{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T20:27:24Z","timestamp":1775766444933,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,4]],"date-time":"2023-06-04T00:00:00Z","timestamp":1685836800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we present and justify a methodology to solve the Monge\u2013Kantorovich mass transfer problem through Haar multiresolution analysis and wavelet transform with the advantage of requiring a reduced number of operations to carry out. The methodology has the following steps. We apply wavelet analysis on a discretization of the cost function level j and obtain four components comprising one corresponding to a low-pass filter plus three from a high-pass filter. We obtain the solution corresponding to the low-pass component in level j\u22121 denoted by \u03bcj\u22121*, and using the information of the high-pass filter components, we get a solution in level j denoted by \u03bc^j. Finally, we make a local refinement of \u03bc^j and obtain the final solution \u03bcj\u03c3.<\/jats:p>","DOI":"10.3390\/axioms12060555","type":"journal-article","created":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T02:57:47Z","timestamp":1685933867000},"page":"555","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Efficient Method to Solve the Monge\u2013Kantarovich Problem Using Wavelet Analysis"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8063-0752","authenticated-orcid":false,"given":"Juan Rafael","family":"Acosta-Portilla","sequence":"first","affiliation":[{"name":"Instituto de Investigaciones y Estudios Superiores Econ\u00f3micos y Sociales, Universidad Veracruzana, Veracruz 94294, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8973-0637","authenticated-orcid":false,"given":"Carlos","family":"Gonz\u00e1lez-Flores","sequence":"additional","affiliation":[{"name":"Escuela Superior de Ingenier\u00eda Mec\u00e1nica y El\u00e9ctrica Zacatenco, Instituto Polit\u00e9cnico Nacional, Mexico City 07738, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4378-2316","authenticated-orcid":false,"given":"Raquiel Rufino","family":"L\u00f3pez-Mart\u00ednez","sequence":"additional","affiliation":[{"name":"Facultad de Matem\u00e1ticas, Universidad Veracruzana, Veracruz 94294, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6258-8477","authenticated-orcid":false,"given":"Armando","family":"S\u00e1nchez-Nungaray","sequence":"additional","affiliation":[{"name":"Facultad de Matem\u00e1ticas, Universidad Veracruzana, Veracruz 94294, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1137\/S1052623497315768","article-title":"Approximation schemes for infinite linear programs","volume":"8","author":"Lasserre","year":"1998","journal-title":"SIAM J. Optim."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1137\/050623991","article-title":"On solutions to the mass transfer problem","volume":"17","year":"2006","journal-title":"SIAM J. Optim."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1121","DOI":"10.1093\/imanum\/drn076","article-title":"Numerical approximations to the mass transfer problem on compact spaces","volume":"30","year":"2010","journal-title":"IMA J. Numer. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1515\/amcs-2016-0053","article-title":"A metaheuristic for a numerical approximation to the mass transfer problem","volume":"26","year":"2016","journal-title":"Int. J. Appl. Math. Comput. Sci."},{"key":"ref_5","first-page":"233","article-title":"An Efficient Numerical Approximation for the Monge\u2013Kantorovich Mass Transfer Problem","volume":"Volume 9432","author":"Pardalos","year":"2016","journal-title":"Machine Learning, Optimization, and Big Data. MOD 2015"},{"key":"ref_6","first-page":"556","article-title":"Multiresolution analysis, Haar bases, and self-similar tilings of Rn","volume":"38","author":"Madych","year":"1992","journal-title":"Inst. Electr. Electron. Eng. Trans. Inf. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4399","DOI":"10.1109\/TMTT.2022.3197593","article-title":"Millimeter-wave E-plane waveguide bandpass filters based on spoof surface plasmon polaritons","volume":"70","author":"Liu","year":"2022","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1049\/hve2.12258","article-title":"Novel detection approach for thermal defects: Study on its feasibility and application to vehicle cables","volume":"8","author":"Liu","year":"2023","journal-title":"High Volt."},{"key":"ref_9","first-page":"1764175","article-title":"Multiresolution Analysis Applied to the Monge\u2013Kantorovich Problem","volume":"2018","year":"2018","journal-title":"Abstr. Appl. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Haker, S., Tannenbaum, A., and Kikinis, R. (2001). Mass Preserving Mappings and Surface Registration, Springer.","DOI":"10.1007\/3-540-45468-3_15"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1023\/B:VISI.0000036836.66311.97","article-title":"Optimal mass transport for registration and warping","volume":"60","author":"Zhu","year":"2004","journal-title":"Int. J. Comput. Vis."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1137\/080730238","article-title":"An efficient numerical method for the solution of the L2 optimal mass transfer problem","volume":"32","author":"Haber","year":"2010","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"108418","DOI":"10.1016\/j.aml.2022.108418","article-title":"Existence of solutions for the (p, q)-Laplacian equation with nonlocal Choquard reaction","volume":"135","author":"Xie","year":"2023","journal-title":"Appl. Math. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"124821","DOI":"10.1016\/j.amc.2019.124821","article-title":"A novel optimal bipartite consensus control scheme for unknown multi-agent systems via model-free reinforcement learning","volume":"369","author":"Peng","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Walnut, D.F. (2004). An Introduction to Wavelet Analysis, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-0001-7"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/j.acha.2005.07.002","article-title":"Wavelets with composite dilations and their MRA properties","volume":"20","author":"Guo","year":"2006","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/BF02922084","article-title":"Some simple Haar-type wavelets in higher dimensions","volume":"17","author":"Krishtal","year":"2007","journal-title":"J. Geom. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Bazaraa-Mokhtar, S., John, J., and Hanif, D. (2010). Linear Programming and Network Flows, John Wiley & Sons.","DOI":"10.1002\/9780471703778"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Billingsley, P. (1999). 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