Dr Anders Hansen

Anders

Anders leads the Applied Functional and Harmonic Analysis group within the Cambridge Centre for Analysis at DAMTP. He is a Reader (Associate Professor) in mathematics  at DAMTP, Professor of Mathematics at the University of Oslo, a Royal Society University Research Fellow and also a Fellow of Peterhouse.

Email: ach70@cam.ac.uk
Tel: +44 1223 760403
Office: F2.01

Resume

Upcoming and Recent Events

  1. Invited speaker at Computational Harmonic Analysis and Data Science, Banff International Research Station (Nov 2019).
  2. Invited speaker at Algorithms and Complexity for Continuous Problems, Dagstuhl (Aug 2019).
  3. Invited speaker at National Academy of Sciences, Arthur M. Sackler Colloquim: The Science of Deep Learning, Washington D.C. (March 2019).
  4. Speaking at Imperial College/University College London, Numerical Analysis Seminar (Feb. 20 2019)
  5. Invited speaker at Variational methods and optimization in imaging, Institut Henri Poincaré (Feb. 2019).
  6. Speaking at Imperial College, Pure Analysis Seminar (Jan. 10 2019).
  7. Invited speaker at Analysis and Computation in High Dimensions, Hausdorff Institute (Oct. 2018).
  8. Invited speaker at Measuring the Complexity of Computational Content: From Combinatorial Problems to Analysis, Dagstuhl (Sept. 2018).
  9. Invited speaker at the Algebraic and geometric aspects of numerical methods for differential equations, Mittag-Leffler Institute (July 5 2018)
  10. Invited speaker at Isaac Newton Institute (May 24 2018)
  11. Speaking at the University of Oslo (May 14-16 2018, slides).
  12. Speaking at the University of Manchester (May 4 2018).
  13. Invited speaker at  Banff Research Station (April 25 2018).
  14. Invited speaker at the Institut Henri Poincaré (Feb 12 2018).
  15. Organizing the program Approximation, sampling and compression in data science, Isaac Newton Institute (Jan-June 2019).
  16. Organizing the workshop Mathematics of data: Structured representations for sensing, approximation and learning, Alan Turing Institute (May 27-May 31, 2019).
  17. Speaking at LMU Munich (Jan 31, 2018).
  18. Organizing the workshop Inverse Problems Network Meeting 2, Isaac Newton Institute (Nov 23-Nov 24, 2017).
  19. Speaking at the University of Warwick (Nov 15, 2017).
  20. Invited speaker at Generative models, parameter learning and sparsity, Isaac Newton Institute (2017).

Prizes and Awards

1. Whitehead Prize 2019.

2. 2018 IMA Prize in Mathematics and Applications.

3. Leverhulme Prize in Mathematics and Statistics 2017.

4. Royal Society University Research Fellow 2012.

News

1. Our paper How to compute spectra with error control is on the cover of the last June edition of Physical Review Letters.

2. Our book "Compressive Imaging" (with B. Adcock) is coming soon on Cambridge University Press.

3. The Sackler Colloquium at the US National Academy of Sciences: "The Science of deep learning". Watch the presentation "On instabilities in deep learning - Does AI come at a cost?" 

4. SIAM News has our work on the Restricted Isometry Property in Levels in compressed sensing on the front page of the October edition: From Global to Local: Getting More from Compressed Sensing.

5. Our new Nature, Sci. Rep. paper Continuous compressed sensing of inelastic and quasielastic Helium Atom Scattering spectra on continuous/infinite-dimensional compressed sensing in surface scattering is now published. This is the first time compressed sensing is used in Helium Atom Scattering and the first time infinite-dimensional compressed sensing is implemented with real data (see the papers Generalized Sampling and Infinite Dimensional Compressed Sensing and Breaking the coherence barrier: A new theory for compressed sensing where the theoretical ideas were launched).  

6. Siemens validated in practice, using a modified MRI machine, the asymptotic sparsity, asymptotic incoherence and high resolution concepts introduced by our work (see Breaking the coherence barrier: A new theory for compressed sensing and also On asymptotic structure in compressed sensing). From their conclusion:

“[...] The image resolution has been greatly improved [...]. Current results practically demonstrated that it is possible to break the coherence barrier by increasing the spatial resolution in MR acquisitions. This likewise implies that the full potential of the compressed sensing is unleashed only if asymptotic sparsity and asymptotic incoherence is achieved.”

