Monitor died after power outage, but then came back, but now dead again?

Please advise me as to whether or not this is hopeless.  We had a power outage while the computer was on and slept.  Afterwards, our as501 monitor still receives power (i.e., the power button is yellow, as if it's not connected to the computer but plugged in and receiving power).   Our computer seems to be having a perfectly normal startup. 

I suggested we just leave the whole rig unplugged for a while, and then plug it back in.  We were thrilled to see the computer and monitor BOTh come back on!  The display was the same photo we had placed on the desktop, and all the icons were there.  The weird thing was that it was sort of "zoomed out," so that we were pulling around on the desktop to see our icons.  We wondered, how do we get the display back to normal and not zoomed out like this?

We figured the best thing would be to just unplug all over again, reset the computer one more time.  But now, the monitor is back to receiving power but seemingly no input from the computer.

I know that ordinarily this would be chalked up to a loss, but the monitor did come back on that second try!  It was clearly communicating with the computer, although not perfectly.  Now it's incommunicative again.

Advice?