Short CV August 2016: Dr. Peter Oswald Position: Professor of Mathematics Scientific degree: Dr. rer. nat. habil. Date of Birth: 13.11.1951 Nationality: German Research areas: Approximation Theory, Numerical Analysis

Scientific Education and Professional Career 1970–1978 1978 1978–1988 1981/82 1982 1988–1994 1989/90 1994–1995 1996–1997 1997–2004 Fall 2003 2004–2015 Fall 2015 2016–17 Fall 2016

Undergraduate and graduate studies of Mathematics at Odessa National University, Soviet Union/Ukraine, Diploma degree 1975. PhD (Kand. fiz. mat. nauk/Dr. rer. nat.), Moscow Institute of Electronic Engineering. Lecturer, Assistant Professor, Department of Mathematics, TU Dresden. Postdoctoral position, Department of Mathematics and Mechanics, Moscow State University. Habilitation (Dr. habil), Mathematics, University of Jena. Associate Professor, Faculty of Mathematics and Informatics, University of Jena. Visiting Professor, Department of Mathematics, Kuwait University. Visiting Professor, Department of Mathematics, Texas A&M University, College Station, USA. Visiting Research Scientist, SCAI, GMD, Bonn/St. Augustin. Member of Technical Staff, Computing Sciences Research Center, Bell Laboratories, Murray Hill, USA. Visiting Professor, IGPM, RWTH Aachen. Professor of Mathematics, Jacobs University Bremen. Visiting Professor, Departement of Mathematics, University of California San Diego. Bonn Research Chair, Hausdorff Center for Mathematics/Institute for Numerical Simulation, University of Bonn. Visiting Professor, Department of Mathematics, Texas A&M Univ., College Station.

Webpage http://www.faculty.jacobs-university.de/poswald http://www.ins.uni-bonn.de/people/oswald/

Research Interests Approximation Theory and Algorithms Function Spaces and Applied Harmonic Analysis Multiscale Methods in Scientific Computing Numerical Methods for PDEs Finite Elements, Splines, Wavelets Mathematical Modeling

Editorial Boards 1993–now 1999–2014 2000–2003 2001–now 2005–2007

Associate Associate Associate Associate Associate

Editor Editor Editor Editor Editor

Constructive Approximation J. Applied and Computational Harmonic Analysis J. Computational Methods in Applied Mathematics J. of Approximation Theory SIAM Journal on Numerical Analysis

Third-party funding since 2009 2009-12 2013

Nonlinear subdivision DFG Research Project: 156 600 EURO SampTA2013 Int. Conf. July 1 – July 5, 2013 organized jointly G¨otz Pfander. Funding DFG: 25 000 EURO.

Recent PhD Theses Stanislav Harizanov

Stability of Nonlinear Subdivision and Multiresolution Analysis (2011).

Recent Research Interests/Ongoing Projects Stable Space Splittings and Applications. Decomposition techniques are a basic tool in analysis and algorithm development. Stable space splittings in a Hilbert space context are generalizations of spectral decompositions, frames, and fusion frames, and have been instrumental in the development of the modern theory of multiscale and domain decomposition methods for large-scale scientific computing applications. The resulting algorithms of POCS type are based on cheaper subproblem solves and cycling through the components of the splitting. As such, they are generic, and can be found in many other application areas. Recent activities include randomization (both random space splittings and random ordering of subproblem solves) and greedy strategies. Concrete projects are to 1) look at randomization and its combined effect on convergence speed and preconditioning, 2) investigate infinite splittings in more detail (also in light of recent theoretical results on Kadison-Singer and paving conjectures), 3) motivated by a flurry of papers on Kaczmarz-type iterations to look at a larger class of inverse and non-symmetric problems, 4) leave the comfort zone of a Hilbert space setting and look at existing and still possible generalizations to convex optimization, and 5) contribute to the topic of robustness (via randomization) of large-scale computation in ”lossy” distributed computing environments. Recent papers: [O15,O19,O23,O25,O26,O28,O30], in collaboration with M. Griebel et al. (INS, Uni Bonn). Topics in finite element theory. I have been working on and off on problems in FE theory. Here is a list of topics that I have looked at in recent years: 1) variable-degree multilevel partition of unity ansatz spaces, related to so-called quarkonial frames (a three-parameter family of basis functions consisting of modulated (by polynomials of arbitrarily high degree) partition of unity basis functions on all scales), [O20] (joint work with S. Dahlke et al.), and their potential for

