Early Friday morning, her boyfriend reported her missing, and a search-and-rescue operation quickly found the vessel in a bay south of Copenhagen. The 26-foot submarine sank just as Mr. Madsen jumped into the water and swam toward a rescue boat, a rescuer said.
As he was brought to shore, Mr. Madsen told the local television station TV2 that he had been on a test drive when he ran into problems with a valve on a ballast tank on the vessel.
“I was toying with various things on the submarine and then an error occurred,” he said, adding that he had only 30 seconds to leave the vessel before it sank.
According to a police statement, Mr. Madsen said he dropped off Ms. Wall in a remote part of the port of Copenhagen around 10:30 p.m. and continued alone.
A family statement sent by Tom Wall, Ms. Wall’s brother, said she had been with Mr. Madsen for an interview when she vanished. “We sincerely hope that she will be found and that she is well,” the statement said.
Ms. Wall is a graduate of Columbia University, and her work has appeared in a wide range of international publications including The New York Times.
Christopher Harres, a Columbia graduate and an investigative journalist in Mobile, Ala., said of Ms. Wall: “She was literally loved everywhere she went. Her smile and charm could get her into events and past police cordons. Her smile filled your heart, her humor and hilarious observations could put you on the floor.”
He added: “Kim embodied everything that was good in journalism and storytelling. She reported from North Korea, China and all over Europe. She had wanderlust in droves and it hung off her every word.”
On Saturday, the authorities raised the vessel from a depth of 22 feet and started taking it to shore. Divers were unable to enter the submarine.
For years Mr. Madsen has led a community of amateurs in Copenhagen working to launch the world’s first manned amateur space rocket. Other members of the community successfully launched an unmanned rocket 1.7 miles into the air in 2011.
Mr. Madsen had planned three launches this summer in preparation for a possible space mission in 2019 with only himself onboard.
He has been described by local news outlets as a maverick and a “Gyro Gearloose” with high ambitions and low social skills.
“He argues with every Tom, Dick and Harry. I’ve argued with him as well. But that’s what it’s like with people driven by deep passion,” Thomas Djursing, an editor who wrote a book about Mr. Madsen, told the daily newspaper BT.
In 2015, the inventor had a severe argument over ownership of the UC3 Nautilus with an association formed to support his plans to build the submarine. In the end, his supporters pulled out.
In a message published on the association’s website two years ago, his associates quoted a text message from Mr. Madsen that rings ominously today: “A curse lies over Nautilus. That curse is me. There will not be calm around Nautilus as long as I exist.”
An earlier version of this article misstated the depth of the water in which the submarine was found. It was 22 feet, not seven feet.