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- W4283712618 abstract "Given a stochastic differential equation (SDE) in <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:msup><mml:mrow><mml:mi mathvariant=double-struck>R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> whose solution is constrained to lie in some manifold <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>M</mml:mi><mml:mo>⊂</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant=double-struck>R</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> , we identify a class of numerical schemes for the SDE whose iterates remain close to <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>M</mml:mi></mml:math> to high order. These schemes approximate a geometrically invariant scheme, which gives perfect solutions for any SDE that is diffeomorphic to <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>n</mml:mi></mml:math> -dimensional Brownian motion. Unlike projection-based methods, they may be implemented without explicit knowledge of <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>M</mml:mi></mml:math> . They can even be implemented if the solution merely remains close to <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML><mml:mi>M</mml:mi></mml:math> , without being exactly confined to it. Our approach does not require simulating any iterated Itô integrals beyond those needed to implement the Euler–Maryuama scheme. We prove that the schemes converge under a standard set of assumptions, and illustrate their geometric advantages in a variety of numerical contexts, including Monte Carlo simulation of the Riemannian Langevin equation." @default.
- W4283712618 created "2022-06-30" @default.
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- W4283712618 date "2022-06-01" @default.
- W4283712618 modified "2023-09-24" @default.
- W4283712618 title "Curved schemes for stochastic differential equations on, or near, manifolds" @default.
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- W4283712618 doi "https://doi.org/10.1098/rspa.2021.0785" @default.
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