Berkeley has at least one cluster, probably more.
https://docs-research-it.berkeley.edu/services/high-performa...
University HPC clusters are typically managed and owned departmentally. My former employer had two clusters in two different departments. I worked directly with a few other universities who also had departmental HPC clusters, and I’ve read a boatload of HPC docs from different unis and labs. Sharing clusters for many types of workloads seems to happen more at the regional level (like Archer cluster in Edinburgh, or PNNL in WA).
Ours is a fairly small cluster of maybe 60 or 80 job nodes. All compute networking travels over Infiniband—-I think we bought HDR for the new cluster, but price may held us to EDR—-which is probably why the cluster was so expensive to build (around $2M). Our storage cluster was another $1-$2M project.
All that is to preface this: cloud usage doesn’t mean they’re only using cloud resources. Our goal, for instance, was to build a hybrid cluster (and their continued goal, as far as I know). The first step was to offload low priority work with low resource requirements and wall times, and comparatively long timelines to the cheapest possible, compliant cloud provider (some of our researchers have specific data sharing and privacy requirements).
Let’s say a job is submitted on Monday morning at 9 am. It needs 2 CPUs and 4 GB of RAM, and a wall time of five minutes. The researcher can wait until next Monday at 10 am for the results. There’s at least a chance that running this job in the on-prem cluster is less cost efficient than offloading the job to another resource, whether the cost is direct (i.e. 5 minutes on-prem costs $0.10, and the cloud costs $0.05 or whatever real values would look like).
Ideally, we would have some method by which you could compare the cost to run a job in the cluster vs another resource, and it would automatically offload jobs of up to $X to the cloud, whether as a very low priority queue, or as overflow in times of full utilization. There are several other conditions that would need to be met, as well, but you get the idea (just for example, one consideration is if the work be interrupted. If the job has to run start to finish, the cloud is likely not suitable for that work).