If someone swears by Allah (wa-Allahi) something is true knowing that it is a lie, what can that person do to repent for what he has done?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Whoever takes a false oath commits a grave sin. To make a false oath means to swear to a lie in order to prove or deny something. Abdullah Bin 'Amr Ibn Al-'Aas related that the Prophet said: "These are major sins: 1) Associating someone with Allah, 2) Disobeying one's parents, 3) Killing someone, 4) and tendering a false oath". [Al-Bukhari]

Ibn Mas’ood narrated that the Prophet said: “A person who swears falsely in order to take possession of a Muslim's property unjustly shall face the wrath of Allah on the Day of Judgment. Another narration reads: “He will be deprived of Paradise and will enter Hell.” Therefore, whoever swears falsely should seek the forgiveness of Allah and he should sincerely repent. If he causes harm or injustice to someone by a false oath, he has to give back the right of those people and remove the injustice from them.

As for the expiation of a false oath, the Muslim Scholars have two different opinions:

1. The majority of the Muslim Scholars say that there is no expiation for such an oath since it is greater than any expiation. It is narrated from Ibn Mas’ood that he said: “We used to consider the oath which can not be expiated to be the false oath”. [Al-Bayhaqi] Sayeed Ibn Al-Musayyib said: "It (the false oath) is among the grave sins. It is greater to be expiated".
2. One should pay the expiation stated in the Saying of Allah in the Quran {Allah will not impose blame upon you for what is meaningless in your oaths, but He will impose blame upon you for [breaking] what you intended of oaths. So its expiation is the feeding of ten needy people from the average of that which you feed your [own] families or clothing them or the freeing of a slave. But whoever cannot find [or afford it] — then a fast of three days [is required]. That is the expiation for oaths when you have sworn. But guard your oaths. Thus does Allah make clear to you His verses [i.e. revealed law] that you may be grateful.} [Quran 5:89]. These scholars believe that the person who swears deliberately while he is lying has to expiate for his oath, so the one who does a false oath has also to expiate for it in the same way.

However, the Second opinion is a preponderant one since it is supported by more evidence as below:

1. The verse mentioned above establishes a general rule that every oath needs expiation except the unintentional oaths.
2. Imposing expiation is better lest people could neglect the    seriousness of taking oaths.
3. The Scholars differed about the soundness of the Hadeeth (mentioned in the first paragraph) which is the basis for those who do not impose expiation on false oaths. Since this Hadeeth is not sound, the general rule concerning oaths applies.

Allah knows best.