{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,24]],"date-time":"2025-10-24T16:49:02Z","timestamp":1761324542098,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,1]],"date-time":"2023-08-01T00:00:00Z","timestamp":1690848000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Higher Education of the Russian Federation","award":["075-15-2020-799"],"award-info":[{"award-number":["075-15-2020-799"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>Problems with interval uncertainties arise in many applied fields. The authors have earlier developed, tested, and proved an adaptive interpolation algorithm for solving this class of problems. The algorithm\u2019s idea consists of constructing a piecewise polynomial function that interpolates the dependence of the problem solution on point values of interval parameters. The classical version of the algorithm uses polynomial full grid interpolation and, with a large number of uncertainties, the algorithm becomes difficult to apply due to the exponential growth of computational costs. Sparse grid interpolation requires significantly less computational resources than interpolation on full grids, so their use seems promising. A representative number of examples have previously confirmed the effectiveness of using adaptive sparse grids with a linear basis in the adaptive interpolation algorithm. The purpose of this paper is to apply adaptive sparse grids with a nonlinear basis for modeling dynamic systems with interval parameters. The corresponding interpolation polynomials on the quadratic basis and the fourth-degree basis are constructed. The efficiency, performance, and robustness of the proposed approach are demonstrated on a representative set of problems.<\/jats:p>","DOI":"10.3390\/computation11080149","type":"journal-article","created":{"date-parts":[[2023,8,1]],"date-time":"2023-08-01T09:06:44Z","timestamp":1690880804000},"page":"149","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Adaptive Sparse Grids with Nonlinear Basis in Interval Problems for Dynamical Systems"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0364-8665","authenticated-orcid":false,"given":"Alexander Yu.","family":"Morozov","sequence":"first","affiliation":[{"name":"Federal Research Center \u201cComputer Science and Control\u201d of the Russian Academy of Sciences, St. Vavilova, 44, Bld. 2, 119333 Moscow, Russia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0998-7975","authenticated-orcid":false,"given":"Dmitry L.","family":"Reviznikov","sequence":"additional","affiliation":[{"name":"Federal Research Center \u201cComputer Science and Control\u201d of the Russian Academy of Sciences, St. Vavilova, 44, Bld. 2, 119333 Moscow, Russia"},{"name":"Moscow Aviation Institute, National Research University, Volokolamskoe Hwy., 4, 125993 Moscow, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Moore, R.E., Kearfott, R.B., and Cloud, M.J. (2009). Introduction to Interval Analysis, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9780898717716"},{"key":"ref_2","unstructured":"Dobronets, B.S. (2007). Interval Mathematics, Krasnoyarsk State University."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Shary, S. (2020). Interval Regularization for Inaccurate Linear Algebraic Equations, Springer. Chapter in the book: Beyond Traditional Probabilistic Data Processing Techniques: Interval, Fuzzy etc. Methods and Their Applications.","DOI":"10.1007\/978-3-030-31041-7_21"},{"key":"ref_4","unstructured":"Herzberger, J. (1994). Topics in Validated Computations, Elsevier."},{"key":"ref_5","unstructured":"Chernousko, F.L. (1998). Evaluation of Phase States of Dynamic Systems. The Method of Ellipsoids, Science."},{"key":"ref_6","unstructured":"Kaucher, E.W., Kulisch, U.W., and Ullrich, C. (1987). Computer Arithmetic: Scientific Computation and Programming Languages, Wiley-Teubner Series in Computer Science."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/978-3-7091-6918-6_14","article-title":"The Wrapping Effect, Ellipsoid Arithmetic, Stability and Confidence Regions","volume":"9","author":"Neumaier","year":"1993","journal-title":"Comput. Suppl."},{"key":"ref_8","unstructured":"Makino, K., and Berz, M. (2017). Numerical Software Verification, Proceedings of the 10th International Workshop, NSV 2017, Heidelberg, Germany, 22\u201323 July 2017, Springer International Publishing."