Computer experiment

A computer experiment or simulation experiment is an experiment used to study a computer simulation, also referred to as an in silico system. This area includes computational physics, computational chemistry, computational biology and other similar disciplines.

Background

Objectives

Computer experiments have been employed with many purposes in mind. Some of those include:

• Uncertainty quantification: Characterize the uncertainty present in a computer simulation arising from unknowns during the computer simulation's construction.
• Inverse problems: Discover the underlying properties of the system from the physical data.
• Bias correction: Use physical data to correct for bias in the simulation.
• Data assimilation: Combine multiple simulations and physical data sources into a complete predictive model.
• Systems design: Find inputs that result in optimal system performance measures.

Computer simulation modeling

Modeling of computer experiments typically uses a Bayesian framework.

. It is natural to see the simulation as a deterministic function that maps these inputs into a collection of outputs. On the basis of seeing our simulator this way, it is common to refer to the collection of inputs as ${\displaystyle x}$, the computer simulation itself as ${\displaystyle f}$, and the resulting output as ${\displaystyle f(x)}$. Both ${\displaystyle x}$ and ${\displaystyle f(x)}$ are vector quantities, and they can be very large collections of values, often indexed by space, or by time, or by both space and time.

Although ${\displaystyle f(\cdot )}$ is known in principle, in practice this is not the case. Many simulators comprise tens of thousands of lines of high-level computer code, which is not accessible to intuition. For some simulations, such as climate models, evaluation of the output for a single set of inputs can require millions of computer hours [3].

Gaussian process prior

The typical model for a computer code output is a Gaussian process. For notational simplicity, assume ${\displaystyle f(x)}$ is a scalar. Owing to the Bayesian framework, we fix our belief that the function ${\displaystyle f}$ follows a Gaussian process, ${\displaystyle f\sim \operatorname {GP} (m(\cdot ),C(\cdot ,\cdot )),}$ where ${\displaystyle m}$ is the mean function and ${\displaystyle C}$ is the covariance function. Popular mean functions are low order polynomials and a popular

${\displaystyle \nu =1/2}$

${\displaystyle \nu \rightarrow \infty }$

Design of computer experiments

The design of computer experiments has considerable differences from

Unlike physical experiments, it is common for computer experiments to have thousands of different input combinations. Because the standard inference requires matrix inversion of a square matrix of the size of the number of samples (${\displaystyle n}$), the cost grows on the ${\displaystyle {\mathcal {O}}(n^{3})}$. Matrix inversion of large, dense matrices can also cause numerical inaccuracies. Currently, this problem is solved by greedy decision tree techniques, allowing effective computations for unlimited dimensionality and sample size patent WO2013055257A1, or avoided by using approximation methods, e.g. [6].

See also

• Simulation
• Uncertainty quantification
• Bayesian statistics
• Gaussian process emulator
• Design of experiments
• Molecular dynamics
• Monte Carlo method
• Surrogate model
• Grey box completion and validation
• Artificial financial market

Further reading

• Santner, Thomas (2003). The Design and Analysis of Computer Experiments. Berlin: Springer.
  ISBN 0-387-95420-1
  .

• Fehr, Jörg; Heiland, Jan; Himpe, Christian; Saak, Jens (2016). "Best practices for replicability, reproducibility and reusability of computer-based experiments exemplified by model reduction software". AIMS Mathematics. 1 (3): 261–281.
  S2CID 14715031
  .