My pet gentler method is to get a lump of cotton wool, dampen it, and add some toothpaste. Then put it in the vase and swirl the vase so the cotton wool slides - and you achieve a gentle grinding action with the toothpaste. I would soak the vase with washing soda or bicarb soda for 24 hours before using the cotton wool.
To get the wool out - if it proves dificult simply add lots of water, swish the woll abit so it becomes dispeersed and then tip upside down rqapidly into the sink. The cotton wool usually comes out as a messy blob.
Ross