U.S. District Court Judge Lucy Koh lowered the damages awarded to Apple Inc. by $450.5 million, saying jurors had not properly followed her instruction in calculating some of the damages. She also concluded that mistakes had been made in determining when Apple had first notified Samsung about the alleged violations of the patents for its trend-setting iPhone and IPad.

The ruling reduced Samsung Electronics’ bill to just under $599 million.

Koh also ordered a new trial on Apple’s allegations that Samsung stole its ideas for more than a dozen different smartphones and tablet computers that include several models in Samsung’s hot-selling Galaxy line-up.

The new trial leaves open the possibility that Samsung’s damages bill could rise back up toward $1 billion, or even higher, depending on the findings of a new jury.

Although Koh denied Apple’s attempt to increase the verdict that the jury reached last August, she agreed that Samsung should be forced to pay an unspecified amount of extra damages that weren’t covered in the first trial. She said these supplemental damages will be applied on sales of certain Samsung devices after Aug. 25, 2012 — the date of the original verdict.

Apple declined to comment on the Koh’s ruling.

After a three-week trial closely followed in Silicon Valley, the jury decided that Samsung ripped off the trailblazing technology and sleek designs used by Apple to create its revolutionary iPhone and iPad and ordered Samsung to pay Apple $1.05 billion in the latest skirmish of a global legal battle between the two tech giants.

Apple filed its patent infringement lawsuit in April 2011 and engaged legions of the country’s highest-paid patent lawyers to demand $2.5 billion from its top smartphone competitor. Samsung Electronics Co. fired back with its own lawsuit seeking $399 million.

The jury found that several Samsung products illegally used such Apple creations as the “bounce-back” feature when a user scrolls to an end image, and the ability to zoom text with a tap of a finger.

The case is ultimately expected to land before the U.S. Court of Appeals for the Federal Circuit, the Washington-based court that decides patent disputes, if not the U.S. Supreme Court.

Nonetheless, Koh is expected to greatly shape the end result.

Samsung has mounted an aggressive post-trial attack on the verdict, raising a number of legal issues that allege the South Korean company was treated unfairly in a federal courtroom a dozen miles from Apple’s Cupertino headquarters. Samsung alleges that some of Apple’s patents shouldn’t have been awarded in the first place and that the jury made mistakes in calculating the damage award.

Samsung has emerged as one of Apple’s biggest rivals and has overtaken as the leading smartphone maker. Samsung’s Galaxy line of phones run on Android, a mobile operating system that Google Inc. has given out for free to Samsung and other phone makers.

Apple and Samsung have filed similar lawsuits in eight other countries, including South Korea, Germany, Japan, Italy, the Netherlands, Britain, France and Australia.