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• W3138128197 abstract "Abstract In this paper we extend the deterministic sublinear FFT algorithm in Plonka et al. (Numer Algorithms 78:133–159, 2018. 10.1007/s11075-017-0370-5 ) for fast reconstruction of M -sparse vectors $${mathbf{x}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>x</mml:mi> </mml:math> of length $$N= 2^J$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>J</mml:mi> </mml:msup> </mml:mrow> </mml:math> , where we assume that all components of the discrete Fourier transform $$hat{mathbf{x}}= {mathbf{F}}_{N} {mathbf{x}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> are available. The sparsity of $${mathbf{x}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>x</mml:mi> </mml:math> needs not to be known a priori, but is determined by the algorithm. If the sparsity M is larger than $$2^{J/2}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> , then the algorithm turns into a usual FFT algorithm with runtime $${mathcal O}(N log N)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>log</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . For $$M^{2} < N$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo><</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> , the runtime of the algorithm is $${mathcal O}(M^2 , log N)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mspace /> <mml:mo>log</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . The proposed modifications of the approach in Plonka et al. (2018) lead to a significant improvement of the condition numbers of the Vandermonde matrices which are employed in the iterative reconstruction. Our numerical experiments show that our modification has a huge impact on the stability of the algorithm. While the algorithm in Plonka et al. (2018) starts to be unreliable for $$M>20$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>></mml:mo> <mml:mn>20</mml:mn> </mml:mrow> </mml:math> because of numerical instabilities, the modified algorithm is still numerically stable for $$M=200$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mn>200</mml:mn> </mml:mrow> </mml:math> ." @default.
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• W3138128197 date "2021-03-11" @default.
• W3138128197 modified "2023-09-26" @default.
• W3138128197 title "Deterministic Sparse Sublinear FFT with Improved Numerical Stability" @default.
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