A mathematicians way into STRONGnet

Transcription

1 A mathematicians way into STRONGnet Nemanja Bozovic Bergische Universität Wuppertal My work is funded by STRONGnet

2 Career so far Nemanja Božović, born on , from Serbia Studied at the University of Novi Sad, Faculty of Sciences, Department of Mathematics and Informatics ( ) Diploma in financial mathematics ( ) Ph.D. student in Novi Sad ( ) Currently, I am a Ph.D. student at the University of Wuppertal (2010-) A mathematicians way into STRONGnet, Nemanja Bozovic 2/13

3 Experiences Work at a German university Part of a research group, good social contact with other fellow Ph.D. scientists Contacts at training events, building a network Bielefeld Summer School; Lattice2011, Squaw Valley, 2011 A look at possible careers in industry (Bielefeld Summer School) Interdisciplinary research Participation at scientific conferences (GAMM Annual Meeting, Graz 2011; ICIAM2011, Vancouver; GAMM Workshop, Bremen 2011) Participation at workshops of the Wuppertal-Regensburg collaboration (SFB-TR 55) A mathematicians way into STRONGnet, Nemanja Bozovic 3/13

4 Get a Ph.D. Further international experience (plan: a few months in Dublin under supervision of Mike Peardon) Helps for a possible career in academia (in Serbia, for example) University systems International network of colleagues Know more about alternatives to the academic career Cooperation with Wuppertal group beyond the end of STRONGnet Improve communication skills in science Gain training experience in English Learn German A mathematicians way into STRONGnet, Nemanja Bozovic 4/13

5 A little bit about my work Working group WG3, University of Wuppertal Algorithmic Innovation A mathematicians way into STRONGnet, Nemanja Bozovic 5/13

6 Restarted GCRO-DR Hybrid Monte Carlo in Lattice QCD Monte Carlo method combined with molecular dynamics Time evolution leads to next configuration in the canonical ensemble Discretized Dirac operator A (i) only changes slightly in each time step Situation: Solve a long sequence of linear systems where A (i) x (i) = b (i), i = 1, 2,... A (i) C n n, b (i) C n change slightly from one system to the next A mathematicians way into STRONGnet, Nemanja Bozovic 6/13

7 Restarted GCRO-DR Krylov subspace methods generate iterates from { } K k (A, b) = span b, Ab,..., A k 1 b minimizes residual norm b Ax for x x 0 + K m While building the Krylov subspace we obtain the Arnoldi relation AV m = V m+1 H m b Ax =... = βe 1 H m y Then we obtain the solution x m = x 0 + V m y m where y m = argmin y βe 1 H m y A mathematicians way into STRONGnet, Nemanja Bozovic 7/13

8 Restarted GCRO-DR Restarted If we are working with systems of large dimensions, is not feasible We have too many vectors to store The total cost of orthogonalization in the Arnoldi process grows We fix m to some value and perform several cycles of Downsides Degradation of the convergence behavior The algorithm can even stagnate A mathematicians way into STRONGnet, Nemanja Bozovic 8/13

9 Restarted GCRO-DR Components of a residual colinear to eigenvectors corresponding to smallest eigenvalues have a big influence on error e = A 1 r We tend to chose augmenting space such that it contains approximations of these eigenvectors This can be done from the Krylov subspace built up in a previous cycle using Harmonic Ritz vectors Harmonic Ritz pair (λ, υ), λ C, υ K, satisfies Aυ λυ AK A mathematicians way into STRONGnet, Nemanja Bozovic 9/13

10 Restarted GCRO-DR GCROT Restarted ignores orthogonality to the discarded space after we restart GCROT retains a subspace between cycles such that the loss of orthogonality with respect to the discarded space is minimized This process is called optimal truncation It maintains matrices U k, C k C n k satisfying the relations AU k = C k C H k C k = I k. A mathematicians way into STRONGnet, Nemanja Bozovic 10/13

11 Restarted GCRO-DR GCROT can be modified to solve a sequence of linear systems by carrying U k to the next system and modifying U k and C k as follows: [Q, R] = reduced QR decomposition of A (i+1) Uk old Ck new = Q Uk new = Uk oldr 1. GCRO-DR combines two ideas of -DR and GCROT A mathematicians way into STRONGnet, Nemanja Bozovic 11/13

12 Preliminary results Preliminary results Wilson discretization of the Dirac operator on a 8 4 lattice, m = 40, k = 20 Restarted GCRO-DR first system second system third system nineth system tenth system eleventh system A mathematicians way into STRONGnet, Nemanja Bozovic 12/13

13 Thank you for your attention A mathematicians way into STRONGnet, Nemanja Bozovic 13/13