creating a new approach to hp-adaptive FE methods, 2) the small overlap situation in partition of unity methods and its relation to discontinuous Galerkin methods extending on [O5] (with A. Schweitzer (INS, Uni Bonn)), 3) properties of projectors onto FE spaces and their sensitivity to shape properties in the underlying partitions, [O7,O8,O21,O27,O31], and others. Nonlinear spectral-type approximation. A relatively unexplored topic is manifold-valued polynomial approximation that I stumbled upon when doing applied research on tunable optical filter architectures for polarization mode dispersion compensation (for references, see below) during my time at Bell Labs. Concretely, for two, closely related Lie matrix groups, namely G=SU(N) and G=SO(N), with rich sets of G-valued trigonometric polynomials (called polynomial loops) of arbitrary degree, we established generalizations of Jackson-type estimates (the smoother a loop, the faster it can be approximated with polynomial loops of increasing degree), see [O10,O13], collaboration with T. Shingel (formerly UCSD). The plan is to explore this topic further, and connect to the geometric integration community, where we see potential for the development of spectral methods. Nonlinear multiscale representations. On a discrete level (grid functions), multiscale methods extract information from point clouds (representing, e.g., sampled information of a continuous object) by coarsening (decimation, restriction) and refinement operations (prolongation, subdivision). There are many instances, when some of these operations need to become nonlinear, e.g., if data is sampled from a nonlinear manifold; more examples can be found in [O14]. We have contributed to the analysis of certain classes of such transforms in the past [O1,O12,O14,O16,O17] in the one-dimensional case (parameter curves), and started a case study on normal multiscale transforms for surfaces [O18,O23]. There is still unexplored territory in the higher-dimensional case, in particular, there is need in further evaluating the potential of these transforms for geometry representation and compression in scientific computation applications (see [O4] for earlier work on multivariate subdivision). Some experimental work with these transforms has been done with J. Moody (UCSD). Unrelated to this well-documented work, I have followed attempts (by Oseledets and Tyrtyshnikov) to connect nonlinear tensor-rank compression methods (QTT transform) using SVDbased coarsening with certain classes of by now classical wavelet transforms. There are many intriguing problems arising from tensor compression algorithms. Applications. I am generally open to cooperations on mathematical modelling questions in any area, if focus is on analytical and numerical aspects. In previous years, I have looked at selected mathematical topics in polarization mode dispersion modeling [O2,O3,O6], iterative decoding techniques, and very recently at Markov-type models for binary decision theory [O24,O29].

Publications since 2004 [O1] P. Oswald, Smoothness of nonlinear median-interpolation subdivision, Adv. Comput. Math. 20, 2004, 401–423. [O2] P. Oswald, C. K. Madsen, R. L. Konsbruck, Analysis of scalable PMD compensators using FIR filters and wavelength-dependent optical power measurements, J. Lightwave Technol., vol. 22, no. 2, Feb. 2004, 647–657. [O3] C. K. Madsen, P. Oswald, M. Capuzzo, E. Chen, L. Gomez, A. Griffin, A. Kasper, E. Laskowski, L.Stulz, A. Wong-Foy, Reset-free integrated polarization controller using phase shifters, IEEE J. Sel. Topics in Quantum Electronics, vol. 11, no. 2, March/April 2005, 431–438. [O4] P. Oswald, Designing composite triangular subdivision schemes, CAGD 22 (2005), 659–679.

[O5] M. Griebel, P. Oswald, M. A. Schweitzer, A particle-partition of unity method-Part VI: A p-robust multilevel solver In Meshfree Methods for Partial Differential Equations II (M. Griebel and M. A. Schweitzer, editors), Lecture Notes in Computational Science and Engineering vol. 43, pages 71–92. Springer, 2005. [O6] P. Oswald, C. K. Madsen, Deterministic analysis of endless tuning of polarization controllers J. Lightwave Technol., vol. 24, no. 7, July 2006, 2932–2939. [O7] P. Oswald, Semiorthogonal linear prewavelets on irregular meshes, In Approximation and Probability, Banach Center Publ. vol. 72, Inst. Math. Polish Acad. Sci., 2006, 221–234. [O8] P. Oswald, A counterexample for the L2 -projector onto linear spline spaces, Mathematics of Computation, 77, 2008, 221–226. [O9] P. Oswald, Optimality of multilevel preconditioning for nonconforming P1 finite elements, Numerische Mathematik, 111(2), 2008, 267–291. [O10] P. Oswald, T. Shingel, Splitting methods for SU(N) loop approximation, J. Approx. Th. 161(1), 2009, 174–186. [O11] J. Maes, P. Oswald, Multilevel finite element preconditioning for sqrt(3) refinement, Mathematics of Computation 78, 2009, 1869–1890. [O12] S. Harizanov, P. Oswald, Stability of nonlinear subdivision and multiscale transforms, Constr. Approx. 31, 2010, 359–393. [O13] P. Oswald, T. Shingel, Close-to-optimal bounds for SU(N) loop approximation, J. Approx. Th. 162(9), 2010, 1511–1517. [O14] N. Dyn, P. Oswald, Univariate subdivision and multiscale transforms: The nonlinear case, in Multiscale, Nonlinear, and Adaptive Approximation (R.A. DeVore, A. Kunoth eds.), pp. 203–247, Springer, Berlin, 2009. [O15] P. Oswald, Stable space splittings and fusion frames, in Wavelets XIII (V.K. Goyal, M. Papadakis, D. Van de Ville, eds.), Proc. SPIE Vol. 7446 (SPIE, Bellingham, 2009), 744611. doi:10.117/12.825303 . [O16] S. Harizanov, P. Oswald, T. Shingel, Normal multi-scale transforms for curves, Found. Comput. Math. 11 (6), 2011, 617–656. [O17] P. Oswald, Nonlinear multi-scale transforms: Lp theory, J. Franklin Inst. 349, 2012, 1619–1636. [O18] P. Oswald, Normal multiscale transforms for surfaces, in Curves and Surfaces 2010 (J.-D. Boissonat et al. (eds.)), LNCS 6920, pp. 527–542, 2011. [O19] M. Griebel, P. Oswald, Greedy and randomized versions of the multiplicative Schwarz method, Lin. Algebra Appl. 437 (7), 2012, 1596–1610. [O20] S. Dahlke, P. Oswald, T. Raasch, A note on quarkonial systems and multilevel partition of unity methods, Math. Nachr. 286 (5-6), 2013, 600–613. [O21] P. Oswald, L∞ bounds for the L2 projection onto linear spline spaces, in Recent Advances in Harmonic Analysis and Applications, Springer Proceedings in Mathematics and Statistics, vol. 25, pp. 303–316, 2013. [O22] P. Oswald, T. Shingel, Commutator estimate for nonlinear subdivision, in Mathematical Methods for Curves and Surfaces, 8th Int. Conf. MMCS 2012, pp. 383-402, LNCS 8177, Springer, 2014. [O23] M. Griebel, A. Hullmann, P. Oswald, Optimal scalings for sparse grid discretizations, Numer. Lin. Alg. Appl. 22 (1), 2014, 76–100, doi: 10.1002/nla.1939.

