Apple had been seeking damages of $2.2bn after accusing Samsung of copying five patents including the “slide to unlock function”.
The US company also filed a permanent injunction to prevent Samsung using the patents it was found to have infringed.
Samsung filed court documents too but the contents have not been made public.
During the original trial Apple had accused Samsung of “systematically” copying features distinctive to its iOS software. Samsung denied that it had copied Apple patents and said it was Apple who was doing the copying.
The jury found that Apple had infringed some Samsung patents and awarded $158,000 in damages to the South Korean firm.
Although the jury found in Apple’s favour in respect of two of the patents, the damages awarded were much lower than the company had been seeking.
The verdict would have been a blow for Apple said the BBC’s North America technology correspondent Richard Taylor, speaking at the time.
“The figure would appear to reflect the jury’s belief that Apple’s settlement claim was unfairly inflated. Samsung argued all along that it should be far lower than the $2.2bn sought, not least because some of the patents were never even incorporated into the iPhone’s software,” he said.
This latest legal step is an attempt by Apple to have its damages award increased but it would require a new trial.
Intellectual property analyst Florian Mueller thinks any new trial is unlikely to happen before an appeal of the original ruling.
“The judge may make some minor amendments to the verdict but then she’ll let the parties appeal the unfavourable parts of the ruling to the Federal Circuit. Thereafter, there may be a retrial.” he said.
These requests by Apple mark the latest instalment in a series of legal battles over intellectual property that the world’s top two smartphone makers have been fighting for years across many countries.
Two years ago, a separate jury ordered Samsung to pay Apple $930m after finding it had used Apple technology. That verdict is still being challenged by Samsung.
But Mr Mueller thinks Apple may be suffering “litigation fatigue”, he pointed out that Apple has not filed a US case against Samsung in more than two years.
Apple recently settled a patent row with Google. The two companies had filed several lawsuits against one another but they agreed to dismiss these cases and said they would work together in “some areas of patent reform”.