Monte-Carlo Walk Research Articles

Dense-sparse quantum Monte Carlo algebraic diagrammatic construction and importance ranking.

Quantum Monte Carlo Algebraic Diagrammatic Construction (QMCADC) has been proposed as a reformulation of the second-order ADC scheme for the polarization propagator within the projection quantum Monte Carlo formalism. Dense-sparse partitioning and importance ranking filtering strategies are now exploited to accelerate its convergence and to alleviate the signproblem inherent in such calculations. By splitting the configuration space into dense and sparse subsets, the corresponding projection operator is decomposed into four distinct blocks. Deterministic calculations handle the dense-to-dense and sparse-to-dense blocks, while the remaining blocks, dense-to-sparse and sparse-to-sparse, are stochastically evaluated. The dense set is efficiently stored in a fixed-size array, and the sparse set is represented through conventional floating random Monte Carlo walks. The stochastic projection is further refined through importance ranking criteria, enabling a reduction in the required number of walkers with a controllable bias. Our results demonstrate the integration of dense-sparse partitioning with importance ranking filtering to significantly enhance the efficiency of QMCADC, enabling large-scale molecular excited-state calculations. Furthermore, this novel approach maximizes the utilization of the sparsity of ADC(2), transforming QMCADC into a tailored framework for ADC calculations.

Monte Carlo Increased-Radius Floating Random Walk Solution For Potential Problems

In this paper, a new Monte Carlo walk method is introduced. The increased radius floating random walk combines of the two classical Monte Carlo methods and derived from fixed-radius floating walk method. In this paper, the method is used to solve typical Laplace’s equations in rectangular region. Also, this method is easily applied to Poisson equations. Lower walk number and hence lower computation time are obtained from new method compared with the fixed random walk, floating random walk and fixed-radius random walk methods. Analyzes were performed on an average computer and the solution time was reduced by 80%. The results are also compared with Finite Element Method. Increased radius walk method’s results are good agreement with other methods.

A geostatistical Markov chain Monte Carlo inversion algorithm for electrical resistivity tomography

ABSTRACTElectrical resistivity tomography is an ill‐posed and nonlinear inverse problem commonly solved through deterministic gradient‐based methods. These methods guarantee a fast convergence towards the final solution, but the local linearization of the inverse operator impedes accurate uncertainty assessments. On the contrary, numerical Markov chain Monte Carlo algorithms allow for accurate uncertainty appraisals, but appropriate Markov chain Monte Carlo recipes are needed to reduce the computational effort and make these approaches suitable to be applied to field data. A key aspect of any probabilistic inversion is the definition of an appropriate prior distribution of the model parameters that can also incorporate spatial constraints to mitigate the ill conditioning of the inverse problem. Usually, Gaussian priors oversimplify the actual distribution of the model parameters that often exhibit multimodality due to the presence of multiple litho‐fluid facies. In this work, we develop a novel probabilistic Markov chain Monte Carlo approach for inversion of electrical resistivity tomography data. This approach jointly estimates resistivity values, litho‐fluid facies, along with the associated uncertainties from the measured apparent resistivity pseudosection. In our approach, the unknown parameters include the facies model as well as the continuous resistivity values. At each spatial location, the distribution of the resistivity value is assumed to be multimodal and non‐parametric with as many modes as the number of facies. An advanced Markov chain Monte Carlo algorithm (the differential evolution Markov chain) is used to efficiently sample the posterior density in a high‐dimensional parameter space. A Gaussian variogram model and a first‐order Markov chain respectively account for the lateral continuity of the continuous and discrete model properties, thereby reducing the null‐space of solutions. The direct sequential simulation geostatistical method allows the generation of sampled models that honour both the assumed marginal prior and spatial constraints. During the Markov chain Monte Carlo walk, we iteratively sample the facies, by moving from one mode to another, and the resistivity values, by sampling within the same mode. The proposed method is first validated by inverting the data calculated from synthetic models. Then, it is applied to field data and benchmarked against a standard local inversion algorithm. Our experiments prove that the proposed Markov chain Monte Carlo inversion retrieves reliable estimations and accurate uncertainty quantifications with a reasonable computational effort.

Observations on variational and projector Monte Carlo methods.

