The offensive follows fighting at the weekend in which thousands of civilians were displaced.
Both sides say they have inflicted heavy casualties on each other, but correspondents say it is impossible to get independent verification.
Officially neither side has formally declared an end to a 2002 ceasefire, but unofficially they are now at war.
Civilians hit
"Sri Lankan army troops undertook a limited operation in the eastern province on Monday morning to neutralise the Tamil Tigers' heavy artillery and mortar positions," a statement from the Sri Lankan Ministry of Defence said.
It said that two "enemy" gun positions had been destroyed.
The army says that 12 soldiers have been killed in the recent fighting, along with 40 rebels.
But the Tigers say that at least 30 soldiers have been killed, with their side losing only three men.
Both sides agree that thousands of civilians have been killed, injured or displaced in the latest confrontations.
The government says that at least 3,000 Sinhalese civilians were forced to seek shelter in Trincomalee district on Saturday following fierce fighting.
The rebels say that 41 civilians were killed in areas bordering Trincomalee and Batticaloa districts, and that dozens of wounded people are being treated in hospital in rebel-held areas.
Talks in Geneva in October between the two sides ended in deadlock, and correspondents say that the fighting looks set to continue.
The rebels want independence for the 2.5m-strong minority Tamil community in the north and east of the country.
At least 3,400 people have been killed in the conflict in the past year, the government says. More than 65,000 lives have been lost since 1972.
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