Their work Novel Sampling Strategies for Sparse MR Image Reconstruction was published in May 2014 in the Proceedings of the International Society for Magnetic Resonance in Medicine.

Previous Events

  1. Plenary speaker at the Fourteenth International Conference on Computability and Complexity in Analysis (2017).
  2. Plenary speaker at SPARS (2017).
  3. Plenary speaker at Structured Regularization for High-Dimensional Data Analysis, Institut Henri Poincaré (2017).
  4. Keynote speaker at FoCM: Approximation Theory Workshop (2017).
  5. Invited speaker at FoCM: Information-Based Complexity Workshop (2017).
  6. Invited speaker at Multiscale and High-Dimensional Problems, Oberwolfach (2017).
  7. Plenary speaker at The 14th International workshop on Quantum Chromodynamics (QCD) in extreme conditions (2016).
  8. Plenary speaker at Strobl16: Time-Frequency Analysis and Related Topics (2016).
  9. Plenary speaker at Computational and Analytic Problems in Spectral Theory (2016).
  10. Invited speaker at Low Complexity Models in Signal Processing, Hausdorff Institute (2016).
  11. Plenary speaker at The Bath/RAL Numerical Analysis Day (2015).
  12. Plenary speaker at UCL-Duke Workshop on Sensing and Analysis of High-Dimensional Data (2014).
  13. Plenary speaker at Pseudospectra of operators: spectral singularities, semiclassics, pencils and random matrices (2014).
  14. Invited speaker at FoCM: Real Number Complexity Workshop (2014).
  15. Plenary speaker at iTWIST'14 (2014).
  16. Plenary speaker at French-German Conference on Mathematical Image Analysis, Institut Henri Poincaré (2014).
  17. Invited speaker at The 5th International Conference on Computational Harmonic Analysis (2014).
  18. Invited speaker at Compressed sensing and its Applications (2013).
  19. Plenary speaker at Sparse Representation of Functions: Analytic and Computational Aspects (2012).
  20. Plenary speaker at Sparsity, Localization and Dictionary Learning (2012).

Teaching

Part III course on Compressed Sensing.

Research interests

Functional Analysis (applied), Foundations of Computations, Artificial Intelligence, Compressed Sensing, Optimisation, Operator/Spectral Theory, Numerical Analysis, Computational Harmonic Analysis, Mathematical Signal Processing, Sampling Theory, Inverse Problems, Medical Imaging, Geometric Integration, Operator Algebras

