation therapy. The main shortcoming in the past, i.e., that MC algorithms are too slow to be acceptable for clinical purposes, could be solved by using faster computers and by introducing new variance reduction (VR) techniques. These techniques decrease the statistical fluctuations without increasing the number of particle histories. Therefore, MC calculation times in the order of a few minutes are possible. A brief overview of VR methods is provided.

Humans↗

PubMed2006

Molecular modeling of porous carbons using the hybrid reverse Monte Carlo method.

We apply a simulation protocol based on the reverse Monte Carlo (RMC) method, which incorporates an energy constraint, to model porous carbons. This method is called hybrid reverse Monte Carlo (HRMC), since it combines the features of the Monte Carlo and reverse Monte Carlo methods. The use of the energy constraint term helps alleviate the problem of the presence of unrealistic features (such as three- and four-membered carbon rings), reported in previous RMC studies of carbons, and also correctly describes the local environment of carbon atoms. The HRMC protocol is used to develop molecular models of saccharose-based porous carbons in which hydrogen atoms are taken into account explicitly in addition to the carbon atoms. We find that the model reproduces the experimental pair correlation function with good accuracy. The local structure differs from that obtained with a previous model (Pikunic, J.; Clinard, C.; Cohaut, N.; Gubbins, K. E.; Guet, J. M.; Pellenq, R. J.-M.; Rannou, I.; Rouzaud, J. N. Langmuir 2003, 19 (20), 8565). We study the local structure by calculating the nearest neighbor distribution, bond angle distribution, and ring statistics.

Carbon↗

PubMed2005

Accurate, efficient, and simple forces computed with quantum Monte Carlo methods.

Computation of ionic forces using quantum Monte Carlo methods has long been a challenge. We introduce a simple procedure, based on known properties of physical electronic densities, to make the variance of the Hellmann-Feynman estimator finite. We obtain very accurate geometries for the molecules H(2), LiH, CH(4), NH(3), H(2)O, and HF, with a Slater-Jastrow trial wave function. Harmonic frequencies for diatomics are also in good agreement with experiment. An antithetical sampling method is also discussed for additional reduction of the variance.

Journal Article↗

PubMed2005

Green's function Monte Carlo method with exact imaginary-time propagation.

We present a general formulation of the Green's function Monte Carlo method in imaginary-time quantum Monte Carlo which employs exact propagators. This algorithm has no time-step errors and is obtained by minimal modifications of the time-independent Green's function Monte Carlo method. We describe how the method can be applied to the many-body Schrödinger equation, lattice Hamiltonians, and simple field theories. Our modification of the Green's function Monte Carlo algorithm is applied to the ground state of liquid 4He. We calculate the zero-temperature imaginary-time diffusion constant and relate that to the effective mass of a mass-four "impurity" atom in liquid 4He.

Journal Article↗

PubMed1995

Exploration of compact protein conformations using the guided replication Monte Carlo method.

We have studied the use of a new Monte Carlo (MC) chain generation algorithm, introduced by T. Garel and H. Orland [(1990) Journal of Physics A, Vol. 23, pp. L621-L626], for examining the thermodynamics of protein folding transitions and for generating candidate C(alpha) backbone structures as starting points for a de novo protein structure paradigm. This algorithm, termed the guided replication Monte Carlo method, allows a rational approach to the introduction of known "native" folded characteristics as constraints in the chain generation process . We have shown this algorithm to be computationally very efficient in generating large ensembles of candidate C(alpha) chains on the face centered cubic lattice, and illustrate its use by calculating a number of thermodynamic quantities related to protein folding characteristics. In particular, we have used this static MC algorithm to compare such temperature-dependent quantities as the ensemble mean energy, ensemble mean free energy, the heat capacity, and the mean-square radius of gyration. We also demonstrate the use of several simple "guide fields" for introducing protein-specific constraints into the ensemble generation process. Several extensions to our current model are suggested, and applications of the method to other folding related problems are discussed.

Algorithms↗

PubMed2005

Self-referential Monte Carlo method for calculating the free energy of crystalline solids.

