-Venus advances in straight sets

-Peng upsets Radwanska, Halep falls to Mladenovic

-Keys set up match against Wozniacki

Second seed Angelique Kerber put on a determined display to rally past France's Pauline Parmentier 7-5 3-6 7-5 in the scorching desert heat to reach the fourth round of the BNP Paribas Open on Monday.

Kerber, who will reclaim the world number one ranking next week from the absent Serena Williams, reeled off the last three games to take the opener after facing a set point, and was a point from trailing 5-2 in the third before surging to victory.

"It was a tough match and a high-level match from both of us," the German said on court after her two-hour, 33-minute victory. "Pauline played unbelievably from the first ball, and we both battled until the end."

Other seeds advancing included Venus Williams, Madison Keys and Caroline Wozniacki, while fifth seed Simona Halep and sixth seed Agnieszka Radwanska were ousted by France's Kristina Mladenovic and China's Peng Shuai, respectively.

The ultra-fit Kerber persevered despite seeing her opponent save 13 break points in the match.

"She is a really tricky opponent," the German said. "She played a lot of spin, especially from the forehand side.

"I'd had great practices in the last few months so I'm ready to stay on court for three hours."

Williams saved three match points in her opening match but the 12th seed faced no such drama in her 6-4 6-2 victory over Lucie Safarova of the Czech Republic.

Next up for the 36-year-old American will be Chinese qualifier Peng, who eliminated Poland's Radwanska 6-4 6-4.

Peng, working to regain form since undergoing back surgery in 2015, knows she is capable of holding her own against the top seeds, having twice beaten Radwanska at the U.S. Open and defeated Williams at last year's China Open.

"I'm really happy that I can come back and play this tournament again -- after my back surgery I almost ended my career," she said in an on-court interview. "But I spent a lot of time and fight hard to be here."

Halep, playing her first event in five weeks due to a knee injury, fell 6-3 6-3 to 28th seed Mladenovic, who is riding high after notching her maiden WTA title at St Petersburg, which she followed up by reaching the Mexican Open final.

Next up for Kerber will be 14th-seeded Russian Elena Vesnina, who recovered from a bout of dizziness to beat 25th seed Timea Babos of Hungary 6-4 1-6 6-4.

American ninth seed Keys routed 19-year-old Naomi Osaka of Japan 6-1 6-4 to reach the Round of 16 against 13th seed Wozniacki, a 6-3 6-1 winner over Czech Katerina Siniakova.