  In this paper, we propose an infinite-dimensional version of the Stein
variational gradient descent (iSVGD) method for solving Bayesian inverse
problems. The method can generate approximate samples from posteriors
efficiently. Based on the concepts of operator-valued kernels and vector-valued
reproducing kernel Hilbert spaces, a rigorous definition is given for the
infinite-dimensional objects, e.g., the Stein operator, which are proved to be
the limit of finite-dimensional ones. Moreover, a more efficient iSVGD with
preconditioning operators is constructed by generalizing the change of
variables formula and introducing a regularity parameter. The proposed
algorithms are applied to an inverse problem of the steady state Darcy flow
equation. Numerical results confirm our theoretical findings and demonstrate
the potential applications of the proposed approach in the posterior sampling
of large-scale nonlinear statistical inverse problems.
