Dr Anders Hansen

Anders

Anders leads the Applied Functional and Harmonic Analysis group within the Cambridge Centre for Analysis at DAMTP. He is a Lecturer 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

News

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.

The major effects and benefits of these concepts are summarised on Page 4 of our latest work On asymptotic structure in compressed sensing, which also includes a large number of example experiments.

Teaching

Part III course on Compressed Sensing.

Research interests

Functional Analysis (applied), Operator/Spectral theory, Complexity Theory, Compressed Sensing, Mathematical Signal Processing, Sampling Theory, Computational Harmonic Analysis, Inverse Problems, Medical Imaging, Geometric Integration, Numerical Analysis, C*-algebras

Editor

Proceedings of the Royal Society Series A

Papers 

  1. J. Ben-Artzi, A. C. Hansen, O. Nevanlinna, M. Seidel, The Solvability Complexity Index - Computer science and logic meet scientific computing.
  2. B. Roman, A. Bastounis, B. Adcock, A. C. Hansen, On fundamentals of models and sampling in compressed sensing.
  3. J. Ben-Artzi, A. C. Hansen, O. Nevanlinna, M. Seidel, Can everything be computed? - On the Solvability Complexity Index and towers of algorithms.
  4. A. Bastounis, A. C. Hansen, On the absence of the RIP in real-world applications of compressed sensing and the RIP in levels.
  5. B. Roman, B. Adcock, A. C. Hansen, On asymptotic structure in compressed sensing.
  6. B. Adcock, A. C. Hansen, A. Jones, Analyzing the structure of multidimensional compressed sensing problems through coherence.
  7. B. Adcock, A. C. Hansen, C. Poon, B. Roman, Breaking the coherence barrier: A new theory for compressed sensing.
  8. A. C. Hansen, C. Wong, On the computation of spectra and pseudospectra of self-adjoint and non-self-adjoint Schrodinger operators.
  9. A. C. Hansen, The infinite dimensional QR-algorithm.
  10. B. Adcock, M. Gataric, A. C. Hansen, Density theorems for nonuniform sampling of bandlimited functions using derivatives or bunched  measurements,
    J. Fourier Anal. Appl.
  11. 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. (to appear)
  12. A. Jones , A. Tamtogl, I. Calvo-Almazan, A. C. Hansen, Continuous compressed sensing of inelastic and quasielastic Helium Atom Scattering spectra,
    (to appear)
  13. 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
  14. B. Adcock, M. Gataric, A. C. Hansen, Recovering piecewise smooth functions from nonuniform Fourier measuremets,
    Springer Lect. Notes in Comp. Sci. and Eng. 2015
  15. 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.
  16. 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. (to appear)
  17. 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.
  18. 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
  19. 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
  20. B. Adcock, A. C. Hansen, Generalized Sampling and Infinite Dimensional Compressed Sensing,
    Found. Comp. Math. (online Aug 2015)
  21. B. Adcock, A. C. Hansen, B. Roman The quest for optimal sampling: computationally efficient, structure-exploiting measurements for compressed sensing,
    Springer
    , 2015
  22. B. Adcock, M. Gataric, A. C. Hansen, On stable reconstructions from univariate nonuniform Fourier measurements,
    SIAM Jour. Imag. Scienc.
    7(3):1690-1723
  23. 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
  24. 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
  25. 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
  26. 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
  27. 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
  28. 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
  29. 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
  30. 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
  31. B. Adcock, A. C. Hansen, Reduced Consistency Sampling in Hilbert Spaces,
    Proc. of the 9th Int. Conf. on Samp. Theory and Appl., 2011
  32. A. C. Hansen, O. Nevanlinna, Complexity Issues in Computing Spectra, Pseudospectra and Resolvents,
    Banach Center Publ.
    (to appear)
  33. 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
  34. B. Adcock, A. C. Hansen, A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases,
    J. Fourier Anal. Appl.
    18, no. 4, 685-716
  35. A. C. Hansen, A theoretical framework for backward error analysis on manifolds,
    J. Geom. Mech. 3, no. 1, 81 - 111
  36. 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
  37. A. C. Hansen, J. Strain, On the order of deferred correction,
    Appl. Numer. Math.
    61, no. 8, 961-973
  38. A. C. Hansen, Infinite dimensional numerical linear algebra; theory and applications,
    Proc. R. Soc. Lond. Ser. A. 466, no. 2124, 3539-3559
  39. A. C. Hansen, On the approximation of spectra of linear operators on Hilbert spaces,
    J. Funct. Anal.
    254, no. 8, 2092--2126
  40. 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.

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.