Probabilistic Solver Research Articles

Probabilistic solvers enable a straight-forward exploration of numerical uncertainty in neuroscience models

Understanding neural computation on the mechanistic level requires models of neurons and neuronal networks. To analyze such models one typically has to solve coupled ordinary differential equations (ODEs), which describe the dynamics of the underlying neural system. These ODEs are solved numerically with deterministic ODE solvers that yield single solutions with either no, or only a global scalar error indicator on precision. It can therefore be challenging to estimate the effect of numerical uncertainty on quantities of interest, such as spike-times and the number of spikes. To overcome this problem, we propose to use recently developed sampling-based probabilistic solvers, which are able to quantify such numerical uncertainties. They neither require detailed insights into the kinetics of the models, nor are they difficult to implement. We show that numerical uncertainty can affect the outcome of typical neuroscience simulations, e.g. jittering spikes by milliseconds or even adding or removing individual spikes from simulations altogether, and demonstrate that probabilistic solvers reveal these numerical uncertainties with only moderate computational overhead.

Neutron Response Modeling to Track Lean Gas Plume in Recycled Gas Cap Reservoir in Concurrent Gas Cap-Oil Rim Development: A Step Forward

Summary Gas cap pressure maintenance while developing the associated oil rim is a critical aspect for optimum recovery. Preventing gas cap pressure dropping below dewpoint by injecting lean gas is essential for concurrent gas cap-oil rim development. Reservoir heterogeneity aggravates lean gas override causing preferential movement of lean gas plume. Thus, it is important to track lean gas plume while recycling and understanding the breakthrough potential of lean gas. This paper demonstrates a new workflow to track lean gas plume by estimating phase saturations with a case study from one of the giant oil and gas fields, Onshore, Abu Dhabi. Pulsed neutron capture (PNC) tools are used for reservoir monitoring and surveillance. However, sigma log evaluation is insufficient to derive individual hydrocarbon phase saturations to monitor lean gas plume. Neutron response modeling (NRM) is devised to differentiate between lean and rich gas. NRM is a probabilistic solver with input of mineral and fluid phase parameters into tool response functions in petrophysical evaluation. To distinguish with discrete neutron fluid response between lean and rich gas, pressure/volume/temperature (PVT) data are utilized to derive hydrogen index, capture cross section, thermal decay length, and neutron macro parameters, such as neutron slowing down length and migration length. Neutron response is investigated for lean and rich gas with sensitivity of invasion effects on neutron log by calibrating to core porosity. The response for each phase under thermal neutron and capture modes with corresponding raw neutron log statistics is reviewed in both openhole and casedhole environments in known lean/rich gas intervals. Thirty-five wells spread across gas cap and oil leg with quality neutron log data are modeled and individual phase saturations are estimated. The target reservoir is under development with over three decades of lean gas injection to support oil production. NRM results and phase saturations are validated with recent formation sampling, which enhanced the confidence in the overall workflow. Later, the results are verified to be in excellent agreement with lean gas injection and production history of the target reservoir. The identified movement of lean gas highlights nonuniform geology and gravity segregation of injected lean gas into upper members of the target reservoir. The results also emphasized the need for better injection support to lower members of the target reservoir where gas cap development is ongoing. The solution presented is unique, particularly for lean gas injection projects by utilizing PVT for NRM based on neutron transport mechanism in pore fluids. Existing workflows require a special nuclear modeling platform with computationally expensive processing on data sets acquired using advanced logging technology. In spite of these prerequisites, existing workflows are not able to distinguish lean gas over rich gas. This paper effectively demonstrates NRM workflow distinguishing lean gas plume from rich gas using neutron logs and reveals compelling reservoir management insights. The sensitivity studies and practicality of this workflow highlight the fundamental importance of neutron logs in mature fields.

Integrated photonic traveling salesman problem probabilistic solver with polynomial calculation time complexity

The traveling salesman problem (TSP) is an NP hard problem, where the solution time becomes unrealistically large as the number of points increases. We propose and simulate a photonic integrated circuit-based TSP solver which uses the delay time of light to express the delay caused by traveling between the points. The solver is comprised of semiconductor optical amplifiers and tunable delay lines, therefore it is reconfigurable. A probabilistic algorithm is used, which enables polynomial calculation and footprint complexity with an error rate of less than 1e-12. The calculation time is 388 ms and footprint is within 10 cm squared for a TSP of 100 points.