Editor

Proceedings of the Royal Society Series A

Papers 

  1.  L. Thesing, V. Antun, A. C. Hansen, What do AI algorithms actually learn - On false structures in deep learning. 
  2. V. Antun, F. Renna, C. Poon, B. Adcock, A. C. Hansen, On instabilities of deep learning in image reconstruction - Does AI come at a cost? 
  3. B. Adcock, V. Antun,  A. C. Hansen, Uniform recovery in infinite-dimensional compressed sensing and applications to structured binary sampling
  4. J. Schoormans, G. J. Strijkers, A. C. Hansen, A. J. Nederveen, B. F. Coolen, Compressed Sensing MRI with Variable Density Averaging (CS-VDA) Outperforms Full Sampling at Low SNR.
  5. J. Ben-Artzi, A. C. Hansen, O. Nevanlinna, M. Seidel, The Solvability Complexity Index - Computer science and logic meet scientific computing.
  6. B. Roman, A. Bastounis, B. Adcock, A. C. Hansen, On fundamentals of models and sampling in compressed sensing.
  7. J. Ben-Artzi, A. C. Hansen, O. Nevanlinna, M. Seidel, Can everything be computed? - On the Solvability Complexity Index and towers of algorithms.
  8. B. Roman, B. Adcock, A. C. Hansen, On asymptotic structure in compressed sensing.
  9. A. Jones, B. Adcock, A. C. Hansen Analyzing the structure of multidimensional compressed sensing problems through coherence.
  10. M. Colbrook, B. Roman, A. C. Hansen, How to compute spectra with error control
    Phys. Rev. Lett. (to appear) 
  11. A. C. Hansen, B. Roman, On structure and optimisation in computational harmonic analysis - The key aspects in sparse regularisation
    Springer (to appear) 
  12. M. Colbrook, A. C. Hansen, On the Infinite-dimensional QR Algorithm, 
    Numerische Mathematik (to appear)
  13. R. Calderbank, A. C. Hansen, L. Thesing, B. Roman On reconstructions from measurements with binary functions,  
    Springer (to appear) 
  14. L. Thesing, A. C. Hansen, Linear reconstructions and the analysis of the stable sampling rate, 
    Sampl. Theory Signal Image Process. (to appear) 
  15. A. C. Hansen, L. Thesing, On the Stable Sampling rate for binary measurements and wavelet reconstruction, 
    Appl. Comput. Harmon. Anal. 
    (to appear) 
  16. A. Bastounis, B. Adcock, A. C. Hansen, From Global to Local: Getting More from Compressed Sensing,
    SIAM News, 50, no. 8 October 2017
  17. A. C. Hansen, L. Thesing, Sampling from binary measurements - On Reconstructions from Walsh coefficients,
    IEEE 2017 Int. Conf. on Samp. Theory and Appl. 256-260 (2017)
  18. A. Bastounis, A. C. Hansen, On the absence of uniform recovery in many real-world applications of compressed sensing and the RIP & nullspace property in levels.
    SIAM Jour. Imag. Scienc.
    10(1):335-371
  19. B. Adcock, A. C. Hansen, C. Poon, B. Roman, Breaking the coherence barrier: A new theory for compressed sensing,
    Forum of Mathematics, Sigma 5(4):1-84
  20. A. C. Hansen, O. Nevanlinna, Complexity Issues in Computing Spectra, Pseudospectra and Resolvents
    Banach Centre Pub. 112:171-194
  21. B. Adcock, M. Gataric, A. C. Hansen, Density theorems for nonuniform sampling of bandlimited functions using derivatives or bunched  measurements,
    J. Fourier Anal. Appl. 23(6):1311-1347
  22. B. Adcock, A. C. Hansen, B. Roman, A note on compressed sensing of structured sparse wavelet coefficients from subsampled Fourier measurements,
    IEEE Signal Process. Lett.  23(5):732 - 736 
  23. A. Jones , A. Tamtogl, I. Calvo-Almazan, A. C. Hansen, Continuous compressed sensing of inelastic and quasielastic Helium Atom Scattering spectra,
    Nature, Sci. Rep. 6, Art. num.: 27776
  24. A. Jones , A. Tamtogl, I. Calvo-Almazan, A. C. Hansen, Continuous compressed sensing of inelastic and quasielastic Helium Atom Scattering spectra (supplementary material),
    Nature, Sci. Rep. 6, Art. num.: 27776
  25. J. Ben-Artzi, A. C. Hansen, O. Nevanlinna, M. Seidel, New barriers in complexity theory: On the Solvability Complexity Index and towers of algorithms,
    C. R. Acad. Sci. Paris Sér. I Math. 353, no. 10, 931-936
  26. B. Adcock, M. Gataric, A. C. Hansen, Recovering piecewise smooth functions from nonuniform Fourier measuremets,
    Springer Lect. Notes in Comp. Sci. and Eng. 2015
  27. A. Bastounis, A. C. Hansen, On random and deterministic compressed sensing and the Restricted Isometry Property in Levels,
    IEEE 2015 Int. Conf. on Samp. Theory and Appl.
