# Prof. Dr. Philipp Grohs

Seminar for Applied Mathematics
ETH Zürich
Rämistrasse 101
8092 Zurich
Switzerland

### Research Interests

• Approximation Theory and Computational Harmonic Analysis:
• Analysis of nonlinear subdivision algorithms
• Nonlinear wavelet type transformations
• Wavelets, Shearlets,...
• Signal processing
• Numerical Methods for PDEs:
• Geodesic finite elements for geometric PDEs (harmonic maps into Riemannian manifolds, ...)
• Numerical solution of kinetic transport PDEs (radiative Transfer, Boltzmann equation, ...)
• Design of optimally convergent adaptive algorithms for PDEs whose solutions possess anisotropic singularities (for instance transport problems or diffusion problems in inhomogeneous media)
• Geometry:
• Discrete differential geometry with applications to architectural geometry
• Lie sphere geometry

### Press

• ETH News press release related to Latsis prize

### Funding

• Intrinsic Approximation of Geometric Data funded by SNF Project 140635 [active]
• Sparse Discretization of Kinetic Transport Problems on High-dimensional Phase Spaces (jointly with R. Hiptmair and Ch. Schwab) funded by SNF Project 146356 [active]

### Book

1. Harmonic and Applied Analysis - From Groups to Signals. (Joint work with S. Dahlke, F. DeMari and D. Labate).

### Scientific Publications

1. Smoothness properties of Lie group subdivision schemes. (Joint work with J. Wallner and E. Nava Yazdani).
SIAM Journal on Multiscale Modeling and Simulation 6 (2007), 493-505.
[MR] [doi] [preprint].
2. Log-exponential analogues of univariate subdivision schemes in Lie groups and their smoothness properties. (Joint work with J. Wallner).
In M. Neamtu and L. L. Schumaker, editors, Approximation Theory XII: San Antonio 2007, pages 181-190. Nashboro Press, 2008, ISBN 978-0-9728482-9-9.
[MR] [preprint].
3. Smoothness analysis of subdivision schemes on regular grids by proximity.
SIAM Journal on Numerical Analysis 46 (2008), 2169-2182.
[MR] [doi] [preprint].
4. Smoothness equivalence properties of univariate subdivision schemes and their projection analogues.
Numerische Mathematik 113/2 (2009), 163-180.
[MR] [doi] [preprint] .
5. Smoothness of interpolatory multivariate subdivision in Lie groups.
IMA Journal of Numerical Analysis 29/3 (2009), 760-772.
[MR] [doi] [preprint] .
6. Interpolatory wavelets for manifold-valued data. (Joint work with J. Wallner).
Applied and Computational Harmonic Analysis 27 (2009), 325-333.
[MR] [doi] [preprint] .
7. Laguerre minimal surfaces, isotropic geometry and linear elasticity. (Joint work with H. Pottmann and N. Mitra).
Advances in Computational Mathematics 31/2 (2009), 391-419.
[MR] [doi] [preprint].
8. Approximation order of interpolatory nonlinear subdivision schemes. (Joint work with N. Dyn and J. Wallner).
Journal of Computational and Applied Mathematics 223 (2010), 1697-1703.
[MR] [doi] [preprint].
9. A general proximity analysis of nonlinear subdivision schemes.
SIAM Journal on Mathematical Analysis 42/2 (2010), 729-750.
[MR] [doi] [preprint].
10. Approximation order from stability of nonlinear subdivision schemes.
Journal of Approximation Theory 162 (2010), 1085-1094.
[MR] [doi] [preprint].
11. Edge offset meshes in Laguerre geometry. (Joint work with H. Pottmann and B. Blaschitz).
Advances in Computational Mathematics 33/1 (2010), 45-73.
[MR] [doi] [preprint].
12. Stability of manifold-valued subdivision schemes and multiscale transformations.
Constructive Approximation 32/3 (2010), 569-596.
[MR] [doi] [preprint].
13. The continuous Shearlet Transform: Representation Formulae and Microlocal Analysis.
Oberwolfach Report 7/4 (2010).
[doi]
14. Continuous shearlet tight frames.
Journal of Fourier Analysis and Applications 17/3 (2011), 506-518.
[doi] [arXiv] [preprint].
15. Continuous Shearlet frames and Resolution of the Wavefront Set.
