SQL Server LocalDB Instance; error restoring backup (MasterDBPath not available)

I had the same problem, and solved it.

You need to add 2 strings in the Windows Registry via regedit.exe at HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Microsoft SQL Server\MSSQL11E.LOCALDB\MSSQLServer:

DefaultData
DefaultLog                     

For me, their values are both C:\Users\neil\AppData\Local\Microsoft\Microsoft SQL Server Local DB\Instances\v11.0.