Applications Based on a Novel Sudoku Solver Algorithm and Grid Based Models

Numerous algorithms for solving sudoku puzzles have been explored, most of which use a backtracking approach. Thus computational efficiency of such algorithms can sometimes yield poor results. We propose a probabilistic solver algorithm which, iteratively fills the sudoku grid and solves the same. In this approach we make use of a dynamic random number set, we identify unassigned sudoku grids for a given puzzle where only one possible value can be filled in and iteratively identify and assign cells with least number of possible values. We not only elaborate on our solver algorithm logic, but also explore application areas based on algorithm devised, after reviewing relevant similar approaches illustrated in the referenced articles. We believe by extension of this algorithm, many combinatorial problems in the field of material characterization, cryptography, cybersecurity can be solved and advanced. We also envision that with application of neural networks, Machine Learning techniques the algorithm will take a very adaptive and robust form, useful for solving complex problems in accurate estimation of missing data, discrete event analysis and prediction. Uniqueness is the ability to use high probability for faster computation and low execution time. With cyberattacks of varied vectors and types, its important to devise a mechanism to create a deliberate mismatch every time a possible attack is detected.

Efficient high-dimensional material reliability analysis with explicit voxel-level stochastic microstructure representation

A novel efficient methodology for probabilistic material reliability analysis considering fine-scale microstructure stochasticity is proposed in this paper. Integrated computational material engineering requires efficient multiscale computational capabilities to enable computational design and validation. Two critical challenges are identified: handling uncertainties from microstructures and material properties; and handling the “curse of dimensionality” for probabilistic solvers. The proposed study addresses these two critical challenges. First, an analytical and hierarchical uncertainty quantification method is proposed for the explicit stochastic microstructure representation at the voxel-level. The hierarchy of uncertainties from both phase maps and uncertainties within each phase is modeled using an explicit Gaussian mixture random field. Analytical approximation for the arbitrary non-Gaussian random field is derived, which can facilitate the computation of gradient information in optimization. Following this, an efficient probabilistic solver using adjoint first-order reliability method combining the importance sampling is derived by formulating the material reliability analysis as a constrained optimization problem. The adjoint method is used to efficiently evaluate the responses and exact gradients with the help of the analytical Gaussian mixture random field. Several numerical examples for material reliability calculation with high-dimensional (voxel-level) random fields are subsequently employed to demonstrate and validate the proposed methodology. The results of the proposed method are quantitatively compared to those obtained via the classical first-order reliability method, direct Monte Carlo simulation, subset simulation, and the sequential importance sampling method. The comparisons indicate that the proposed method possesses high efficiency for high-dimensional material reliability problems.

Empirical Random Kick Model and Casing Reliability

Summary The development of exploratory wells is based on the interpretation of seismic readings by geotechnical field experts. Correlation wells, when they exist, are located very far, leading to large uncertainties and to different interpretations by different experts. Exploratory wells are expensive and risky endeavors, in particular deep wells with high-temperature and high-pressure conditions. Reaching reservoirs at record depths often requires challenging usual safety margins, which should be based on reliability analyses. In this paper, we present an empirical random kick model to combine the opinions of different specialists and the reliability analysis of casing subject to kick loads. The empirical kick model has a random kick volume and intensity, gas gravity, and influx depth. Three tubular strength models are considered: Barlow formula, von Mises stress, and Klever-Stewart model (Klever and Stewart 1998; Stewart and Klever 1998), with random yield stress and tube thickness. A typical 19,000 ft well is considered as a case study example, and the First-Order Reliability Method (FORM) is used as a probabilistic solver. Results show the differences between safety factors and reliability indices obtained for the three strength models. Sensitivity factors show that yield stress and model error are the most relevant random variable at shallow depths. Kick volume and influx depth are found to be the most relevant random variables of the kick model, especially at greater depths. The proposed empirical kick model is shown to have a significant impact on casing reliability, in comparison with a kick in which uncertain kick variables are assumed with uniform distribution.

Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective

We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with nonlinear measurement functions. This is achieved by defining the measurement sequence to consist of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP—which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduced to a nonlinear Bayesian filtering problem and all widely used approximations to the Bayesian filtering and smoothing problems become applicable. Furthermore, all previous GP-based ODE solvers that are formulated in terms of generating synthetic measurements of the gradient field come out as specific approximations. Based on the nonlinear Bayesian filtering problem posed in this paper, we develop novel Gaussian solvers for which we establish favourable stability properties. Additionally, non-Gaussian approximations to the filtering problem are derived by the particle filter approach. The resulting solvers are compared with other probabilistic solvers in illustrative experiments.