Monatshefte für Mathematik 164/4 (2011), 393-426.
[doi] [arXiv] [preprint].
16. Interpolating composite systems.
In M. Neamtu and L. L. Schumaker, editors, Approximation Theory XIII: San Antonio 2010, Springer Proceedings in Mathematics 13 (2012), 111-120.
[doi] [preprint].
17. Shearlets and Microlocal Analysis.
In: Shearlets: Multiscale Analysis for Multivariate Data, Birkhäuser-Springer (2012).
[doi] [book] [preprint].
18. Ridgelet-type Frame Decompositions for Sobolev Spaces related to linear Transport.
Journal of Fourier Analysis and Applications 18/2 (2012), 309-325.
[doi] [preprint].
19. Definability and stability of multiscale decompositions for manifold-valued data. (Joint work with J. Wallner).
Journal of the Franklin Institute 349/5 (2012), 1648-1664.
[doi] [arXiv] [preprint].
20. Tree Approximation with anisotropic decompositions.
Applied and Computational Harmonic Analysis 33 (2012), 44-57.
[doi] [preprint].
21. Tree Approximation and optimal Image Coding with Shearlets.
Proceedings SampTA: Singapore 2011 (2012).
[preprint].
22. On the Structure of Anisotropic Frames.
Oberwolfach Report 29 (2012).
[doi]
23. Ruled Laguerre minimal surfaces. (Joint work with H. Pottmann and M. Skopenkov).
Mathematische Zeitschrift 272/1 (2012), 645-674.
[doi] [arXiv] [preprint].
24. Bandlimited Shearlet Frames with nice Duals.
Journal of Computational and Applied Mathematics 243 (2013), 139-151.
[doi] [preprint].
25. Refinable functions for dilation families.
Advances in Computational Mathematics 38/3 (2013), 531-561.
[doi] [arXiv] [preprint].
26. Intrinsic Localization of Anisotropic Frames.
Applied and Computational Harmonic Analysis 35/2 (2013), 264-283.
[doi] [preprint].
27. Finite Elements of Arbitrary Order and Quasiinterpolation for Riemannian Data.
IMA Journal of Numerical Analysis 33/3 (2013), 849-874.
[doi] [preprint].
28. Geometric Multiscale Decompositions of Dynamic Low-Rank Matrices.
Computer Aided Geometric Design 30 (2013), 805-826.
[doi] [preprint].
29. Parabolic Molecules. (Joint work with G. Kutyniok).
Foundations of Computational Mathematics 14/2 (2014), 299-337.
[doi] [preprint].
30. Parabolic Molecules: Curvelets, Shearlets, and Beyond. (Joint work with S. Keiper, G. Kutyniok, M. Schaefer).
In Approximation Theory XIV: San Antonio 2013, Springer Proceedings in Mathematics (2014), to appear.
[preprint].
31. $\alpha$-Molecules: Curvelets, Shearlets, Ridgelets, and Beyond. (Joint work with S. Keiper, G. Kutyniok, M. Schaefer).
In Wavelets and Sparsity XV (San Diego, CA, 2013), SPIE Proc. 8858, SPIE, Bellingham, WA, (2014), to appear.
[preprint].
32. Wolfowitz's Theorem and Convergence of Consensus Algorithms in Hadamard Spaces.
Proceedings of the American Mathematical Society (2014), to appear.
[preprint].
33. Hyperbolic Cross Approximation for the Spatially Homogenous Boltzmann Equation. (Joint work with E. Fonn and R. Hiptmair).
IMA Journal of Numerical Analysis (2014), to appear.
[doi] [preprint].
34. Intrinsic Localization of Anisotropic Frames II: $\alpha$-molecules (Joint work with S. Vigogna).
Journal of Fourier Analysis and Applications (2014), to appear.
[doi] [preprint].
35. Cartoon Approximation with $\alpha$-Curvelets (Joint work with S. Keiper, G. Kutyniok and M. Schaefer).
Journal of Fourier Analysis and Applications (2014), to appear.
[preprint].
36. Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements. (Joint work with H. Hardering and O. Sander).
Foundations of Computational Mathematics (2014), to appear.
[doi] [preprint].
37. FFRT-A Fast Finite Ridgelet Transform for Radiative Transport (Joint work with S. Etter and A. Obermeier).
SIAM Journal on Multiscale Modeling and Simulation (2014), to appear.
[preprint] [MATLAB codes].
38. $\varepsilon$-Subgradient Algorithms for Locally Lipschitz Functions on Riemannian Manifolds (Joint work with S. Hosseini).