  28. B. Adcock, A. C. Hansen, M. Gataric, Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples,
    Appl. Comput. Harmon. Anal. 
    42(3):508-535
  29. B. Adcock, M. Gataric, A. C. Hansen, Stable nonuniform sampling with weighted Fourier frames and recovery in arbitrary spaces,
    IEEE 2015 Int. Conf. on Samp. Theory and Appl.
  30. B. Adcock, A. C. Hansen, A. Jones, On asymptotic incoherence and its implications for compressed sensing for inverse problems,
    IEEE Trans. Inf. Theory,
    62, no. 2, 1020-1032
  31. B. Adcock, G. Kutyniok, A. C. Hansen, J. Ma, Linear Stable Sampling Rate: Optimality of 2D Wavelet Reconstructions from Fourier Measurements,
    SIAM J. Math. Anal.
    47(2), 1196–1233
  32. B. Adcock, A. C. Hansen, Generalized Sampling and Infinite Dimensional Compressed Sensing,
    Found. Comp. Math. 16, no. 5, 1263-1323
  33. B. Adcock, A. C. Hansen, B. Roman The quest for optimal sampling: computationally efficient, structure-exploiting measurements for compressed sensing,
    Springer
    , 2015
  34. B. Adcock, M. Gataric, A. C. Hansen, On stable reconstructions from univariate nonuniform Fourier measurements,
    SIAM Jour. Imag. Scienc.
    7(3):1690-1723
  35. B. Adcock, A. C. Hansen, B. Roman, G. Teschke, Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum,
    Adv. in Imag. and Electr. Phys.
    vol 182, 187-279, Elsevier, 2014
  36. B. Adcock, A. C. Hansen, A. Shadrin, A stability barrier for reconstructions from Fourier samples,
    SIAM Jour. on Num. Anal. 
    52, no. 1, 125-139
  37. B. Adcock, A. C. Hansen, C. Poon, B. Roman, Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing,
    Proc. of the 10th Int. Conf. on Samp. Theory and Appl., 2013
  38. B. Adcock, A. C. Hansen, C. Poon, Optimal wavelet reconstructions from Fourier samples via generalized sampling,
    Proc. of the 10th Int. Conf. on Samp. Theory and Appl., 2013
  39. B. Adcock, A. C. Hansen, C. Poon, Beyond Consistent Reconstructions: Optimality and Sharp Bounds for Generalized Sampling, and Application to the Uniform Resampling Problem,
    SIAM J. Math. Anal. 
    45, no. 5, 3132-3167
  40. B. Adcock, A. C. Hansen, C. Poon, On optimal wavelet reconstructions from Fourier samples: linearity and universality of the stable sampling rate,
    Appl. Comput. Harmon. Anal.
     36, no. 3, 387-415
  41. B. Adcock, A. C. Hansen, Generalized sampling and the stable and accurate reconstruction of piecewise analytic functions from their Fourier coefficients,
    Math. Comp. 84, 237-270
  42. B. Adcock, A. C. Hansen, E. Herrholz, G. Teschke, Generalized Sampling: Extensions to Frames and Inverse and Ill-Posed Problems,
    Inverse Prob.
    29, no 1, 015008
  43. B. Adcock, A. C. Hansen, Reduced Consistency Sampling in Hilbert Spaces,
    Proc. of the 9th Int. Conf. on Samp. Theory and Appl., 2011
  44. B. Adcock, A. C. Hansen, Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon,
    Appl. Comput. Harmon. Anal.
    32, no. 3, 357-388
  45. B. Adcock, A. C. Hansen, A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases,
    J. Fourier Anal. Appl.
    18, no. 4, 685-716
  46. A. C. Hansen, A theoretical framework for backward error analysis on manifolds,
    J. Geom. Mech. 3, no. 1, 81 - 111
  47. A. C. Hansen, On the Solvability Complexity Index, the n-Pseudospectrum and Approximations of Spectra of Operators,
    J. Amer. Math. Soc.
    24, no. 1, 81-124
  48. A. C. Hansen, J. Strain, On the order of deferred correction,
    Appl. Numer. Math.
    61, no. 8, 961-973
  49. A. C. Hansen, Infinite dimensional numerical linear algebra; theory and applications,
    Proc. R. Soc. Lond. Ser. A. 466, no. 2124, 3539-3559
  50. A. C. Hansen, On the approximation of spectra of linear operators on Hilbert spaces,
    J. Funct. Anal.
    254, no. 8, 2092--2126
  51. A. C. Hansen, J. Strain, Convergence theory for spectral deferred correction,
    Preprint, UC Berkeley

Thesis

A. C. Hansen, On the approximation of spectra of linear Hilbert space operators, PhD Thesis.

Student Awards

  1. Smith-Knight/Rayleigh-Knight Prize 2007, On the approximation of spectra and pseudospectra of linear operators on Hilbert spaces
  2. John Butcher Award 2007 (joint with T. Schmelzer (Oxford)), A theoretical framework for backward error analysis on manifolds.