  In this work, we investigate a class of elliptic inverse problems and aim to
simultaneously recover multiple inhomogeneous inclusions arising from two
different physical parameters, using very limited boundary Cauchy data
collected only at one or two measurement events. We propose a new fast, stable
and highly parallelable direct sampling method (DSM) for the simultaneous
reconstruction process. Two groups of probing and index functions are
constructed, and their desired properties are analyzed. In order to identify
and decouple the multiple inhomogeneous inclusions of different physical
nature, we introduce a new concept of mutually almost orthogonality property
that generalizes the important concept of almost orthogonality property in
classical DSMs for inhomogeneous inclusions of same physical nature. With the
help of this new concept, we develop a reliable strategy to distinguish two
different types of inhomogeneous inclusions with noisy data collected at one or
two measurement events. We further improve the decoupling effect by choosing an
appropriate boundary influx. Numerical experiments are presented to illustrate
the robustness and efficiency of the proposed method.
