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added Ralston and EulerHeun
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‎docs/devdocs/sde_example.ipynb‎

Lines changed: 38 additions & 36 deletions
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"metadata": {},
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"source": [
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"```\n",
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"+----------------+-------+------------+----------------------------------+-------------------+----------------+------------------------------------------+\n",
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"| | SDE | Lévy | Strong/weak order per noise type | VF evaluations | Embedded error | Recommended for |\n",
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"| | type | area +---------+-------------+----------+-------+-----------+ estimation | (and other notes) |\n",
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"| | | | General | Commutative | Additive | Drift | Diffusion | | |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| Euler | Itô | BM only | 0.5/1.0 | 0.5/1.0 | 1.0/1.0 | 1 | 1 | No | Itô SDEs when a cheap solver is needed |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| Heun | Strat | BM only | 0.5/1.0 | 1.0/1.0 | 1.0/1.0 | 2 | 2 | Yes | Stratonovich SDEs without space-time LA. |\n",
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"| | | | | | | | | | Has weak order 2 for constant noise. |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| ItoMilstein | Itô | BM only | 0.5/1.0 | 1.0/1.0 | 1.0/1.0 | 1 | 1 + deriv | No | Better than Euler for Itô SDEs, but |\n",
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"| | | | | | | | ative | | comuptes the derivative of diffusion VF |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| Stratonovich | Strat | BM only | 0.5/1.0 | 1.0/1.0 | 1.0/1.0 | 1 | 1 + deriv | No | For commutative Stratonovich SDEs when |\n",
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"| Milstein | | | | | | | ative | | space-time Lévy area is not available. |\n",
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"| | | | | | | | | | Computes derivative of diffusion VF. |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| ReversibleHeun | Strat | BM only | 0.5/1.0 | 1.0/1.0 | 1.0/1.0 | 1 | 1 | Yes | When a reversible solver is needed, e.g. |\n",
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"| | | | | | | | | | for Neural SDEs. This method is FSAL. |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| Midpoint | Strat | BM only | 0.5/1.0 | 1.0/1.0 | 1.0/1.0 | 2 | 2 | Yes | Usually Heun should be preferred. |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| ShARK | Addit | space-time | / | / | 1.5/2.0 | 2 | 2 | Yes | Additive noise SDEs |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| SRA1 | Addit | space-time | / | / | 1.5/2.0 | 2 | 2 | Yes | Only slightly worse than ShARK |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| SEA | Addit | space-time | / | / | 1.0/1.0 | 1 | 1 | No | Cheap solver for additive noise SDEs |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| SPaRK | Strat | space-time | 0.5/1.0 | 1.0/1.0 | 1.5/2.0 | 3 | 3 | Yes | General SDEs when embedded error |\n",
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"| | | | | | | | | | estimation is needed |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| GeneralShARK | Strat | space-time | 0.5/1.0 | 1.0/1.0 | 1.5/2.0 | 2 | 3 | No | General SDEs when embedded error |\n",
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"| | | | | | | | | | estimaiton is not needed |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| SlowRK | Strat | space-time | 0.5/1.0 | 1.5/2.0 | 1.5/2.0 | 2 | 5 | No | Commutative noise SDEs. |\n",
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"+----------------+-------+------------+---------+-------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"+----------------+-------+------------+------------------------------------+-------------------+----------------+------------------------------------------+\n",
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"| | SDE | Lévy | Strong/weak order (per noise type) | VF evaluations | Embedded error | Recommended for |\n",
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"| | type | area +----------+--------------+----------+-------+-----------+ estimation | (and other notes) |\n",
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"| | | | General | Commutative | Additive | Drift | Diffusion | | |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| Euler | Itô | BM only | 0.5/1.0 | 0.5/1.0 | 1.0/1.0 | 1 | 1 | No | Itô SDEs when a cheap solver is needed. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| Heun | Strat | BM only | 0.5/1.0 | 1.0/1.0 | 1.0/1.0 | 2 | 2 | Yes | Standard solver for Stratonovich SDEs. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| EulerHeun | Strat | BM only | 0.5/1.0 | 0.5/1.0 | 1.0/1.0 | 1 | 2 | No | Stratonovich SDEs with expensive drift. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| ItoMilstein | Itô | BM only | 0.5/1.0 | 1.0/1.0 | 1.0/1.0 | 1 | 1 | No | Better than Euler for Itô SDEs, but |\n",
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"| | | | | | | | | | comuptes the derivative of diffusion VF. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| Stratonovich | Strat | BM only | 0.5/1.0 | 1.0/1.0 | 1.0/1.0 | 1 | 1 | No | For commutative Stratonovich SDEs when |\n",
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"| Milstein | | | | | | | | | space-time Lévy area is not available. |\n",
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"| | | | | | | | | | Computes derivative of diffusion VF. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| ReversibleHeun | Strat | BM only | 0.5/1.0 | 1.0/1.0 | 1.0/1.0 | 2 | 2 | Yes | When a reversible solver is needed. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| Midpoint | Strat | BM only | 0.5/1.0 | 1.0/1.0 | 1.0/1.0 | 2 | 2 | Yes | Usually Heun should be preferred. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| Ralston | Strat | BM only | 0.5/1.0 | 1.0/1.0 | 1.0/1.0 | 2 | 2 | Yes | Usually Heun should be preferred. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| ShARK | Strat | space-time | / | / | 1.5/2.0 | 2 | 2 | Yes | Additive noise SDEs. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| SRA1 | Strat | space-time | / | / | 1.5/2.0 | 2 | 2 | Yes | Only slightly worse than ShARK. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| SEA | Strat | space-time | / | / | 1.0/1.0 | 1 | 1 | No | Cheap solver for additive noise SDEs. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| SPaRK | Strat | space-time | 0.5/1.0 | 1.0/1.0 | 1.5/2.0 | 3 | 3 | Yes | General SDEs when embedded error |\n",
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"| | | | | | | | | | estimation is needed. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| GeneralShARK | Strat | space-time | 0.5/1.0 | 1.0/1.0 | 1.5/2.0 | 2 | 3 | No | General SDEs when embedded error |\n",
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"| | | | | | | | | | estimaiton is not needed. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"| SlowRK | Strat | space-time | 0.5/1.0 | 1.5/2.0 | 1.5/2.0 | 2 | 5 | No | Commutative noise SDEs. |\n",
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"+----------------+-------+------------+----------+--------------+----------+-------+-----------+----------------+------------------------------------------+\n",
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"```"
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]
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}

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