Once you become a U.S. resident under the IRS rules, you become a dual-status alien unless you qualify as a resident of a treaty country within the meaning of an income tax treaty between the United States and that country. If applicable to you, you can claim a treaty benefit (as a nonresident of the United States) so as to reduce your U.S. income tax liability. Otherwise, absent such a tax treaty, you cannot choose to file as a nonresident for that portion of the year you either have a green card or have met the substantial presence test.
You may, however, have the First-Year Choice option explained in IRS Publication 519. If you are eligible for this choice, then you do have the option to choose to be a nonresident for tax purposes in that tax year, when you could otherwise elect to be treated as a resident for a portion of that year (the year prior to the time you obtain a green card or meet the substantial presence test).
In the two situations where you can choose (tax treaty or first-year choice), the lowest tax depends on your particular circumstances. This IRS web page has a good overview of the taxation of dual-status aliens.
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