Algorithms & Methods

Mathematics in and of Planet Earth



Optimization



Parameterization

Where a process of Nature is not completely understood or can only be partially measured with today's technology, then one approach within a larger mathematical model is to substitute a few variables in lieu of a true representation of the 'missing' process, and then to 'tune' these variables as best possible, based on the 'skill' of the larger model to achieve a measurable macro-level performance. This method of approximation within a model is commonly known as 'parameterization' - and is subject to continuous improvement as more knowledge of the 'missing processes' is gained. Read more...


Assimilation



Adaptive Mesh Refinement (AMR)

Performance benchmarking and profiling Read more...


Grids



Finite-Volume Methods



Spectral Nudging



Spectral Transforms



Solvers



Computational Fluid Dynamics (non-turbulent)

Both compressible and incompressible fluids Read more...


Convection



Vorticity, Helicity & Turbulence



Non-fluid flows



Stochastic Physics

Wherever we have errors in a model (initialization errors, sub-grid parameterization errors, or model structural errors), it may be possible to inject an element of noise and randomness, a 'stochastic' approach, and obtain additional outcomes and computational accuracy. For example, adding stochastic perturbations to a model may ensure it does not remain in one particular regime. Read more...


Uncertainty and Error Management

Errors can occur at all levels in the modelling and simulation paradigm. Models themselves can have structural errors, parameters within the model can have errors, and data used to initialiize or to update the simulation computations can have errors. In every case, the main issue is to estimate the magnitude of the error and to stop the error from growing with each iterative time-step of the computational process. Read more...


Variability



Linear equations



Randomness



Statistics



Ensemble Averaging



Slow Feature Analysis



Fracture Modelling



Data Mining



Predictive Analytics



Graph Analytics



Stochastic Gradient Descent



Bayesian Methods



Parallelization



Downscaling



Complex Network Analysis



Stretching Time



Sparse Matrices & Tensors



Sparse Fourier Transform



Quantum systems modelling



Machine Learning



Biological Methods