  For solving Bayesian inverse problems governed by large-scale forward
problems, we present an infinite-dimensional version of the Stein variational
gradient descent (iSVGD) method, which has the ability to generate approximate
samples from the posteriors efficiently. Specifically, we introduce the concept
of the operator-valued kernel and the corresponding function-valued reproducing
kernel Hilbert space (RKHS). Through the properties of RKHS, we give an
explicit meaning of the infinite-dimensional objects (e.g., the Stein operator)
and prove that the infinite-dimensional objects are indeed the limit of
finite-dimensional items. Furthermore, by generalizing the change of variables
formula, we construct iSVGD with preconditioning operators, yielding more
efficient iSVGD. During these generalizations, we introduce a regularity
parameter $s\in[0,1]$. Our analysis shows that the intuitive trivial version
(i.e., by directly taking finite-dimensional objects as infinite-dimensional
items) of iSVGD with preconditioning operators ($s=0$) will yield inaccurate
estimates, and the parameter $s$ should be chosen larger than $0$ and smaller
than $0.5$. Finally, the proposed algorithms are applied to an inverse problem
governed by the Helmholtz equation. Numerical results confirm the correctness
of our theoretical findings and demonstrate the potential usefulness of the
proposed approach in the posterior sampling of large-scale nonlinear
statistical inverse problems.
