Thinking about the basics.
Resolve the velocity of the rain into a vertical component, a horizontal component parallel to your path and a horizontal component orthogonal to your path.
Vertical component:
The amount landing on the top of your head will be proportional to time. Running reduces that.
The amount of rain you walk into will be proportional to the distance travelled, your velocity is irrelevant to that.
Parallel Horizontal component:
The rate at which rain hits you from this factor will be proportional to the magnitude of your horizontal velocity relative to the rain.
If the rain direction is with you, then matching its speed is best (reducing this factor to zero). If it is against you, you are better going faster (doubling your speed will halve the time, but it will not double the rate at which the rain hits).
Orthogonal Horizontal Component
This factor only affects the rain that hits you from the side.The rate will be constant. Running is better from this point of view.
From this reasoning it looks as if running is usually better, in terms of the sheer amount of water that will hit you in most cases. It is always better for the amount of rain hitting you from above or the side. If the wind is against you it also reduces the amount that will hit your front.
Of course there are plenty of other factors (does the velocity with which the rain hits have an effect ?)
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