# Dr Anders Hansen

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

## News

1. 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).

2. 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

- J. Ben-Artzi, A. C. Hansen, O. Nevanlinna, M. Seidel, The Solvability Complexity Index - Computer science and logic meet scientific computing.
- B. Roman, A. Bastounis, B. Adcock, A. C. Hansen,
*On fundamentals of models and sampling in compressed sensing.* - J. Ben-Artzi, A. C. Hansen, O. Nevanlinna, M. Seidel, Can everything be computed? - On the Solvability Complexity Index and towers of algorithms.
- B. Roman, B. Adcock, A. C. Hansen,
*On asymptotic structure in compressed sensing.* - A. Jones, B. Adcock, A. C. Hansen
*Analyzing the structure of multidimensional compressed sensing problems through coherence*. - A. C. Hansen, C. Wong,
*On the computation of spectra and pseudospectra of self-adjoint and non-self-adjoint Schrodinger operators.* - A. C. Hansen,
*The infinite dimensional QR-algorithm*. - 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.*(to appear)

SIAM Jour. Imag. Scienc. - B. Adcock, A. C. Hansen, C. Poon, B. Roman,
*Breaking the coherence barrier: A new theory for compressed sensing,*

*Forum of Mathematics, Sigma (to appear)* - B. Adcock, M. Gataric, A. C. Hansen,
*Density theorems for nonuniform sampling of bandlimited functions using derivatives or bunched measurements,*

*J. Fourier Anal. Appl. (to appear)*

*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*

*A. Jones , A. Tamtogl, I. Calvo-Almazan, A. C. Hansen,**Continuous compressed sensing of inelastic and quasielastic Helium Atom Scattering spectra**,*

*Nature, Sci. Rep. (online June 2016)*

*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. (online June 2016)**J. Ben-Artzi, A. C. Hansen, O. Nevanlinna, M. Seidel,**New barriers in complexity theory: On the Solvability Complexity Index and towers of algorithms**,*

353, no. 10, 931-936*C. R. Acad. Sci. Paris Sér. I Math.**B. Adcock, M. Gataric, A. C. Hansen, Recovering piecewise smooth functions from nonuniform Fourier measuremets,*

Springer Lect. Notes in Comp. Sci. and Eng. 2015- 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.* - 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) - 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.* - B. Adcock, A. C. Hansen, A. Jones,
*On asymptotic incoherence and its implications for compressed sensing for inverse problems,*62, no. 2, 1020-1032

IEEE Trans. Inf. Theory, - B. Adcock, G. Kutyniok, A. C. Hansen, J. Ma,
*Linear Stable Sampling Rate: Optimality of 2D Wavelet Reconstructions from Fourier Measurements*,47(2), 1196–1233

SIAM J. Math. Anal. - B. Adcock, A. C. Hansen,
*Generalized Sampling and Infinite Dimensional Compressed Sensing*,

*Found. Comp. Math. (online Aug 2015)* - B. Adcock, A. C. Hansen, B. Roman
*The quest for optimal sampling: computationally efficient, structure-exploiting measurements for compressed sensing*,

Springer2015*,* - B. Adcock, M. Gataric, A. C. Hansen,
*On stable reconstructions from univariate nonuniform Fourier measurements*,7(3):1690-1723

SIAM Jour. Imag. Scienc. - 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. - B. Adcock, A. C. Hansen, A. Shadrin,
*A stability barrier for reconstructions from Fourier samples**,*52, no. 1, 125-139

SIAM Jour. on Num. Anal. - 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 - 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 - 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. - B. Adcock, A. C. Hansen, C. Poon,
*On optimal wavelet reconstructions from Fourier samples: linearity and universality of the stable sampling rate**,*36, no. 3, 387-415

Appl. Comput. Harmon. Anal. - 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 - B. Adcock, A. C. Hansen, E. Herrholz, G. Teschke,
*Generalized Sampling: Extensions to Frames and Inverse and Ill-Posed Problems*,

Inverse Prob. - B. Adcock, A. C. Hansen,
*Reduced Consistency Sampling in Hilbert Spaces*,

*Proc. of the 9th Int. Conf. on Samp. Theory and Appl., 2011* - A. C. Hansen, O. Nevanlinna,
*Complexity Issues in Computing Spectra, Pseudospectra and Resolvents*,

Banach Center Publ. - B. Adcock, A. C. Hansen,
*Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon*,

Appl. Comput. Harmon. Anal. - B. Adcock, A. C. Hansen,
*A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases*,

J. Fourier Anal. Appl. - A. C. Hansen,
*A theoretical framework for backward error analysis on manifolds*,

*J. Geom. Mech.*3, no. 1, 81 - 111 - A. C. Hansen,
*On the Solvability Complexity Index, the n-Pseudospectrum and Approximations of Spectra of Operators*,

J. Amer. Math. Soc. - A. C. Hansen, J. Strain,
*On the order of deferred correction*,

Appl. Numer. Math. - A. C. Hansen,
*Infinite dimensional numerical linear algebra; theory and applications*,

*Proc. R. Soc. Lond. Ser. A.*466, no. 2124, 3539-3559 - A. C. Hansen,
*On the approximation of spectra of linear operators on Hilbert spaces*,

J. Funct. Anal. - 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

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