A self-referential Monte Carlo method is described for calculating the free energy of crystalline solids. All Monte Carlo methods for the free energy of classical crystalline solids calculate the free-energy difference between a state whose free energy can be calculated relatively easily and the state of interest. Previously published methods employ either a simple model crystal, such as the Einstein crystal, or a fluid as the reference state. The self-referential method employs a radically different reference state; it is the crystalline solid of interest but with a different number of unit cells. So it calculates the free-energy difference between two crystals, differing only in their size. The aim of this work is to demonstrate this approach by application to some simple systems, namely, the face centered cubic hard sphere and Lennard-Jones crystals. However, it can potentially be applied to arbitrary crystals in both bulk and confined environments, and ultimately it could also be very efficient.

Journal Article↗

PubMed2003

Bilinear diffusion quantum Monte Carlo methods.

The standard method of quantum Monte Carlo for the solution of the Schrödinger equation in configuration space can be described quite generally as devising a random walk that generates-at least asymptotically-populations of random walkers whose probability density is proportional to the wave function of the system being studied. While, in principle, the energy eigenvalue of the Hamiltonian can be calculated with high accuracy, estimators of operators that do not commute the Hamiltonian cannot. Bilinear quantum Monte Carlo (BQMC) is an alternative in which the square of the wave function is sampled in a somewhat indirect way. More specifically, one uses a pair of walkers at positions x and y and introduces stochastic dynamics to sample phi(i)(x)t(x,y)phi(j)(y), where phi(i)(x) and phi(j)(y) are eigenfunctions of (possibly different) Hamiltonians, and t(x,y) is a kernel that correlates positions x and y. Using different Hamiltonians permits the accurate computation of small energy differences. We review the conceptual basis of BQMC, discuss qualitatively and analytically the problem of the fluctuations in the branching, and present partial solutions to that problem. Finally we exhibit numerical results for some model systems including harmonic oscillators and the hydrogen and helium atoms. Further research will be necessary to make this a practical and generally applicable scheme.

Journal Article↗

PubMed1991

Protonation of interacting residues in a protein by a Monte Carlo method: application to lysozyme and the photosynthetic reaction center of Rhodobacter sphaeroides.

We used Monte Carlo methods to treat statistical problem of electrostatic interactions among many titrating amino acids and applied these methods to lysozyme and the photosynthetic reaction center of Rhodobacter sphaeroides, including all titrating sites. We computed the average protonation of residues as a function of pH from an equilibrium distribution of states generated by random sampling. Electrostatic energies were calculated from a finite difference solution to the linearized Poisson-Boltzmann equation using the coordinates from solved protein structures. For most calculations we used the Metropolis algorithm to sample protonation states; for strongly coupled sites, we substantially reduced sampling errors by using a modified algorithm that allows multiple site transitions. The Monte Carlo method agreed with calculations for a small test system, lysozyme, for which the complete partition function was calculated. We also calculated the pH dependence of the free energy change associated with electron transfer from the primary to the secondary quinone in the photosynthetic reaction center. The shape of the resulting curve agreed fairly well with experiment, but the proton uptake from which the free energy was calculated agreed only to within a factor of two with the observed values. We believe that this discrepancy resulted from errors in the individual electrostatic energy calculations rather than from errors in the Monte Carlo sampling.

Electron Transport↗

PubMed1992

A Monte Carlo method for combined segregation and linkage analysis.

We introduce a Monte Carlo approach to combined segregation and linkage analysis of a quantitative trait observed in an extended pedigree. In conjunction with the Monte Carlo method of likelihood-ratio evaluation proposed by Thompson and Guo, the method provides for estimation and hypothesis testing. The greatest attraction of this approach is its ability to handle complex genetic models and large pedigrees. Two examples illustrate the practicality of the method. One is of simulated data on a large pedigree; the other is a reanalysis of published data previously analyzed by other methods.

Algorithms↗

PubMed2000

Monte Carlo methods in electronic structures for large systems.

Quantum Monte Carlo methods have recently made it possible to calculate the electronic structure of relatively large molecular systems with very high accuracy. These large systems range from positron complexes [NH(2),Ps] with approximately 10 electrons to C(20) isomers with 120 electrons, to silicon crystal structures of 250 atoms and 1000 valence electrons. The techniques for such calculations and a sampling of applications are reviewed.

