How to build a high-frequency (>20kHz) driver for LEDs?
Just change the timer prescaler to a lower value. 

E.g. timer2 (PWM pins 3 and 11) has the prescaler set to clock/64 and runs at around 500Hz. So changing it to clock/8 will result in 4kHz. (Sorry, no clock/16 available).

TCCR2 &amp;= ~_BV(CS22);//clear CS22 bit
TCCR2 |= _BV(CS21);//set CS21 bit