Stephen Shing-Toung Yau (Life Fellow, IEEE) received the Ph.D. degree in mathematics from the State University of New York, Stony Brook, NY, USA, in 1976. He was a Member of the Institute of Advanced Study, Princeton, NJ, USA, from 1976 to 1977 and 1981 to 1982. He was a Benjamin Pierce Assistant Professor with Harvard University, Cambridge, MA, USA, from 1977 to 1980. He then joined the Department of Mathematics, Statistics and Computer Science (MSCS), University of Illinois at Chicago (UIC), Chicago, IL, USA, and served for more than 30 years. From 2005 to 2011, he was a Joint Professor with the Department of Electrical and Computer Engineering, MSCS, UIC. After retiring in 2011, he joined the Department of Mathematical Sciences at Tsinghua University in Beijing, China, where he served for over 10 years. In 2022, he became a research fellow at the Beijing Institute of Mathematical Sciences and Applications (BIMSA) in Beijing, China, to begin his new research.  His research interests include nonlinear filtering, bioinformatics, complex algebraic geometry, Cauchy–Riemann geometry, and singularities theory. Dr. Yau has been the Managing Editor and Founder of Journal of Algebraic Geometry since 1991 and the Editor-in-Chief and Founder of Communications in Information and Systems since 2000. He was the General Chairman of the 1995 IEEE International Conference on Control and Information. He received the Sloan Fellowship in 1980, the Guggenheim Fellowship in 2000, and the American Mathematical Society Fellow Award in 2013. In 2005, he was entitled the UIC Distinguished Professor. In 2019, He won the Chern Prize of Lifetime Achievement in Mathematics.Xiuqiong Chen received the B.S. degree in the School of Mathematical Sciences, Beihang University, Beijing, China, in 2014, and the Ph.D. degree in applied mathematics from the Department of Mathematical Sciences, Tsinghua University, Beijing, China in 2019. After her graduation, she was a Postdoctoral Scholar with Yau Mathematical Sciences Center, Tsinghua University, Beijing, China, from 2019 to 2021. She joined in Renmin University of China, Beijing, China, since 2021. She is currently an Assistant Professor with School of Mathematics, Renmin University of China. Her research interests include nonlinear filtering and deep learning.Xiaopei Jiao received his Bachelor's degree from Shanghai Jiao Tong University in 2017 and completed his Ph.D. from the Department of Mathematics at Tsinghua University in 2022. From 2022 to 2024, he worked as a postdoctoral researcher at the Beijing Institute of Mathematical Science and Application. He is currently employed at the University of Twente in the Netherlands as postdoctoral researcher. His research focuses on nonlinear filtering, Lie estimation algebra, physics-informed deep learning, and bioinformatics.Jiayi Kang received the B.S. degree from the college of mathematics, Sichuan University, Chengdu, China, in 2019 and Ph.D. degree from Department of Mathematical Sciences at Tsinghua University, China in 2024. He is currently an assistant professor at the Beijing Institute of Mathematical Sciences and Applications in Beijing, China.Zeju Sun received the B.S. degree from Department of Mathematical Sciences, Tsinghua University, Beijing, China, in 2020. He is currently pursuing the Ph.D. degree in mathematics with the department of Mathematical Sciences, Tsinghua University, Beijing, China.Yangtianze Tao received the B.S degree in College of Mathematics, Sichuan University, Sichuan, China in 2019. Now he is pursuing Ph.D. degree with Department of Mathematical Sciences, Tsinghua University, Beijing, China, under the supervision of Prof. Stephen Yau in the field of applied mathematics. His research interests include deep learning, machine learning and nonlinear filtering.

“This book will be an invaluable resource for researchers and practitioners.” (Mykhailo Moklyachuk , zbMATH 1562.93004, 2025)

• Preface.- I. Preliminary knowledge.- 1. Probability theory.- 2. Stochastic processes.- 3. Stochastic differential equations.- 4. Optimization.- II. Filtering theory.- 5. The filtering equations.- 6. Estimation algebra.-  III. Numerical algorithms.- 7.  Yau-Yau algorithm.- 8. Direct methods.- 9. Classical filtering methods.- 10. Estimation algorithms based on deep learning.

