Optimization and regularization of the inverse problem for stochastic differential equation using graphics accelerators

The problem of drift and volatility parameters identification in stochastic differential equations (SDEs) using additional measurement of single trajectory of stochastic process is investigated. The classical way for solving such a problem is to reduce it to a Fokker-Planck equation and minimize a suitable data fidelity functional, what is carried out sequentially. In addition to that, such inverse problems are ill-posed,i.e. their solutions are unstable. For higher-dimensional systems of SDEs, the numerical solution of the Fokker-Planck equations becomes infeasible since the solution depends on n + 1 variables for drift coefficient and (n + 1)2 for covariance matrix, and computational complexity is O(n3). We propose regularized Landweber iteration algorithm for easier paralleling of problem. The key idea is introduction of solution-dependant parameters, what allows us to introduce implicit time dependency. The benefit of such approach consists in that the adjoint problem is deterministic, gradient of fidelity functional has the integral form and consists of mathematical expectations, that allow us to effectively parallelize algorithm with Monte-Carlo approach. Moreover, dependency is implemented with Fourier series that helps to reduce number of variables and compute gradient independently for each basis function. The algorithm determines the stable space of parameters. We conduct this process on synthetic data for validation of an algorithm and regularization for variety of input data.