Just to follow up on this. After I last posted I continued to have the problem intermittently although it was better.
So the other day I was using the computer, left the room, and a little while later the UPS (UPS is less than 2 months old) started screeching. Came in and found the computer totally dead. I was at home so know that the power to the house didn't go out. This was something internal to the computer.
Based upon the fact the computer was dead, decided to replace the power supply. Replaced the power supply and computer immediately powered on but wouldn't boot. Having exhausted my capabilities, I took it in to the shop (we've moved so not the shop that built it). They verified that the power supply was connected properly and had the same problem and set it aside to work on it the next morning.
The next morning they called and told us (1) the new power supply was dead and (2) a connector to the hard drive bays was broken. That connector was a proprietary Thermaltake connector so Thermaltake was contacted to send a new one.
Days pass and connector was received. Now get a call that the good news is that with the connector put it and a new power supply they think they get my computer to work again. However, all SATA cables need to be replaced and both of my optical drives and both my hard drives are dead. I had my OS and some programs on an SSD drive (a new larger one that I just bought in June) and the other programs and data on a 1 TB HDD. The comment was that the HDD would not spin up.
Supposedly the motherboard is fine.
A few years ago I had a desktop computer go dead after the power went out in the house (this was pre having a UPS) and over a period of 18 months I had to replace virtually many parts in the computer as each slowly failed. I said I would never do that again.
Obviously by the time I replace the power supply, the cables, both optical drives, the SSD drive and the HDD I am looking at serious money. So, I will probably just replace the computer.