### Abstract

An underdetermined linear algebraic equation system (Formula presented.), where Φ is an (m00D7n)m003Cn rectangular constant matrix with rank (Formula presented,) and (Formula presented.), has an infinite number of solutions. Diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression seeks a solution satisfying the extra requirement of minimizing a chosen cost function, K. A wide variety of choices of the cost function makes it possible to achieve diverse goals, and hence D-MORPH regression has been successfully applied to solve a range of problems. In this paper, D-MORPH regression is extended to determine a sparse or a nonnegative sparse solution of the vector x. For this purpose, recursive reweighted least-squares (RRLS) minimization is adopted and modified to construct the cost function K for D-MORPH regression. The advantage of sparse and nonnegative sparse D-MORPH regression is that the matrix Φ does not need to have row-full rank, thereby enabling flexibility to search for sparse solutions x with ancillary properties in practical applications. These tools are applied to (a) simulation data for quantum-control-mechanism identification utilizing high dimensional model representation (HDMR) modeling and (b) experimental mass spectral data for determining the composition of an unknown mixture of chemical species.

Original language | English (US) |
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Pages (from-to) | 1885-1914 |

Number of pages | 30 |

Journal | Journal of Mathematical Chemistry |

Volume | 53 |

Issue number | 8 |

DOIs | |

State | Published - Sep 13 2015 |

### All Science Journal Classification (ASJC) codes

- Chemistry(all)
- Applied Mathematics

### Keywords

- D-MORPH regression
- IRLS
- Least-squares regression
- Mass spectrum analysis
- Quantum-control-mechanism identification
- RRLS
- Underdetermined system

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## Cite this

*Journal of Mathematical Chemistry*,

*53*(8), 1885-1914. https://doi.org/10.1007/s10910-015-0527-1