Does a triple divorce issued in a single session by the husband to his wife who left the marital home take effect, knowing that the husband regrets it and wishes to reconcile with his wife?

You did not mention the reason for the dispute. If the wife desires a dwelling independent of your family, it is her right, and you are mistaken if you do not provide it. However, if she has no valid excuse for leaving and refuses to return, she is disobedient (nashiz) and has no right to maintenance (nafaqa). The wife's family erred by refusing reconciliation and assisting her in her disobedience. The Imam erred by his misconduct and bigotry. Regarding the divorce, if you intended confirmation with only the first phrase, then one divorce takes effect. Otherwise, three divorces take effect. Sheikh Ibn Uthaymin stated in Al-Sharh Al-Mumti: "If he repeats the entire phrase (You are divorced, you are divorced), the number takes effect. If he repeats only the predicate (You are divorced, divorced, divorced), then it is one unless he intended more." If one divorce has occurred, you may reconcile with her before the end of her waiting period (idda). If three divorces have taken effect, then she is unlawful for you until she marries another husband. If your anger was severe at the time of uttering the divorce, to the extent that you were unaware of what you were saying, then nothing is incumbent upon you.