Million-Iteration Monte Carlo Equivalent on a Fluid Flow Model in Under 600 Microseconds
Monte Carlo simulation can be used to estimate the uncertainty of the output of a computational kernel as a function of the uncertainty of the kernel's inputs, by transforming a large number of samples from an input probability distribution. The accuracy of the resulting probability distribution depends on the number of samples used; improving the accuracy of the result by a factor k requires a number of samples (or Monte Carlo iterations) that grows quadratically with k. This can make Monte Carlo simulation prohibitively expensive. The Signaloid's UxHw technology can be used to estimate the uncertainty of the output of a computational kernel as a function of the uncertainty of the kernel's inputs, but without resorting to samples needed in Monte Carlo simulation. This allows platforms such as the cloud-based Signaloid Cloud Compute Engine and hardware module implementations of Signaloid's UxHw technology to estimate the uncertainty of the output of a computational kernel as a function of the uncertainty of the kernel's inputs, with fidelity that is equivalent to a million-iteration Monte Carlo simulation, but doing so in significantly less time than is required for a Monte Carlo running on a high-end processor configured with the same amount of hardware parallelism (and using the same amount of energy) as the Signaloid UxHw implementation.
Why It Matters
Monte Carlo simulations are a foundation for many business-critical tasks. Signaloid's UxHw technology provides an alternative to solving the same problems today addressed using Monte Carlo simulation, while providing significantly easier engineering implementation and much higher execution speeds. These benefits make it possible, for the first time, to achieve the same kinds of results as Monte Carlo methods, but using Signaloid's UxHw to achieve execution times that make real-time deployment possible.
The Technical Details
Poiseuille's Law is a mathematical model relating the laminar flow rate of a viscous fluid through a pipe, to the geometry of the pipe, the viscosity of the fluid, and the pressure difference between the ends of the pipe. Each of the model's parameters will in practice have uncertainties as they are physical parameters (e.g., diameter of the pipe) that must be measured for a given system, and the resulting distribution of the flow rate therefore also has uncertainties. The conventional method for reliably estimating the uncertainty in such a model is to use Monte Carlo simulation. When running such a model on a computing platform implementing Signaloid's UxHw technology, the sample-based iterative Monte Carlo simulation can be replaced with a single-shot execution, where the probability distributions representing the uncertainties of the model parameters can be specified using the UxHw programming interface and where the uncertainty of the model's output is determined automatically by the UxHw-enabled code execution technology.
The Wasserstein distance is a way of measuring the difference between two probability distributions and can be used measure the equivalency of the outputs of different methods of carrying of uncertainty quantification. As a ground truth reference to quantify the fidelity two competing methods, one natural choice is to use a very large Monte Carlo simulation (e.g., a 100 million iterations). The outcomes of two competing methods, such as a 1 million iteration Monte Carlo simulation and the Signaloid Cloud Compute Engine, can then be compared against this reference.
Compared to a 100 million iteration Monte Carlo simulation, the 1 million iteration Monte Carlo simulation obtained an average Wasserstein distance of 0.00012 with a standard deviation of 0.00004, across 100 repetitions of the 1 million iteration runs. Each of these 100 repetitions took on average 41.65 milliseconds. When running on the Signaloid Cloud Compute Engine, Signaloid's cloud-based computing platform which implements our UxHw technology, and core with a representation size of 64, the resulting distribution has a Wasserstein distance of 0.00016 compared to the 100 million iteration Monte Carlo simulation ground truth and completes in 547 microseconds. The plots below show the distribution outputs of the uncertainty quantification of the output of the Poiseuille's law model for the execution on Signaloid's platform, execution with a 1 million iteration Monte Carlo on a traditional (Intel-based) computing platform, as well as the ground truth distribution obtained from a 100 million iteration Monte Carlo.
Relevant Code Example
Figure 1: The output distribution of Poiseuille's Law using the UxHw API. It took 520 microseconds to compute this distribution (Signaloid Cloud Compute Engine with core type Athens Pro with a representation size of 64).
Figure 2: The output distribution of Poiseuille's Law using a Monte Carlo simulation with 1 million iterations. It took 46.78 milliseconds to compute this distribution (Macbook Pro with M3 Pro).
Figure 3: The ground truth output distribution of Poiseuille's Law using a Monte Carlo simulation with 100 million iterations. It took 2.28 seconds to compute this distribution (Macbook Pro with M3 Pro).
The Takeaway
Monte Carlo evaluations are a foundation for many business-critical tasks. Signaloid's UxHw technology provides an alternative to solving the same problems today addressed using Monte Carlo evaluation, while providing significantly easier engineering implementation and much higher execution speeds. For an example computational task of characterizing the uncertainty in the output of a model for laminar flow rate of a viscous fluid through a pipe, execution on the Signaloid Cloud Compute Engine takes 547 microseconds while a Monte Carlo evaluation of equal fidelity is a factor of 76 slower (takes 41.65 milliseconds).