Journal Article↗

PubMed1990

Monte Carlo methods of neutron beam design for neutron capture therapy at the MIT Research Reactor (MITR-II).

Monte Carlo methods of coupled neutron/photon transport are being used in the design of filtered beams for Neutron Capture Therapy (NCT). This method of beam analysis provides segregation of each individual dose component, and thereby facilitates beam optimization. The Monte Carlo method is discussed in some detail in relation to NCT epithermal beam design. Ideal neutron beams (i.e., plane-wave monoenergetic neutron beams with no primary gamma-ray contamination) have been modeled both for comparison and to establish target conditions for a practical NCT epithermal beam design. Detailed models of the 5 MWt Massachusetts Institute of Technology Research Reactor (MITR-II) together with a polyethylene head phantom have been used to characterize approximately 100 beam filter and moderator configurations. Using the Monte Carlo methodology of beam design and benchmarking/calibrating our computations with measurements, has resulted in an epithermal beam design which is useful for therapy of deep-seated brain tumors. This beam is predicted to be capable of delivering a dose of 2000 RBE-cGy (cJ/kg) to a therapeutic advantage depth of 5.7 cm in polyethylene assuming 30 micrograms/g 10B in tumor with a ten-to-one tumor-to-blood ratio, and a beam diameter of 18.4 cm. The advantage ratio (AR) is predicted to be 2.2 with a total irradiation time of approximately 80 minutes. Further optimization work on the MITR-II epithermal beams is expected to improve the available beams.

Brain Neoplasms↗

PubMed2007

Efficient combination of Wang-Landau and transition matrix Monte Carlo methods for protein simulations.

An efficient combination of the Wang-Landau and transition matrix Monte Carlo methods for protein and peptide simulations is described. At the initial stage of simulation the algorithm behaves like the Wang-Landau algorithm, allowing to sample the entire interval of energies, and at the later stages, it behaves like transition matrix Monte Carlo method and has significantly lower statistical errors. This combination allows to achieve fast convergence to the correct values of density of states. We propose that the violation of TTT identities may serve as a qualitative criterion to check the convergence of density of states. The simulation process can be parallelized by cutting the entire interval of simulation into subintervals. The violation of ergodicity in this case is discussed. We test the algorithm on a set of peptides of different lengths and observe good statistical convergent properties for the density of states. We believe that the method is of general nature and can be used for simulations of other systems with either discrete or continuous energy spectrum.

Algorithms↗

PubMed1993

Achieving irreducibility of the Markov chain Monte Carlo method applied to pedigree data.

Markov chain Monte Carlo (MCMC) methods have been explored by various researchers as an alternative to exact probability computation in statistical genetics. The objective is to simulate a Markov chain with the desired equilibrium distribution. If the transition kernel is aperiodic and irreducible, then convergence to the equilibrium distribution is guaranteed; realizations of the Markov chain can thus be used to estimate desired probabilities. Aperiodicity is easily satisfied, but, although it has been shown that irreducibility is satisfied for a diallelic locus, reducibility is a potential problem for a multiallelic locus. This is a particularly serious problem in linkage analysis, because multiallelic markers are much more informative than diallelic markers and thus highly preferred. In this paper, the authors propose a new algorithm to achieve irreducibility of the Markov chain of interest by introducing an irreducible auxiliary chain. The irreducibility of the auxiliary chain is obtained by assigning positive probabilities to a small subset of the genotypic configurations inconsistent with the data, to bridge the gap between the irreducible sets.

Algorithms↗

PubMed2000

Diffusion monte carlo methods with a fixed number of walkers

In this paper we discuss various aspects of diffusion Monte Carlo methods using a fixed number of walkers. First, a rigorous proof of the divergence of pure diffusion Monte Carlo (PDMC) methods (DMC without branching in which the weights are carried along trajectories) is given. Second, a bias-free Monte Carlo method combining DMC and PDMC approaches, and based on a minimal stochastic reconfiguration of the population, is discussed. Finally, some illustrative calculations for a system of coupled quantum rotators are presented.

Journal Article↗