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, a Royal Society University Research Fellow and also a Fellow of Peterhouse College.

Tel: +44 1223 760403
Email: ach70 [at] cam.ac.uk

Resume

News

Siemens validated in practice, using a modified MRI machine, the asymptotic sparsity, asymptotic incoherence and high resolution ideas 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:

“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 ideas are summarised in Section 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 (2014). For last year's course and slides, click here.

Research interests

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

Papers

  1. B. Roman, B. Adcock, A. Hansen, On asymptotic structure in compressed sensing
  2. B. Adcock, A. C. Hansen, A. Jones, Analyzing the Structure of Multidimensional Compressed Sensing Problems through Coherence.
  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. B. Adcock, A. C. Hansen, C. Poon, B. Roman, Breaking the coherence barrier: A new theory for compressed sensing.
  5. B. Adcock, A. C. Hansen, C. Poon, M. Gataric, Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples.
  6. B. Adcock, A. C. Hansen, A. Jones, On asymptotic incoherence and its implications for compressed sensing for inverse problems.
  7. A. C. Hansen, C. Wong, On the computation of spectra and pseudospectra of self-adjoint and non-self-adjoint Schrodinger operators.
  8. A. C. Hansen, The infinite dimensional QR-algorithm.
  9. B. Adcock, G. Kutyniok, A. C. Hansen, J. Ma, Linear Stable Sampling Rate: Optimality of 2D Wavelet Reconstructions from Fourier Measurements, SIAM Jour. on Math. Anal.
  10. B. Adcock, A. C. Hansen, Generalized Sampling and Infinite Dimensional Compressed Sensing, Found. Comp. Math.
  11. B. Adcock, A. C. Hansen, The quest for optimal sampling: computationally efficient, structure-exploiting measurements for compressed sensing, Springer, (to appear)
  12. B. Adcock, M. Gataric, A. C. Hansen, On stable reconstructions from univariate nonuniform Fourier measurements, SIAM Jour. Imag. Scienc. (to appear)
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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 Jour. on Math. Anal. 45, no. 5, 3132-3167
  18. 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
  19. B. Adcock, A. C. Hansen, Generalized sampling and the stable and accurate reconstruction of piecewise analytic functions from their Fourier coefficients, Math. Comp. (to appear)
  20. 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
  21. B. Adcock, A. C. Hansen, Reduced Consistency Sampling in Hilbert Spaces, Proc. of the 9th Int. Conf. on Samp. Theory and Appl., 2011
  22. A. C. Hansen, O. Nevanlinna, Complexity Issues in Computing Spectra, Pseudospectra and Resolvents, Banach Center Publ. (to appear)
  23. 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
  24. B. Adcock, A. C. Hansen, A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases, J. Fourier Anal. Appl. 18, no. 4, 685-716
  25. A. C. Hansen, A theoretical framework for backward error analysis on manifolds, J. Geom. Mech. 3, no. 1, 81 - 111
  26. 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
  27. A. C. Hansen, J. Strain, On the order of deferred correction, Appl. Numer. Math. 61, no. 8, 961-973
  28. A. C. Hansen, Infinite dimensional numerical linear algebra; theory and applications, Proc. R. Soc. Lond. Ser. A. 466, no. 2124, 3539-3559
  29. A. C. Hansen, On the approximation of spectra of linear operators on Hilbert spaces, J. Funct. Anal. 254, no. 8, 2092--2126
  30. 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.