  The scalar auxiliary variable (SAV)-type methods are very popular techniques
for solving various nonlinear dissipative systems. Compared to the
semi-implicit method, the baseline SAV method can keep a modified energy
dissipation law but doubles the computational cost. The general SAV approach
does not add additional computation but needs to solve a semi-implicit solution
in advance, which may potentially compromise the accuracy and stability. In
this paper, we construct a novel first- and second-order unconditional energy
stable and positivity-preserving stabilized SAV (PS-SAV) schemes for $L^2$ and
$H^{-1}$ gradient flows. The constructed schemes can reduce nearly half
computational cost of the baseline SAV method and preserve its accuracy and
stability simultaneously. Meanwhile, the introduced auxiliary variable is
always positive while the baseline SAV cannot guarantee this
positivity-preserving property. Unconditionally energy dissipation laws are
derived for the proposed numerical schemes. We also establish a rigorous error
analysis of the first-order scheme for the Allen-Cahn type equation in
$l^{\infty}(0,T; H^1(\Omega) ) $ norm. In addition we propose an energy
optimization technique to optimize the modified energy close to the original
energy. Several interesting numerical examples are presented to demonstrate the
accuracy and effectiveness of the proposed methods.
