MINNEAPOLIS — Jerome Pate, a homeless alcoholic, went to the emergency room when he was cold. He went when he needed a safe place to sleep. He went when he was hungry, or drunk, or suicidal.

“I’d go sometimes just to have a place to be,” he said.

He made 17 emergency room visits in just four months last year, a costly spree that landed him in the middle of an experiment to reinvent health care for the hardest-to-help patients here in Hennepin County.

More than 11 million Americans have joined the Medicaid rolls since the major provisions of the Affordable Care Act went into effect, and health officials are searching for ways to contain the costs of caring for them. Some of the most expensive patients have medical conditions that are costly no matter what. But a significant share of them — so-called super utilizers like Mr. Pate — rack up costs for avoidable reasons. Many are afflicted with some combination of poverty, homelessness, mental illness, addiction and past trauma.

A patchwork of experiments across the country are trying to better manage these cases. The Center for Health Care Strategies, a policy center in New Jersey, has documented such efforts in 26 states. Some are run by private insurers and health care providers, while others are part of broader state overhaul efforts. The federal government is supporting some, too, through its $10 billion Innovation Center, set up under the Affordable Care Act.