Sparse spikes super-resolution on thin grids II: the continuous basis pursuit. (English) Zbl 1392.35332
The article investigates the continuous basis pursuit (C-BP) method for sparse super-resolution of spikes. As for the basis persuit (BP) method (Lasso) which solves an inverse problem of an integral equation in Hilbert space by using discrete \(l^1\) approximations (Fourier coefficients), the authors use a first order Taylor approximation of the integral kernel (C-BP) for improving the BP. C-BP turns out to be a constrained Lasso. Optimality conditions are derived and used in the proofs together with duality arguments. C-BP identifies under some constellations the spikes locations with sub-grid accuracy and can avoid that a spike appears by a pair of two dirac masses. However for too thin grids the pair of dirac masses cannot be avoided as known from Lasso. All results are detailed proven. Several illustrative examples underline the presented results and the improvement of the new approach.
For Part I, see [the authors, ibid. 33, No. 5, Article ID 055008, 29 p. (2017; Zbl 1373.65039)].
For Part I, see [the authors, ibid. 33, No. 5, Article ID 055008, 29 p. (2017; Zbl 1373.65039)].
Reviewer: Armin Hoffmann (Ilmenau)
MSC:
| 35R30 | Inverse problems for PDEs |
| 90C25 | Convex programming |
| 90C46 | Optimality conditions and duality in mathematical programming |
| 90C59 | Approximation methods and heuristics in mathematical programming |
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 49N45 | Inverse problems in optimal control |
Keywords:
super-resolution; sparsity; Lasso; convex optimization; spikes recognition; first order approximation of the kernel; continuous basis pursuit (C-BP)Citations:
Zbl 1373.65039Software:
PDCOReferences:
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