  We address the experimentally relevant problem of robust mitigation of
dephasing noise acting on a qubit. We first present an extension of a method
for representing $1/\omega^{\alpha}$ noise developed by Kuopanportti et al. to
the efficient representation of arbitrary Markovian noise. We then add qubit
control pulses to enable the design of numerically optimized, two-dimensional
control functions with bounded amplitude, that are capable of decoupling the
qubit from the dephasing effects of a broad variety of Markovian noise spectral
densities during arbitrary one qubit quantum operations. We illustrate the
method with development of numerically optimized control pulse sequences that
minimize decoherence due to a combination of $1/\omega$ and constant offset
noise sources. Comparison with the performance of standard dynamical decoupling
protocols shows that the numerically optimized pulse sequences are considerably
more robust with respect to the noise offset, rendering them attractive for
application to situations where homogeneous dephasing noise sources are
accompanied by some extent of heterogeneous dephasing. Application to the
mitigation of dephasing noise on spin qubits in silicon indicates that high
fidelity single qubit gates are possible with current pulse generation
technology.
