Bilinear quantum Monte Carlo: Expectations and energy differences

We propose a bilinear sampling algorithm in the Green's function Monte Carlo for expectation values of operators that do not commute with the Hamiltonian and for differences between eigenvalues of different Hamiltonians. The integral representations of the Schrödinger equations are transformed into two equations whose solution has the form _a(x) t(x, y)_b(y), where _a and _b are the wavefunctions for the two related systems andt(x, y) is a kernel chosen to couplex andy. The Monte Carlo process, with random walkers on the enlarged configuration spacex y, solves these equations by generating densities whose asymptotic form is the above bilinear distribution. With such a distribution, exact Monte Carlo estimators can be obtained for the expectation values of quantum operators and for energy differences. We present results of these methods applied to several test problems, including a model integral equation, and the hydrogen atom.     

A new look at correlations in atomic and molecular systems. I. Application of fermion monte carlo variational method

We are engaged in research directed toward the development of compact and accurate correlation functions for many-electron systems. Our computational tool is the variational method in which the many-electron integrals are calculated by Monte Carlo using the fermion Metropolis sampling algorithm. That is, a many-fermion system is simulated by sampling the square of a correlated antisymmetric wave function. The principal advantage of the method is that interelectronic distance r_ij may be included directly in the wave function without adding significant computational complexity. In addition, other quantities of physical and theoretical interest such as electron correlation functions and representations of Coulomb and Fermi “holes” are very easily obtained. Preliminary results are reported for He, H₂, and Li₂.     

Optimizing head/media electrical characterization in the presence of skew at high track density

The skew angle causes a discrepancy in determining the reader-to-writer offset (RWO) when using different periodical patterns in track profile tests. It also separates the peak overwrite (OW) from the peak high frequency amplitude HFA, (1 T periodical pattern) on corresponding track profiles. Furthermore, higher track density and larger skew angle exacerbate the skew effect and induce more RWO error, thus impacting the parametric performance optimization. Simulation studies are used to interpret the skew effect on the RWO determination and OW cross-track characteristics. Based on experimental investigations and simulation analyses, using the HFA, track profile for deriving the optimal RWO is proposed for spin-stand tests. Actual parametric characterization has proven that the optimal RWO minimized the skew effect and the RWO error, thus improving the parametric performance and reducing the test variation. The method is beneficial and necessary for the high track density characterization.     

Adaptive importance sampling Monte Carlo simulation of rare transition events

We develop a general theoretical framework for the recently proposed importance sampling method for enhancing the efficiency of rare-event simulations [W. Cai, M. H. Kalos, M. de Koning, and V. V. Bulatov, Phys. Rev. E 66, 046703 (2002)], and discuss practical aspects of its application. We define the success/fail ensemble of all possible successful and failed transition paths of any duration and demonstrate that in this formulation the rare-event problem can be interpreted as a "hit-or-miss" Monte Carlo quadrature calculation of a path integral. The fact that the integrand contributes significantly only for a very tiny fraction of all possible paths then naturally leads to a "standard" importance sampling approach to Monte Carlo (MC) quadrature and the existence of an optimal importance function. In addition to showing that the approach is general and expected to be applicable beyond the realm of Markovian path simulations, for which the method was originally proposed, the formulation reveals a conceptual analogy with the variational MC (VMC) method. The search for the optimal importance function in the former is analogous to finding the ground-state wave function in the latter. In two model problems we discuss practical aspects of finding a suitable approximation for the optimal importance function. For this purpose we follow the strategy that is typically adopted in VMC calculations: the selection of a trial functional form for the optimal importance function, followed by the optimization of its adjustable parameters. The latter is accomplished by means of an adaptive optimization procedure based on a combination of steepest-descent and genetic algorithms.     

Model fermion Monte Carlo with correlated pairs. II

We study the fundamental challenge of fermion Monte Carlo for continuous systems: the fermion “sign problem.” In particular, we describe methods that depend upon the use of correlated dynamics for ensembles of correlated sets of walkers that carry opposite signs. We explain the concept of marginally correct dynamics, and show that marginally correct dynamics that produce a stable overlap with an antisymmetric trial function give the correct fermion ground state. Many-body harmonic oscillator problems are particularly tractable: their stochastic dynamics permits the use of regular geometric structures for the ensembles, structures that are stable when appropriate correlations are introduced, and avoid the decay of signal-to-noise that is a normal characteristic of the sign problem. This approach may be a guide in the search for algorithmic approaches to calculations of physical interest.     

Exact monte carlo method for continuum fermion systems

We offer a new proposal for the Monte Carlo treatment of many-fermion systems in continuous space. It is based upon diffusion Monte Carlo with significant modifications: correlated pairs of random walkers that carry opposite signs, different functions "guide" walkers of different signs, the Gaussians used for members of a pair are correlated, and walkers can cancel so as to conserve their expected future contributions. We report results for free-fermion systems and a fermion fluid with 14 3He atoms, where it proves stable and correct. Its computational complexity grows with particle number, but slowly enough to make interesting physics within the reach of contemporary computers.