The　relationship　between　material　properties　and　independent　variables　such　as　temperature,　external　field,　or　time　is　usually　represented　by　a　curve　or　surface　in　a　multidimensional　space.　Determining　such　a　curve　or　surface　requires　a　series　of　experiments　or　calculations　which　are　often　time　and　cost　consuming.　A　general　strategy　uses　an　appropriate　utility　function　to　sample　the　space　to　recommend　the　next　optimal　experiment　or　calculation　within　an　active　learning　loop.　However,　knowing　what　optimal　sampling　strategy　to　use　to　minimize　the　number　of　experiments　is　an　outstanding　problem.　We　compare　a　number　of　strategies　based　on　directed　exploration　on　several　materials　problems　of　varying　complexity　using　a　Kriging-based　model.　These　include　one-dimensional　curves　such　as　the　fatigue　life　curve　for　304L　stainless　steel　and　the　Liquidus　line　of　the　Fe-C　phase　diagram,　surfaces　such　as　the　Hartmann　3　function　in　three-dimensional　space　and　the　fitted　intermolecular　potential　for　Ar-SH,　and　a　four-dimensional　data　set　of　experimental　measurements　for　BaTiO3-based　ceramics.　We　also　consider　the　effects　of　experimental　noise　on　the　Hartmann　3　function.　We　find　that　directed　exploration　guided　by　maximum　variance　provides　better　performance　overall,　converging　faster　across　several　data　sets.　However,　for　certain　problems,　the　tradeoff　methods　incorporating　exploitation　can　perform　at　least　as　well,　if　not　better　than　maximum　variance.　Thus,　we　discuss　how　the　choice　of　the　utility　function　depends　on　the　distribution　of　the　data,　the　model　performance　and　uncertainties,　additive　noise,　as　well　as　the　budget.　©　2021　American　Physical　Society.