Variational Monte Carlo and various projector Monte Carlo (PMC) methods are presented in a unified manner. Similarities and differences between the methods and choices made in designing the methods are discussed. Both methods where the Monte Carlo walk is performed in a discrete space and methods where it is performed in a continuous space are considered. It is pointed out that the usual prescription for importance sampling may not be advantageous depending on the particular quantum Monte Carlo method used and the observables of interest, so alternate prescriptions are presented. The nature of the sign problem is discussed for various versions of PMC methods. A prescription for an exact PMC method in real space, i.e., a method that does not make a fixed-node or similar approximation and does not have a finite basis error, is presented. This method is likely to be practical for systems with a small number of electrons. Approximate PMC methods that are applicable to larger systems and go beyond the fixed-node approximation are also discussed.

Improved trial wave functions in quantum Monte Carlo: Application to acetylene and its dissociation fragments

Recent quantum Monte Carlo (QMC) studies of electronic structure have considered various trial function enhancements directed at improved fixed-node energies. In this study we investigate complete active space self-consistent field (CASSCF) trial functions in the diffusion Monte Carlo (DMC) method. We study longer CASSCF expansions than typically used in QMC studies and optimize correlation function parameters, basis function coefficients, and configuration state function mixing coefficients. To perform a stable, efficient wavefunction optimization, sample points are analytically obtained from an integrable probability density function or a Monte Carlo walk guided by a positive definite function. The approach is applied to acetylene and its dissociation fragments (C, CH, C2, C2H, C2H2). For these systems 70%–90% of the correlation energy is recovered with variational MC and 91%–98% with DMC.

Computation of transition dipole moments by Monte Carlo

Three Monte Carlo methods for computing transition dipole moments are presented. Two of these approaches are based on the use of multiple Monte Carlo ‘‘random walks’’ to sample different probability distributions. The remaining technique employs a single Monte Carlo walk and averages an analytic approximation to the Green’s function to sample other distributions. The accuracy and efficiency of each method is investigated by computing the transition dipole moment between the 1s and 2px states of the hydrogen atom. Monte Carlo parameters, such as the time step size and the convergence time, are varied in order to study their effect on computed results. It is found that the approach based on a guided Metropolis walk with quantum Monte Carlo ‘‘side walks’’ and also the approach based on Green’s function averages yield accurate transition dipole moments efficiently. These two methods also yield accurate energies and expectation values for the individual eigenstates. The approach based on two equivalent quantum Monte Carlo walks, one for each state, is found to be least satisfactory.

Structure of water and counterions in DNA

Monte Carlo simulation have been performed to study the hydration of two DNA conformers: the right-handed B-DNA and the left-handed Z-DNA, both in the double helix conformation. The models used to represent the polyelectrolytes consist of three full pitches (60 and 72 nucleotides units for B- and Z-DNA, respectively). The simulations have been carried out at a temperature of 300 K with 500 water molecules per pitch and as many counterions as necessary to neutralize DNA. In the present studies three different mono-charged counterions have been considered: Li +, Na + and K +. During the Monte Carlo process either the water molecules or the counterions are subject to displacements [1]. The Monte Carlo walks, for these six simulations, have been analyzed in such way to give, for the counterions, information about their position (not yet determined by experiments), and the number of water molecules in their first hydration shell. Some of the counterions have, in the first hydration shell, the same number of water molecules as determined for the same ion in solution [2], but some other lose one or even two water molecules and complete their coordination by binding DNA atoms. There is preference for the three kind of ions here analyzed, to bind either the PO − 4 groups of the N7 atoms in guanine and adenine [3]. The water structure in the first hydration shell of the DNA is analyzed for specific sites (for example for the nitrogen or the nitrogen or the oxygen atoms at he bases) or for groups of sites (for example the bases of the PO − 4). For both DNA conformers, the polyelectrolytes loses water from the first hydration shell if K + is replaced with Na + or Na + with Li +. As known in B-DNA, there are two grooves, the minor and major. The water was found to be highly structured in both grooves. In the minor groove there is either long filament of water molecules running along the groove or small filaments connecting two phosphate groups belonging to the same strand. In the major groove the water molecules form filaments connecting phosphate groups belonging to two different strands [4]. The first and the last type of filaments are stable for a sufficiently long time as to be detected by X-ray diffraction, as shown recently [5]. These results have been obtained performing, for each of the six simulations, 10 6 displacements (after equilibration). As known, after each displacement the interaction energy of the system, namely, the interaction energy between water-DNA, water-ions, ion-ion, ion-DNA and water-water is computed. In the present study the interaction energies are calculated with potentials obtained fitting thousands of ab initio computations [6]; for the water-water the MCY potential [7], obtained from Configuration Interaction computations, has been used.