91 research outputs found
Projected particle methods for solving McKean-Vlasov stochastic differential equations
We propose a novel projection-based particle method for solving the
McKean-Vlasov stochastic differential equations. Our approach is based on a
projection-type estimation of the marginal density of the solution in each time
step. The projection-based particle method leads in many situation to a
significant reduction of numerical complexity compared to the widely used
kernel density estimation algorithms. We derive strong convergence rates and
rates of density estimation. The convergence analysis in the case of linearly
growing coefficients turns out to be rather challenging and requires some new
type of averaging technique. This case is exemplified by explicit solutions to
a class of McKean-Vlasov equations with affine drift. The performance of the
proposed algorithm is illustrated by several numerical examples
Convergence analysis of an explicit method and its random batch approximation for the McKean-Vlasov equations with non-globally Lipschitz conditions
In this paper, we present a numerical approach to solve the McKean-Vlasov
equations, which are distribution-dependent stochastic differential equations,
under some non-globally Lipschitz conditions for both the drift and diffusion
coefficients. We establish a propagation of chaos result, based on which the
McKean-Vlasov equation is approximated by an interacting particle system. A
truncated Euler scheme is then proposed for the interacting particle system
allowing for a Khasminskii-type condition on the coefficients. To reduce the
computational cost, the random batch approximation proposed in [Jin et al., J.
Comput. Phys., 400(1), 2020] is extended to the interacting particle system
where the interaction could take place in the diffusion term. An almost half
order of convergence is proved in sense. Numerical tests are performed to
verify the theoretical results
Projected particle methods for solving McKean--Vlaslov equations
We study a novel projection-based particle method to the solution of the corresponding McKean-Vlasov equation. Our approach is based on the projection-type estimation of the marginal density of the solution in each time step. The projection-based particle method can profit from additional smoothness of the underlying density and leads in many situation to a significant reduction of numerical complexity compared to kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The case of linearly growing coefficients of the McKean-Vlasov equation turns out to be rather challenging and requires some new type of averaging technique. This case is exemplified by explicit solutions to a class of McKean-Vlasov equations with affine drift
Euler simulation of interacting particle systems and McKean-Vlasov SDEs with fully superlinear growth drifts in space and interaction
We consider in this work the convergence of a split-step Euler type scheme
(SSM) for the numerical simulation of interacting particle Stochastic
Differential Equation (SDE) systems and McKean-Vlasov Stochastic Differential
Equations (MV-SDEs) with full super-linear growth in the spatial and the
interaction component in the drift, and non-constant Lipschitz diffusion
coefficient.
The super-linear growth in the interaction (or measure) component stems from
convolution operations with super-linear growth functions allowing in
particular application to the granular media equation with multi-well confining
potentials. From a methodological point of view, we avoid altogether functional
inequality arguments (as we allow for non-constant non-bounded diffusion maps).
The scheme attains, in stepsize, a near-optimal classical (path-space) root
mean-square error rate of for and an optimal
rate in the non-path-space mean-square error metric. Numerical examples
illustrate all findings. In particular, the testing raises doubts if taming is
a suitable methodology for this type of problem (with convolution terms and
non-constant diffusion coefficients).Comment: 40 pages, 3 figures; Final author accepted version (to appear in IMA
J. of Num. Analysis
Monte-Carlo based numerical methods for a class of non-local deterministic PDEs and several random PDE systems
In this thesis, we will investigate McKean-Vlasov SDEs (McKV-SDEs) on Rd,:
; where coefficient functions b and σ satisfy sufficient regularity conditions and
fWtgt2[0;T ] is a Wiener process. These SDEs correspond to a class of deterministic
non-local PDEs.
The principal aim of the first part is to present Multilevel Monte Carlo (MLMC)
schemes for the McKV-SDEs. To overcome challenges due to the dependence
of coefficients on the measure, we work with Picard iteration. There are two
different ways to proceed. The first way is to address the McKV-SDEs with
interacting kernels directly by combining MLMC and Picard. The MLMC is used
to represent the empirical densities of the mean-fields at each Picard step. This
iterative MLMC approach reduces the computational complexity of calculating
expectations by an order of magnitude.However, we can also link the McKV-SDEs with interacting kernels to that with
non-interacting kernels by projection and then iteratively solve the simpler by
MLMC method. In each Picard iteration, the MLMC estimator can approximate
a few mean-fields directly. This iterative MLMC approach via projection reduces
the computational complexity of calculating expectations hugely by three orders
of magnitude.
In the second part, the main purpose is to demonstrate the plausibility of applying
deep learning technique to several types of random linear PDE systems. We
design learning algorithms by using probabilistic representation and Feynman-
Kac formula is crucial for deriving the recursive relationships on constructing the
loss function in each training session
Hybrid PDE solver for data-driven problems and modern branching
The numerical solution of large-scale PDEs, such as those occurring in
data-driven applications, unavoidably require powerful parallel computers and
tailored parallel algorithms to make the best possible use of them. In fact,
considerations about the parallelization and scalability of realistic problems
are often critical enough to warrant acknowledgement in the modelling phase.
The purpose of this paper is to spread awareness of the Probabilistic Domain
Decomposition (PDD) method, a fresh approach to the parallelization of PDEs
with excellent scalability properties. The idea exploits the stochastic
representation of the PDE and its approximation via Monte Carlo in combination
with deterministic high-performance PDE solvers. We describe the ingredients of
PDD and its applicability in the scope of data science. In particular, we
highlight recent advances in stochastic representations for nonlinear PDEs
using branching diffusions, which have significantly broadened the scope of
PDD.
We envision this work as a dictionary giving large-scale PDE practitioners
references on the very latest algorithms and techniques of a non-standard, yet
highly parallelizable, methodology at the interface of deterministic and
probabilistic numerical methods. We close this work with an invitation to the
fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European
Journal of Applied Mathematics (EJAM
Projected particle methods for solving McKean-Vlasov equations
We study a novel projection-based particle method to the solution of
the corresponding McKean-Vlasov equation. Our approach is based on the
projection-type estimation of the marginal density of the solution in each
time step. The projection-based particle method can profit from additional
smoothness of the underlying density and leads in many situation to a
signficant reduction of numerical complexity compared to kernel density
estimation algorithms. We derive strong convergence rates and rates of
density estimation. The case of linearly growing coefficients of the
McKean-Vlasov equation turns out to be rather challenging and requires some
new type of averaging technique. This case is exemplified by explicit
solutions to a class of McKean-Vlasov equations with affine drift
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