Stephen Shing-Toung Yau (Life Fellow, IEEE) received the Ph.D. degree in mathematics from the State University of New York, Stony Brook, NY, USA, in 1976. He was a Member of the Institute of Advanced Study, Princeton, NJ, USA, from 1976 to 1977 and 1981 to 1982. He was a Benjamin Pierce Assistant Professor with Harvard University, Cambridge, MA, USA, from 1977 to 1980. He then joined the Department of Mathematics, Statistics and Computer Science (MSCS), University of Illinois at Chicago (UIC), Chicago, IL, USA, and served for more than 30 years. From 2005 to 2011, he was a Joint Professor with the Department of Electrical and Computer Engineering, MSCS, UIC. After retiring in 2011, he joined the Department of Mathematical Sciences at Tsinghua University in Beijing, China, where he served for over 10 years. In 2022, he became a research fellow at the Beijing Institute of Mathematical Sciences and Applications (BIMSA) in Beijing, China, to begin his new research.  His research interests include nonlinear filtering, bioinformatics, complex algebraic geometry, Cauchy–Riemann geometry, and singularities theory. Dr. Yau has been the Managing Editor and Founder of Journal of Algebraic Geometry since 1991 and the Editor-in-Chief and Founder of Communications in Information and Systems since 2000. He was the General Chairman of the 1995 IEEE International Conference on Control and Information. He received the Sloan Fellowship in 1980, the Guggenheim Fellowship in 2000, and the American Mathematical Society Fellow Award in 2013. In 2005, he was entitled the UIC Distinguished Professor. In 2019, He won the Chern Prize of Lifetime Achievement in Mathematics.Xiuqiong Chen received the B.S. degree in the School of Mathematical Sciences, Beihang University, Beijing, China, in 2014, and the Ph.D. degree in applied mathematics from the Department of Mathematical Sciences, Tsinghua University, Beijing, China in 2019. After her graduation, she was a Postdoctoral Scholar with Yau Mathematical Sciences Center, Tsinghua University, Beijing, China, from 2019 to 2021. She joined in Renmin University of China, Beijing, China, since 2021. She is currently an Assistant Professor with School of Mathematics, Renmin University of China. Her research interests include nonlinear filtering and deep learning.Xiaopei Jiao received his Bachelor's degree from Shanghai Jiao Tong University in 2017 and completed his Ph.D. from the Department of Mathematics at Tsinghua University in 2022. From 2022 to 2024, he worked as a postdoctoral researcher at the Beijing Institute of Mathematical Science and Application. He is currently employed at the University of Twente in the Netherlands as postdoctoral researcher. His research focuses on nonlinear filtering, Lie estimation algebra, physics-informed deep learning, and bioinformatics.Jiayi Kang received the B.S. degree from the college of mathematics, Sichuan University, Chengdu, China, in 2019 and Ph.D. degree from Department of Mathematical Sciences at Tsinghua University, China in 2024. He is currently an assistant professor at the Beijing Institute of Mathematical Sciences and Applications in Beijing, China.Zeju Sun received the B.S. degree from Department of Mathematical Sciences, Tsinghua University, Beijing, China, in 2020. He is currently pursuing the Ph.D. degree in mathematics with the department of Mathematical Sciences, Tsinghua University, Beijing, China.Yangtianze Tao received the B.S degree in College of Mathematics, Sichuan University, Sichuan, China in 2019. Now he is pursuing Ph.D. degree with Department of Mathematical Sciences, Tsinghua University, Beijing, China, under the supervision of Prof. Stephen Yau in the field of applied mathematics. His research interests include deep learning, machine learning and nonlinear filtering.

“This book will be an invaluable resource for researchers and practitioners.” (Mykhailo Moklyachuk , zbMATH 1562.93004, 2025)

• Preface.- I. Preliminary knowledge.- 1. Probability theory.- 2. Stochastic processes.- 3. Stochastic differential equations.- 4. Optimization.- II. Filtering theory.- 5. The filtering equations.- 6. Estimation algebra.-  III. Numerical algorithms.- 7.  Yau-Yau algorithm.- 8. Direct methods.- 9. Classical filtering methods.- 10. Estimation algorithms based on deep learning.