
Semester  Spring Semester 2015 
Lecturers  Dr. Peng Chen and Prof. Christoph Schwab 
Time  Thursday 15:1517:00 
Place  HG F 26.3 
Abstract  The seminar is concerned with the recently developed computational reduction methods, including Reduced Basis Method (RBM), Proper Orthogonal Decomposition (POD), Empirical Interpolation Method (EIM), and their application in solving highfidelity and highdimensional parametric and stochastic Partial Differential Equations (PDEs). 
Objective  During the semester each participant is supposed to prepare a twohours lecture, which is to be given in May 2015. It should be based on at least two of the recent research papers listed below. Depending on the student's preferences and study program, the focus may be made on the fundamentals, implementation aspects or application of these methods. The students are encouraged to conduct some original research. Help and guidance will be provided during the whole semester by the lecturers. 
Prerequisites / Notice 
The number of participants of the seminar is limited to 6. The preference will be given to ETH students of the following programs: 1. ETH MSc Applied Math, 2. ETH MSc RW/CSE, 3. ETH MSc MATH. 4. ETH BSc MATH, The prerequisites are: (*) for students taking the seminar for ETH BSc MATH: completed BSc examinations in Numerische Mathematik I+II, Numerical Methods for Elliptic and Parabolic PDEs. (*) for students taking the seminar for ETH MSc Math, Applied Math, RW/CSE: completed exam in courses Numerical Methods for Elliptic and Parabolic PDEs, OR NumPDEs for RW/CSE, Numerical Analysis of Stochastic PDEs. Topics will be discussed and selected for each participant during the first meeting on Thursday, Feb 19th, from 15h15 to 17h00, in the room HG F 26.3. 
References 
[1] M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera. An empirical interpolation method: application to efficient reducedbasis discretization of partial differential equations. Comptes Rendus Mathematique, Analyse Num ́erique, 339(9):667–672, 2004. [2] P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk. Convergence rates for greedy algorithms in reduced basis methods. SIAM Journal of Mathematical Analysis, 43(3):1457–1472, 2011. [3] S. Boyaval, C. Le Bris, T. Leli`evre, Y. Maday, N.C. Nguyen, and A.T. Patera. Reduced basis techniques for stochastic problems. Archives of Computational Methods in Engineering, 17:435–454, 2010. [4] A. Buffa, Y. Maday, A. Patera, C. Prudhomme, and G. Turinici. A priori convergence of the greedy algorithm for the parametrized reduced basis. ESAIM: Mathematical Modelling and Numerical Analysis, 46:595–603, 2011. [5] T. BuiThanh, K. Willcox, and O. Ghattas. Model reduction for largescale systems with highdimensional parametric input space. SIAM Journal on Scientific Computing, 30(6):3270–3288, 2008. [6] K. Carlberg, C. BouMosleh, and C. Farhat. Efficient nonlinear model reduction via a leastsquares Petrov–Galerkin projection and compressive tensor approximations. International Journal for Numerical Methods in Engineering, 86(2):155–181, 2011. [7] S. Chaturantabut and D.C. Sorensen. Nonlinear model reduction via discrete empirical interpolation. SIAM Journal on Scientific Computing, 32(5):2737–2764, 2010. [8] P. Chen. Model order reduction techniques for uncertainty quantification problems. PhD thesis, EPFL, Number 6118, 2014. [9] P. Chen and A. Quarteroni. Accurate and efficient evaluation of failure probability for partial differential equations with random input data. Computer Methods in Applied Mechanics and Engineering, 267(0):233–260, 2013. [10] P. Chen and A. Quarteroni. A new algorithm for highdimensional uncertainty problems based on dimensionadaptive and reduced basis methods. EPFL, MATHICSE Report 09, submitted, 2014. [11] P. Chen, A. Quarteroni, and G. Rozza. A weighted reduced basis method for elliptic partial differential equations with random input data. SIAM Journal on Numerical Analysis, 51(6):3163 – 3185, 2013. [12] P. Chen, A. Quarteroni, and G. Rozza. Comparison of reduced basis and stochastic collocation methods for elliptic problems. Journal of Scientific Computing, 59:187–216, 2014. [13] P. Chen and C. Schwab. Sparse grid, reduced basis Bayesian inversion. ETH Zurich, SAM Report 36, submitted, 2014. [14] A. Cohen and R. DeVore. Kolmogorov widths under holomorphic mappings. manuscript, 2014. [15] A. Cohen, R. DeVore, and Ch. Schwab. Convergence rates of best Nterm Galerkin approximations for a class of elliptic SPDEs. Foundations of Computational Mathematics, 10(6):615–646, 2010. [16] A. Cohen, R. Devore, and Ch. Schwab. Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s. Analysis and Applications, 9(01):11–47, 2011. [17] R. DeVore, G. Petrova, and P. Wojtaszczyk. Greedy algorithms for reduced bases in Banach spaces. Constructive Approximation, 37(3):455–466, 2013. [18] M. Drohmann, B. Haasdonk, and M. Ohlberger. Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM Journal on Scientific Computing, 34(2):A937–A969, 2012. [19] H. Elman and Q. Liao. Reduced basis collocation methods for partial differential equations with random coefficients. SIAM/ASA Journal on Uncertainty Quantification, 1(1):192–217, 2013. [20] R. Everson and L. Sirovich. Karhunen–Lo`eve procedure for gappy data. Journal of the Optical Society of America A, 12(8):1657–1664, 1995. [21] M.A. Grepl, Y. Maday, N.C. Nguyen, and A.T. Patera. Efficient reducedbasis treatment of nonaffine and nonlinear partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis, 41(03):575–605, 2007. [22] J.S. Hesthaven, B. Stamm, and S. Zhang. Efficient greedy algorithms for highdimensional parameter spaces with applications to empirical interpolation and reduced basis methods. ESAIM: Mathematical Modelling and Numerical Analysis, 48(01):259–283, 2014. [23] T. Lassila and G. Rozza. Parametric freeform shape design with PDE models and reduced basis method. Computer Methods in Applied Mechanics and Engineering, 199(23):1583–1592, 2010. [24] Y. Maday, N.C. Nguyen, A.T. Patera, and G.S.H. Pau. A general, multipurpose interpolation procedure: the magic points. Communications on Pure and Applied Analysis, 8(1):383–404, 2009. [25] Y. Maday, A.T. Patera, and G. Turinici. Global a priori convergence theory for reduced basis approximations of singleparameter symmetric coercive elliptic partial differential equations. Comptes Rendus Mathematique, 335(3):289–294, 2002. [26] N.C. Nguyen, A.T. Patera, and J. Peraire. A best points interpolation method for efficient approximation of parametrized functions. International Journal for Numerical Methods in Engineering, 73(4):521–543, 2008. [27] A.T. Patera and G. Rozza. Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Copyright MIT, http://augustine.mit.edu, 2007. [28] A. Quarteroni. Numerical Models for Differential Problems. Springer, Milano, 2nd ed., 2013. [29] A. Quarteroni, G. Rozza, and A. Manzoni. Certified reduced basis approximation for parametrized partial differential equations and applications. Journal of Mathematics in Industry, 1(1):1–49, 2011. [30] G. Rozza, D.B.P. Huynh, and A.T. Patera. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Archives of Computational Methods in Engineering, 15(3):229–275, 2008. 
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