Optimally Stopped Optimization

We combine the fields of heuristic optimization and optimal stopping. We propose a strategy for benchmarking randomized optimization algorithms that minimizes the expected total cost for obtaining a good solution with an optimal number of calls to the solver. To do so, rather than letting the objective function alone define a cost to be minimized, we introduce a further cost-per-call of the algorithm. We show that this problem can be formulated using optimal stopping theory. The expected cost is a flexible figure of merit for benchmarking probabilistic solvers that can be computed when the optimal solution is not known, and that avoids the biases and arbitrariness that affect other measures. The optimal stopping formulation of benchmarking directly leads to a real-time, optimal-utilization strategy for probabilistic optimizers with practical impact. We apply our formulation to benchmark simulated annealing on a class of MAX2SAT problems. We also compare the performance of a D-Wave 2X quantum annealer to the HFS solver, a specialized classical heuristic algorithm designed for low tree-width graphs. On a set of frustrated-loop instances with planted solutions defined on up to N = 1098 variables, the D-Wave device is two orders of magnitude faster than the HFS solver, and, modulo known caveats related to suboptimal annealing times, exhibits identical scaling with problem size.

Advanced Workflow Package for Shale Assets Developed

Technology Update The growth in unconventional resource plays in the past several years has produced a burgeoning need for new software tools for organic shales. Geoscientists need tools to help them understand complex hydrocarbon generation, storage capacity, and migration paths in source rock reservoirs, enabling them to flag and map optimized pay. Engineers need tools to help them define optimum techniques to deliver the most shale gas and oil to the market and enable them to build the best reservoir models to exploit these resources. And for unconventional resource development to proceed as it should, these tools must work together in a common framework. An advanced integrated petrophysical evaluation software package, based on a calibrated workflow, was recently developed by Halliburton for organic shales. The concept behind it was to bring all the requisite pieces of an exploration shale play analysis into a single vantage point for an asset team. This is critical when very few vertical exploration wells are used to define the economics of these resource plays before full-scale horizontal development begins. The software’s workflow modules encompass the following capabilities: total organic carbon (TOC) and organic maturity estimation; fluid and minerals evaluation; advanced saturation modeling; mechanical properties and brittleness; 3D stress and stress orientation; permeability; and pay analysis. TOC Estimation and Organic Maturity To define the resource volume, one needs to determine an accurate volume of organic kerogen present in the rock. To determine potential hydrocarbon type, the level of thermal maturity must be established. To solve for kerogen, the TOC measured by core pyrolysis can be calibrated to logs, using eight industry accepted correlations. Organic maturity, VRo, is measured by actual vitrinite reflectance or calculated from pyrolysis-derived Tmax (temperature between 300°C and 600°C that generates peak hydrocarbons from existing kerogen). This maturity value is used to make the final TOC calibration and predict hydrocarbon type. Fluid and Minerals Evaluation The heart of the volumetric analysis is its probabilistic solver. Total porosity in organic shales can only be resolved by logs when relative amounts of geochemically derived minerals are measured and combined with the TOC calculation. Minimum requirements for this type of analysis include a triple combo log, neutron capture spectroscopy, and natural gamma spectroscopy. The software uses a probabilistic error minimization methodology to determine formation fluid and mineral volumes. The idea is to construct theoretical logs that closely replicate actual logs. Tool response equations are expressed in terms of fluid and mineral volumes and their corresponding tool response parameters. Most response equations are linear. Some, such as neutron, conductivity, and certain acoustic equations, are nonlinear. The inclusion of additional evaluation tools, such as the dipole sonic travel time curves DTC (compressional velocity) and DTS (shear velocity), helps add coherence to the analysis, as long as the correct acoustic equations are used for harder rock-clay shales.

Reliability approach for fibre-reinforced composites design

This study aims at investigating the mechanical reliability of unidirectional fibre-reinforced composites in view of their design optimization. Owing to the manufacturing process, many uncertainties affect inherently the properties of these materials. Besides, the brittle character of the fibre failure leads to important safety factors that limit their development for engineering applications. The objective is to get a realistic evaluation of the mechanical reliability of such structures which accounts for the various uncertainties involved. The effects of random design parameters on the composite failure are then investigated for different kinds of reinforcement (nature of fibre, mono-material or hybrid structures) with finite-element probabilistic solver PERMAS.