What you want is a type of switching power supply called a boost converter. These convert a low DC voltage at high current into a high DC voltage at low current. In theory with ideal parts, they are 100% efficient. For what you want to do, real efficiency of 80% would be relatively easy. Over 90% is doable, but requires tricks like synchronous rectification (a method of working around the fact that the diode is not ideal and has a small voltage drop across it when conducting).
Here is the basic concept of a boost converter:
When the switch is closed, current builds up linearly with time thru the inductor, storing energy in the inductor. When the switch is opened, the current must continue flowing somewhere in the short term, which is thru the diode onto the output. Now the voltage across the inductor is reversed and the current decreases linearly with time. When it reaches 0, the diode stops conducting and everything is off until the next time the switch is closed again. This is the discontinuous mode operation case. In practise, the switch is often closed and opened so fast that the current doesn't die down to 0 during the brief time it is open. That is called continuous mode operation.
In any case, a control system watches Vout and adjusts the timing of the switch being opened and closed to dump more or less current onto Vout depending on whether it is low or high.
Boost converters are quite common, and there are many chips out there that integrate a lot of the function of them. For your voltages and currents, you should be easily able to find a chip that needs little more than the inductor and the output cap externally. I would start by checking the offerings from Microchip, ST, and TI. In the unlikely event I don't find something suitable there, I'd check Linear and others that are generally more pricey.