After upgrading from 15.10 to 16.04 my computer fans have started making much more noise.
I think this could be related to the graphics card, which is an ATI Radeon:
07:00.0 VGA compatible controller: Advanced Micro Devices, Inc. [AMD/ATI] Redwood XT [Radeon HD 5670/5690/5730]
07:00.1 Audio device: Advanced Micro Devices, Inc. [AMD/ATI] Redwood HDMI Audio [Radeon HD 5000 Series]
08:00.0 VGA compatible controller: Advanced Micro Devices, Inc. [AMD/ATI] Redwood XT [Radeon HD 5670/5690/5730]
08:00.1 Audio device: Advanced Micro Devices, Inc. [AMD/ATI] Redwood HDMI Audio [Radeon HD 5000 Series]
How can I reduce the fan speed to reduce the noise?
You need to establish which fans are making the noise by running the PC with the sides off and listening to each fan. Is it the CPU fan, the GPU fans (I see you have two graphics cards), or the case fan(s)? You can test if a fan is the one making the noise by gently touching its centre with a finger to stop it.
You also need to establish whether the increase in noise is accompanied by a rise in temperature and where.
To do this install lm-sensors and configure it. Then install psensors and study the fan speeds and temperatures. Instructions are here lm-sensor and fanspeed instructions
Once you have established which fans are running faster and making noise you need to determine whether they are running faster because the relevant system component is heavily loaded. Then the solution will be to reduce the load on the component by adjusting the power management settings or finding a bug.
If the fans are running faster than you like and the relevant system component is not getting too hot you can slow the fans. There are two routes to doing this. If the relevant fan can be controlled by your BIOS I recommend doing this. If not then install and configure fanspeed.
This is a problem related to drivers. I can tell you because I have an ATI HD5750 and I always had issues regarding fan noise using Open Source AMD Drivers. Easiest solution was just to use the ATI Catalyst drivers but that is not an option now.
Since Ubuntu 16.04 AMD is dropping support of proprietary drivers for future releases. Good news is that they are focusing on Open Source drivers, so for now this is a transitioning period.
Solution for now is just stick to Ubuntu 15.10 or find a way to control power profile for Open Source Drivers but I honestly never found an easy solution for this.