Verification method of Monte Carlo codes for transport processes with arbitrary accuracy

In this work, we present a robust and powerful method for the verification, with arbitrary accuracy, of Monte Carlo codes for simulating random walks in complex media. Such random walks are typical of photon propagation in turbid media, scattering of particles, i.e., neutrons in a nuclear reactor or animal/humans’ migration. Among the numerous applications, Monte Carlo method is also considered a gold standard for numerically “solving” the scalar radiative transport equation even in complex geometries and distributions of the optical properties. In this work, we apply the verification method to a Monte Carlo code which is a forward problem solver extensively used for typical applications in the field of tissue optics. The method is based on the well-known law of average path length invariance when the entrance of the entities/particles in a medium obeys to a simple cosine law, i.e., Lambertian entrance, and annihilation of particles inside the medium is absent. By using this law we achieve two important points: (1) the invariance of the average path length guarantees that the expected value is known regardless of the complexity of the medium; (2) the accuracy of a Monte Carlo code can be assessed by simple statistical tests. We will show that we can reach an arbitrary accuracy of the estimated average pathlength as the number of simulated trajectories increases. The method can be applied in complete generality versus the scattering and geometrical properties of the medium, as well as in presence of refractive index mismatches in the optical case. In particular, this verification method is reliable to detect inaccuracies in the treatment of boundaries of finite media. The results presented in this paper, obtained by a standard computer machine, show a verification of our Monte Carlo code up to the sixth decimal digit. We discuss how this method can provide a fundamental tool for the verification of Monte Carlo codes in the geometry of interest, without resorting to simpler geometries and uniform distribution of the scattering properties.

Simultaneous bicolor interrogation in thulium optical clock providing very low systematic frequency shifts

Optical atomic clocks have already overcome the eighteenth decimal digit of instability and uncertainty, demonstrating incredible control over external perturbations of the clock transition frequency. At the same time, there is an increasing demand for atomic (ionic) transitions and new interrogation and readout protocols providing minimal sensitivity to external fields and possessing practical operational wavelengths. One of the goals is to simplify the clock operation while maintaining the relative uncertainty at a low 10−18 level achieved at the shortest averaging time. This is especially important for transportable and envisioned space-based optical clocks. Here, we demonstrate implementation of a synthetic frequency approach for a thulium optical clock with simultaneous optical interrogation of two clock transitions. Our experiment shows suppression of the quadratic Zeeman shift by at least three orders of magnitude. The effect of the tensor lattice Stark shift in thulium can also be reduced to below 10−18 in fractional frequency units. This makes the thulium optical clock almost free from hard-to-control systematic shifts. The “simultaneous” protocol demonstrates very low sensitivity to the cross-talks between individual clock transitions during interrogation and readout.

On The Implementation of Densely Packed Decimal Number System Based Adder: Prospects and Challenges

Decimal digit number computation, through bit compression methodology, offers space and time saving, which can be incurred by the Chen-Ho and Densely Packed Decimal (DPD) coding techniques. Such coding techniques have a property of bit compression, like, three decimal digits can be represented by 10 bits instead of 12 bits in binary coded decimal (BCD) format. The compression has been obtained through the elimination of the redundant 0’s from BCD representation. This manuscript reports the pros and cons of the techniques mentioned above. The logic level functionalities have been examined through MATLAB, whereas circuit simulation has been erified through Cadence Spectre. Performance parameters (such as delay, power consumption) have been evaluated through CMOS gpdk45 nm technology. Furthermore, the best design has been chosen from them, and the decimal adder design technique has been incorporated in this paper.

Bit Grooming: Statistically accurate precision-preserving quantization with compression, evaluated in the netCDF Operators (NCO, v4.4.8+)

Lossy compression schemes can help reduce the space required to store the false precision (i.e, scientifically meaningless data bits) that geoscientific models and measurements generate. We introduce, implement, and characterize a new lossy compression scheme suitable for IEEE floating-point data. Our new Bit Grooming algorithm alternately shaves (to zero) and sets (to one) the least significant bits of consecutive values to preserve a desired precision. This is a symmetric, two-sided variant of an algorithm sometimes called Bit Shaving which quantizes values solely by zeroing bits. Our variation eliminates the artificial low-bias produced by always zeroing bits, and makes Bit Grooming more suitable for arrays and multi-dimensional fields whose mean statistics are important. Bit Grooming relies on standard lossless compression schemes to achieve the actual reduction in storage space, so we tested Bit Grooming by applying the DEFLATE compression algorithm to bit-groomed and full-precision climate data stored in netCDF3, netCDF4, HDF4, and HDF5 formats. Bit Grooming reduces the storage space required by uncompressed and compressed climate data by up to 50 % and 20 %, respectively, for single-precision data (the most common case for climate data). When used aggressively (i.e., preserving only 1–3 decimal digits of precision), Bit Grooming produces storage reductions comparable to other quantization techniques such as linear packing. Unlike linear packing, Bit Grooming works on the full representable range of floating-point data. Bit Grooming reduces the volume of single-precision compressed data by roughly 10 % per decimal digit quantized (or "groomed") after the third such digit, up to a maximum reduction of about 50 %. The potential reduction is greater for double-precision datasets. Data quantization by Bit Grooming is irreversible (i.e., lossy) yet transparent, meaning that no extra processing is required by data users/readers. Hence Bit Grooming can easily reduce data storage volume without sacrificing scientific precision or imposing extra burdens on users.

Introduction

This introductory chapter provides an overview of Benford' law. Benford's law, also known as the First-digit or Significant-digit law, is the empirical gem of statistical folklore that in many naturally occurring tables of numerical data, the significant digits are not uniformly distributed as might be expected, but instead follow a particular logarithmic distribution. In its most common formulation, the special case of the first significant (i.e., first non-zero) decimal digit, Benford's law asserts that the leading digit is not equally likely to be any one of the nine possible digits 1, 2, … , 9, but is 1 more than 30 percent of the time, and is 9 less than 5 percent of the time, with the probabilities decreasing monotonically in between. The remainder of the chapter covers the history of Benford' law, empirical evidence, early explanations and mathematical framework of Benford' law.