[O24] A. Diederich, P. Oswald, Sequential sampling model for multiattribute choice alternatives with random attention time and processing order, Front. Hum. Neurosci. 8:697, 2014, doi: 10.3389/fnhum.2014.00697. [O25] P. Oswald, W. Zhou, Convergence analysis for Kaczmarz-type methods in a Hilbert space framework, Lin. Algebra Appl. 478, 2015, 131-161, doi: 10.1016/j.laa.2015.03.028. [O26] M. Griebel, P. Oswald, Schwarz Iterative Methods: Infinite Space Splittings, Constr. Approx. 44 (1), 2016, 121-139, also arXiv:1501.00938. [O27] P. Oswald, Divergence of FEM: Babuˇska-Aziz triangulations revisited, Appl. Math. 60, 5, 2015, 473-484. [O28] P. Oswald, W. Zhou, Random reordering in SOR-type methods, Numer. Math., 2016, doi:10.1007/s00211016-0829-7, also arXiv:1510.04727. [O29] A. Diederich, P. Oswald, Multi-stage sequential sampling models with finite or infinite time horizon and variable boundaries, J. Math. Psychology, 2016, doi:10.1016/j.jmp.2016.02.010. [O30] M. Griebel, P. Oswald, Hilbert function space splittings on domains with infinitely many variables, J. Complexity (submitted), INS Preprint No. 1617, Bonn Univ., also arxiv:1607.05978. [O31] P. Oswald, Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error, INS Preprint No. 1618, Bonn Univ., also arxiv:1607.05983.

Teaching Here is a list of my teaching activities at Jacobs University Bremen in the years 2004–2015, all classes were held in English, audiences varied from 5 to 150 students. My philosophy is to provide enough challenge from the instructor’s side, reward those who live up to the challenge, and help those along (by, e.g., offering makeup opportunities and second chances) who try but have difficulties. I have taught on all levels, undergrad Math service, applied and pure mathematics courses and lab units, and (fewer) graduate classes. Links to many of the still existing course pages can be found on my personal webpage. Before 1997, I have also taught at Technical University Dresden, University of Jena (in German), University of Kuwait, Texas A&M University (in English). While visiting UCSD in the Fall quarter 2015, I taught the second-year undergraduate class Math20F Linear Algebra with about 250 students. 110211 Numerical Methods I (Fall 2004) 110212 Numerical Methods II (Spring 2005, Spring 2006) 110311 Computational PDE (Spring 2005) 100261 Introduction to Mathematical Modeling (Spring 2005) 110112 NatSciLab Math Numerical Software (Spring 2012, Spring 2011, Fall 2005) 110221 Derivatives Lab (Fall 2014, Fall 2013) 100371 Introductory Harmonic Analysis (Spring 2006) 100262 Applied Differential Equations and Modeling (Spring 2006, Spring 2007, Spring 2009, Spring 2011) 120101 ESM1A Single Variable Calculus (Fall 2006, Fall 2007, Fall 2012) 120121 ESM1C Introduction to Calculus (Fall 2008) 110311 Numerical Analysis (Fall 2008) 100201/2 Analysis I/II (Fall 2009/Spring 2010) 120201 ESM3A Advanced Linear Algebra/Stochastic Processes (Fall 2010, Fall 2011, Fall 2012)

100313 Real Analysis (Fall 2014, Fall 2013, Fall 2010) 120112 ESM2B Linear Algebra, Fourier, Probability (Spring 2014) USC Mathematical Modeling of Social-Economic Behavior (Spring 2008, Spring 2010) 110411 Topics in Applied Analysis (Fall 2006, Fall 2011) 100411 Real Analysis (Fall 2007) 100471 Functional Analysis (Spring 2014)