Michael Griebel received his education at the Technical University of Munich, Germany. He is a professor at the Institute for Numerical Simulation at the University of Bonn, Germany, where he holds the Chair of Scientific Computing and Numerical Simulation. Additionally, he is the director of Fraunhofer SCAI (Institute for Algorithms and Scientific Computing), Sankt Augustin, Germany. His research interests include numerical simulation, scientific computing, machine learning, and high-dimensional approximation. Since 2002, he has served as the Editor-in-Chief of the Springer journal Numerische Mathematik.Peter Oswald received his education at Odessa State University and Moscow State University. He has held research, teaching, and professorship positions at various institutions, including TU Dresden, FSU Jena, Kuwait University, Texas A&M University, Bell Laboratories, Jacobs University Bremen, and the University of Bonn. His research interests include approximation theory, function spaces, and numerical analysis.

• 1 Introduction.- 2 Hilbert space splittings: Abstract theory.- References.- 3 Hilbert space splittings: Examples and extensions.- References.- 4 Schwarz iterative methods: Finite Omega.- References.- 5 Special topics and extensions.- References.- 6 Schwarz approximation methods: Infinite Omega.- References.- 7 Applications to PDE solvers.- References.- A Hilbert space basics.- A.1 Spaces: Basic notation, definitions and properties.- A.2 Operators between Hilbert spaces.- A.3 Linear equations and variational problems.- A.4 Constructions on Hilbert spaces.- A.5 Sobolev spaces on domains.- A.6 Reproducing kernel Hilbert spaces (RKHS).- References.- Index.

Michael Griebel received his education at the Technical University of Munich, Germany. He is a professor at the Institute for Numerical Simulation at the University of Bonn, Germany, where he holds the Chair of Scientific Computing and Numerical Simulation. Additionally, he is the director of Fraunhofer SCAI (Institute for Algorithms and Scientific Computing), Sankt Augustin, Germany. His research interests include numerical simulation, scientific computing, machine learning, and high-dimensional approximation. Since 2002, he has served as the Editor-in-Chief of the Springer journal Numerische Mathematik.Peter Oswald received his education at Odessa State University and Moscow State University. He has held research, teaching, and professorship positions at various institutions, including TU Dresden, FSU Jena, Kuwait University, Texas A&M University, Bell Laboratories, Jacobs University Bremen, and the University of Bonn. His research interests include approximation theory, function spaces, and numerical analysis.

• 1 Introduction.- 2 Hilbert space splittings: Abstract theory.- References.- 3 Hilbert space splittings: Examples and extensions.- References.- 4 Schwarz iterative methods: Finite Omega.- References.- 5 Special topics and extensions.- References.- 6 Schwarz approximation methods: Infinite Omega.- References.- 7 Applications to PDE solvers.- References.- A Hilbert space basics.- A.1 Spaces: Basic notation, definitions and properties.- A.2 Operators between Hilbert spaces.- A.3 Linear equations and variational problems.- A.4 Constructions on Hilbert spaces.- A.5 Sobolev spaces on domains.- A.6 Reproducing kernel Hilbert spaces (RKHS).- References.- Index.

Michael Griebel received his education at the Technical University of Munich, Germany. He is a professor at the Institute for Numerical Simulation at the University of Bonn, Germany, where he holds the Chair of Scientific Computing and Numerical Simulation. Additionally, he is the director of Fraunhofer SCAI (Institute for Algorithms and Scientific Computing), Sankt Augustin, Germany. His research interests include numerical simulation, scientific computing, machine learning, and high-dimensional approximation. Since 2002, he has served as the Editor-in-Chief of the Springer journal Numerische Mathematik.Peter Oswald received his education at Odessa State University and Moscow State University. He has held research, teaching, and professorship positions at various institutions, including TU Dresden, FSU Jena, Kuwait University, Texas A&M University, Bell Laboratories, Jacobs University Bremen, and the University of Bonn. His research interests include approximation theory, function spaces, and numerical analysis.

• 1 Introduction.- 2 Hilbert space splittings: Abstract theory.- References.- 3 Hilbert space splittings: Examples and extensions.- References.- 4 Schwarz iterative methods: Finite Omega.- References.- 5 Special topics and extensions.- References.- 6 Schwarz approximation methods: Infinite Omega.- References.- 7 Applications to PDE solvers.- References.- A Hilbert space basics.- A.1 Spaces: Basic notation, definitions and properties.- A.2 Operators between Hilbert spaces.- A.3 Linear equations and variational problems.- A.4 Constructions on Hilbert spaces.- A.5 Sobolev spaces on domains.- A.6 Reproducing kernel Hilbert spaces (RKHS).- References.- Index.