  Bayesian inverse problems highly rely on efficient and effective inference
methods for uncertainty quantification (UQ). Infinite-dimensional MCMC
algorithms, directly defined on function spaces, are robust under refinement of
physical models. Recent development of this class of algorithms has started to
incorporate the geometry of the posterior informed by data so that they are
capable of exploring complex probability structures. However, the required
geometric quantities are usually expensive to obtain in high dimensions. On the
other hand, most geometric information of the unknown parameter space in this
setting is concentrated in an \emph{intrinsic} finite-dimensional subspace. To
mitigate the computational intensity and scale up the applications of
infinite-dimensional geometric MCMC ($\infty$-GMC), we apply geometry-informed
algorithms to the intrinsic subspace to probe its complex structure, and
simpler methods like preconditioned Crank-Nicolson (pCN) to its geometry-flat
complementary subspace. In this work, we take advantage of dimension reduction
techniques to accelerate the original $\infty$-GMC algorithms. More
specifically, partial spectral decomposition of the (prior or posterior)
covariance operator is used to identify certain number of principal
eigen-directions as a basis for the intrinsic subspace. The combination of
dimension-independent algorithms, geometric information, and dimension
reduction yields more efficient implementation, \emph{(adaptive)
dimension-reduced infinite-dimensional geometric MCMC}. With a small amount of
computational overhead, we can achieve over 70 times speed-up compared to pCN
using a simulated elliptic inverse problem and an inverse problem involving
turbulent combustion. A number of error bounds comparing various MCMC proposals
are presented to predict the asymptotic behavior of the proposed
dimension-reduced algorithms.
