First Evaluation Report
I have completed the implementation of ESERK4 method. I also worked towards making the ROCK and SERK solvers GPU compatible. This allows using DifferentialEquations.jl problem type mixed with neural networks. These methods are a very promising option in training data generation, for example ROCK2 taking 3 minutes against 40 minutes with BS3.[1]
I also started the implementation of Stochastic-ROCK solvers. These solvers boast similar benifits for SDEs like ROCK methods for ODEs. The three methods that I’ve implemented so far are SROCK1 both for Stratonovich and Ito type problems and SROCK2 for Ito type problems.
| Solver | Description |
|---|---|
| SROCK1{:Ito} | This is an explicit strong order 1 solver for stiff Ito type problems. It gives strong order 1 for 1-Dimensional Noise and Diagonal Noise and strong order 1/2 for General Noise case. It can be configured to get strong order 1 for Commutative Noise with some extra calculation but this is not yet implemented. Also this methods gives weak order 1 convergence for all the above Noise cases. |
| SROCK1{:Stratonovich} | This is an explicit strong order 1 solver for stiff Stratonovich type problems. It has convergence of strong order 1 for 1-Dimensional, Diagonal and Commutative Noise and strong order 1/2 for General Noise cases. Also we get weak order 1 for the above mentioned noise cases. |
| SROCK2 | This is an explicit solver with weak order 2 for Ito type problems. |
The above mentioned solvers are compatible with 1-D, Diagonal, Commutative and General Noise. Note that SROCK1{:Ito} is also compatible with Commutative Noise but with strong order 1/2.
Apart from these I’ve also added Three stage split-step Explicit Milstein Methods mentioned in StochasticDiffEq.jl Issue #65. These solvers are compatible with 1-D noise and Diagonal Noise.
In addition to the above solver I’ve added Benchmark files for ROCK2 and ROCK4 ODE solvers.
List of Contributions during Weeks 1 - 4
OrdinaryDiffEq.jl
| Description | Pull Requests | Current Status |
|---|---|---|
| Implemented ESERK4 | #737 | Merged |
| RKC_utils GPU Compatibility | #765 | Merged |
StochasticDiffEq.jl
| Description | Pull Requests | Current Status |
|---|---|---|
| Implemented Three Stage Split-Step Milstein Methods | #150 | Merged |
| Implemented SROCK Method for Stratonovich-type Problems | #153 | Merged |
| Optimised Stage-selection for SROCK1 method | #159 | Merged |
| Implemented SROCK Method for Ito-type Problems | #161 | Merged |
| Bug Fix in RKMilCommute | #164 | Merged |
| Implemented SROCK2 Method for Ito-type Problems | #166 | Merged |
DiffEqBenchmarks.jl
| Description | Pull Requests | Current Status |
|---|---|---|
| Added Benchmark files for ROCK2 and ROCK4 | #39 | Merged |
Work Precision Diagrams for Stabilized ODE-Solvers including ESERK4
-
Non-Stiff Problems
-
Stiff Problems
Future Work
Right now I’m focusing on implementing and optimising all the SROCK solvers mentioned in Issue #73 of StochasticDiffEq.jl. I also plan to add benchmarks for these solvers.
All the problems for benchmarking purposes are taken from DiffEqBenchmarks.jl