Reduced Basis Methods

With target applications characterized by computationally intensive parametrized problems that require repeated evaluation, it is clear that we need to seek alternatives to simply solving the full problem many times. This is exactly where reduced models have its place and we are now ready to dive deeper into a discussion of central elements of the certified reduced basis method.

[1]  Wolfgang Dahmen,et al.  Convergence Rates for Greedy Algorithms in Reduced Basis Methods , 2010, SIAM J. Math. Anal..

[2]  D. Rovas,et al.  A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations , 2003 .

[3]  Stefan Volkwein,et al.  Error estimates for abstract linear–quadratic optimal control problems using proper orthogonal decomposition , 2008, Comput. Optim. Appl..

[4]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[5]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[6]  M. Grepl.  Reduced-basis approximation a posteriori error estimation for parabolic partial differential equations , 2005 .

[7]  Y. Makovoz,et al.  On $n$-widths of certain functional classes defined by linear differential operators , 1983 .

[8]  I. Jolliffe.  Principal Component Analysis , 2002 .

[9]  Stefan Volkwein,et al.  Galerkin proper orthogonal decomposition methods for parabolic problems , 2001, Numerische Mathematik.

[10]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[11]  Michel Loève,et al.  Probability Theory I , 1977 .

[12]  H. Hotelling.  Analysis of a complex of statistical variables into principal components. , 1933 .

[13]  A. Patera,et al.  A PRIORI CONVERGENCE OF THE GREEDY ALGORITHM FOR THE PARAMETRIZED REDUCED BASIS METHOD , 2012 .

[14]  Alexander J. Smola,et al.  Learning with Kernels: support vector machines, regularization, optimization, and beyond , 2001, Adaptive computation and machine learning series.

[15]  Ronald DeVore,et al.  Greedy Algorithms for Reduced Bases in Banach Spaces , 2012, Constructive Approximation.

[16]  Kari Karhunen,et al.  Über lineare Methoden in der Wahrscheinlichkeitsrechnung , 1947 .