LED's use current, not voltage.
LED's are not linear, or rather the emitted light from 1ma is not proportional to the emitted light at 10mA.
What you want may not be what you are asking for. we call this an XY problem.
What you are asking for is a variable voltage, but that will not work.
what you think you are searching for is an adjustable current source. to vary the current, you can vary the output of an LED.
to vary the voltage, you are attempting to alter one side of OHM's law, and relying on the selection of the resistor to deliver the changing current needed to change power in the LED to effect a change in output.
since this is not the correct approach, you are having a very difficult time finding it. kinda like looking for a 5 wheel car to have a softer ride when you should be looking at shock absorbers.
there are lots of LED driver chips from all the name brand suppliers, TI, fairchild, OM, you name it.
that said, most LED data sheets offer that you can pulse and LED about 2.5 times more power than you can deliver continuous.
as you mentioned, POV or persistence of vision, means your perception of the brightness of the light is based on it's highest value. so pulsing a light with much higher power, but for a brief time, will deliver a much higher perceived light.
go into a dark room with a cigarette lighter and flick the flint. you can see what is in the room from the very bright light.
any of the chips that can do lots and lots of LED's with independent brightness control, rely on the POV effect and by delivering a higher power pulse, they conserve time so they can do more LED's.
but this also means that a constant power LED will have to be powered to the specs on the data sheet or it will overheat.
Since you appear to have an XY problem, would you offer why you do not want to use the industry standard and universally accepted method of lighting LED's ?
TV refresh times are 60 hz movie frame rates are about 72hz. new HD TV's claim 120hz and even 240hz.
in comparison, the PWM output of a 16mhz Atmel 328 chip as used in the Arduino's is about 976hz .
the pulse is not perfect, but at those speeds, few things have the ability to notice.
you can wave correct the pulse stream to get a more uniform pulse and that consumes clock cycles, you might drop to 694hz or 347hz..... still 10 times faster than a TV set.
don't confuse the plasma TV pulse of 600hz to charge the screen with the 60hz refresh rate for the picture. it uses the POV to charge the individual pixel at a pulse of 600hz, but only changes the value at 60hz.
Please enlighten us why you want to drive an LED at a constant power.