Advances in Computational Mathematics (2015), to appear.
[preprint].
39. Optimal Adaptive Ridgelet Schemes for Linear Transport Equations (Joint work with A. Obermeier).
Applied and Computational Harmonic Analysis (2015), to appear.
[preprint].
40. Ridgelet Methods for Linear Transport Equations (Joint work with A. Obermeier).
In Curves and Surfaces: Paris 2014, Springer Proceedings in Mathematics (2015), to appear.
[preprint]
41. Nonsmooth Trust Region Algorithms for Locally Lipschitz Functions on Riemannian Manifolds (Joint work with S. Hosseini).
IMA Journal of Numerical Analysis (2015), to appear.
[preprint].
42. $\alpha$-Molecules (Joint work with S. Keiper, G. Kutyniok and M. Schaefer).
Applied and Computational Harmonic Analysis (2015), to appear.
[preprint].
43. Total Variation Regularization by Iteratively Reweighted Least Squares on Hadamard Spaces and the Sphere (Joint work with M. Sprecher).
IMA Journal on Information and Inference (2015), to appear.
[preprint]
44. Deep convolutional neural networks on cartoon functions (Joint work with T. Wiatowski, and H. Bölcskei).
Proc. of IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, (2016), to appear.
[preprint]
45. Discrete Deep Feature Extraction: A Theory and New Architectures (Joint work with T. Wiatowski, M. Tschannen, A. Stanic and H. Boelcskei).
Journal of Machine Learning Research (Proc. ICML 2016) (2016), to appear.
[preprint]
46. A Shearlet-Based Fast Thresholded Landweber Algorithm for Deconvolution (Joint work with Z. Kereta and U. Wiesmann).
International Journal of Wavelets, Multiresolution and Information Processing (2016), to appear.
[preprint].
47. Scattered Manifold-Valued Data Approximation (Joint work with M. Sprecher and T. P.-Y. Yu).
Numerische Mathematik (2016), accepted subject to revision.
[preprint].
48. On the maxima of cardinal refinable functions with positive symbol.
Geometry preprint 2009/03, TU Graz, August 2009.
[preprint].
49. Sparse Twisted Tensor Frame Discretizations of Parametric Transport Equations. (Joint work with Ch. Schwab).
SAM Report 2011-41, ETH Zurich, May 2011.
[preprint].
50. Projection-based Quasiinterpolation in Manifolds. (Joint work with M. Sprecher).
SAM Report 2013-23, ETH Zurich, July 2013.
[preprint] [MATLAB codes].
51. Polar Spectral Scheme for the Spatially Homogenous Boltzmann Equation (Joint work with E. Fonn and R. Hiptmair).
SAM Report 2014-13, ETH Zurich, April 2014.
[preprint].
52. Continuous Parabolic Molecules (Joint work with Z. Kereta).
SAM Report 2015-17, ETH Zurich, July 2015.
[preprint]
53. Anisotropic Multiscale Systems on Bounded Domains (Joint work with G. Kutyniok, J. Ma and P. Petersen).
SAM Report 2015-30, ETH Zurich, September 2015.
[preprint]
54. Tensor-product discretization for the spatially inhomogeneous and transient Boltzmann equation in 2D (Joint work with R. Hiptmair and S. Pintarelli).
SAM Report 2015-38, ETH Zurich, November 2015.
[preprint]
55. On the Approximation of Functions with Line Singularities by Ridgelets (Joint work with A. Obermeier).
SAM Report 2016-04, ETH Zurich, January 2016.
[preprint]
56. Reconstructing real-valued functions from unsigned coefficients with respect to wavelet and other frames (Joint work with R. Al-Aifari, I. Daubechies and G. Thakur).
SAM Report 2016-08, ETH Zurich, January 2016.
[preprint]
57. Phase Retrieval in the general Setting of Continuous Banach Frames (Joint work with R. Al-Aifari).
SAM Report 2016-21, ETH Zurich, April 2016.
[preprint]

### Theses

1. Approximation Theory in Manifolds.
Habilitation Thesis, TU Graz, 2010.
[full thesis] [introduction].
2. Smoothness Analysis of nonlinear Subdivison Schemes on regular grids.
Phd Thesis, TU-Wien, 2007.
[pdf].
3. Asymptotik von Goppa Codes (in German).
Diploma Thesis, TU-Wien, 2006.
[pdf].
4. Beweis der Bieberbachschen Vermutung (in German).
Semester Thesis, TU-Wien, 2005.
[pdf].