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Register a new userMonte-Carlo Algorithms for Forward Feynman-Kac type representation for semilinear nonconservative Partial Differential Equations
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The paper is devoted to the construction of a probabilistic particle algorithm. This is related to nonlin-ear forward Feynman-Kac type equation, which represents the solution of a nonconservative semilinear parabolic Partial Differential Equations (PDE). Illustrations of the efficiency of the algorithm are provided by numerical experiments.
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This paper presents a partial state of the art about the topic of representation of generalized Fokker-Planck Partial Differential Equations (PDEs) by solutions of McKean Feynman-Kac Equations (MFKEs) that generalize the notion of McKean Stochastic Differential Equations (MSDEs). While MSDEs can be related to non-linear Fokker-Planck PDEs, MFKEs can be related to non-conservative non-linear PDEs. Motivations come from modeling issues but also from numerical approximation issues in computing the solution of a PDE, arising for instance in the context of stochastic control. MFKEs also appear naturally in representing final value problems related to backward Fokker-Planck equations.
We provide a representation result of parabolic semi-linear PD-Es, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod (1964), Watanabe (1965) and McKean (1975), by allowing for polynomial nonlinearity in the pair $(u, Du)$, where $u$ is the solution of the PDE with space gradient $Du$. Similar to the previous literature, our result requires a non-explosion condition which restrict to small maturity or small nonlinearity of the PDE. Our main ingredient is the automatic differentiation technique as in Henry Labordere, Tan and Touzi (2015), based on the Malliavin integration by parts, which allows to account for the nonlinearities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation. Indeed, this provides the first numerical method for high dimensional nonlinear PDEs with error estimate induced by the dimension-free Central limit theorem. The complexity is also easily seen to be of the order of the squared dimension. The final section of this paper illustrates the efficiency of the algorithm by some high dimensional numerical experiments.
Let $mathcal{X}$ be a real separable Hilbert space. Let $Q$ be a linear, self-adjoint, positive, trace class operator on $mathcal{X}$, let $F:mathcal{X}rightarrowmathcal{X}$ be a (smooth enough) function and let ${W(t)}_{tgeq 0}$ be a $mathcal{X}$-valued cylindrical Wiener process. For $alphain [0,1/2]$ we consider the operator $A:=-(1/2)Q^{2alpha-1}:Q^{1-2alpha}(mathcal{X})subseteqmathcal{X}rightarrowmathcal{X}$. We are interested in the mild solution $X(t,x)$ of the semilinear stochastic partial differential equation begin{gather} left{begin{array}{ll} dX(t,x)=big(AX(t,x)+F(X(t,x))big)dt+ Q^{alpha}dW(t), & t>0; X(0,x)=xin mathcal{X}, end{array} right. end{gather} and in its associated transition semigroup begin{align} P(t)varphi(x):=E[varphi(X(t,x))], qquad varphiin B_b(mathcal{X}), tgeq 0, xin mathcal{X}; end{align} where $B_b(mathcal{X})$ is the space of the real-valued, bounded and Borel measurable functions on $mathcal{X}$. In this paper we study the behavior of the semigroup $P(t)$ in the space $L^2(mathcal{X}, u)$, where $ u$ is the unique invariant probability measure of eqref{Tropical}, when $F$ is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincare inequalities and we study the maximal Sobolev regularity for the stationary equation [lambda u-N_2 u=f,qquad lambda>0, fin L^2(mathcal{X}, u);] where $N_2$ is the infinitesimal generator of $P(t)$ in $L^2(mathcal{X}, u)$.
In this paper we consider sequential joint state and static parameter estimation given discrete time observations associated to a partially observed stochastic partial differential equation (SPDE). It is assumed that one can only estimate the hidden state using a discretization of the model. In this context, it is known that the multi-index Monte Carlo (MIMC) method of [11] can be used to improve over direct Monte Carlo from the most precise discretizaton. However, in the context of interest, it cannot be directly applied, but rather must be used within another advanced method such as sequential Monte Carlo (SMC). We show how one can use the MIMC method by renormalizing the MI identity and approximating the resulting identity using the SMC$^2$ method of [5]. We prove that our approach can reduce the cost to obtain a given mean square error (MSE), relative to just using SMC$^2$ on the most precise discretization. We demonstrate this with some numerical examples.
Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity $mathrm{poly}(1/epsilon)$, where $epsilon$ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be $mathrm{poly}(d, log(1/epsilon))$, where $d$ is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.
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