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1137\/050638448","article-title":"On Taylor model based integration of ODEs","volume":"45","author":"Neher","year":"2007","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_10","first-page":"102","article-title":"Guaranteed Methods of Ordinary Differential Equations Solution on the Basis of Transformation of Analytical Formulas","volume":"8","author":"Rogalev","year":"2003","journal-title":"Vychisl. Tekhnol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"012165","DOI":"10.1088\/1742-6596\/1441\/1\/012165","article-title":"Estimates of the accuracy of numerical solutions using regularization","volume":"1441","author":"Rogalev","year":"2020","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_12","unstructured":"Ermakov, S.M., and Mikhailov, G.A. (1982). Statistical Modeling, Science."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1134\/S0012266118070121","article-title":"Adaptive Interpolation Algorithm Based on a kd-Tree for Numerical Integration of Systems of Ordinary Differential Equations with Interval Initial Conditions","volume":"54","author":"Morozov","year":"2018","journal-title":"Differ. Equ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1016\/j.cnsns.2018.08.004","article-title":"Steady-state response analysis of cracked rotors with uncertain-but-bounded parameters using a polynomial surrogate method","volume":"68","author":"Fu","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"115047","DOI":"10.1016\/j.jsv.2019.115047","article-title":"Response analysis of an accelerating unbalanced rotating system with both random and interval variables","volume":"466","author":"Fu","year":"2020","journal-title":"J. Sound Vib."},{"key":"ref_16","unstructured":"Nickel, K. (1985). Interval Mathematics, Proceedings of the International Symposium, Freiburg i.Br., Federal Republic of Germany, 23\u201326 September 1985, Springer."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"935","DOI":"10.1134\/S0012266120070125","article-title":"Analysis and Optimization of an Adaptive Interpolation Algorithm for the Numerical Solution of a System of Ordinary Differential Equations with Interval Parameters","volume":"56","author":"Morozov","year":"2020","journal-title":"Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1134\/S2070048219040100","article-title":"Adaptive Interpolation Algorithm Based on a kd-Tree for the Problems of Chemical Kinetics with Interval Parameters","volume":"11","author":"Morozov","year":"2019","journal-title":"Math. Model. Comput. Simul."},{"key":"ref_19","first-page":"479","article-title":"Modeling of Dynamic Systems with Interval Parameters in the Presence of Singularities","volume":"16","author":"Morozov","year":"2020","journal-title":"Russ. J. Nonlinear Dyn."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Morozov, A.Y., Zhuravlev, A.A., and Reviznikov, D.L. (2021). Sparse Grid Adaptive Interpolation in Problems of Modeling Dynamic Systems with Interval Parameters. Mathematics, 9.","DOI":"10.3390\/math9040298"},{"key":"ref_21","first-page":"1042","article-title":"Quadrature and Interpolation Formulae on Tensor Products of Certain Classes of Functions","volume":"148","author":"Smoliak","year":"1963","journal-title":"Dokl. Akad. Nauk. Sssr"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1134\/S0012266121070107","article-title":"Adaptive Interpolation Algorithm on Sparse Meshes for Numerical Integration of Systems of Ordinary Differential Equations with Interval Uncertainties","volume":"57","author":"Morozov","year":"2021","journal-title":"Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1387","DOI":"10.1134\/S0965542521090098","article-title":"Adaptive Interpolation Algorithm Using TT-Decomposition for Modeling Dynamical Systems with Interval Parameters","volume":"61","author":"Gidaspov","year":"2021","journal-title":"Comput. Math. Math. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2295","DOI":"10.1137\/090752286","article-title":"Tensor-train decomposition","volume":"33","author":"Oseledets","year":"2011","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.laa.2009.07.024","article-title":"TT-cross approximation for multidimensional arrays","volume":"432","author":"Oseledets","year":"2010","journal-title":"Linear Algebra Its Appl."},{"key":"ref_26","unstructured":"Yserentant, H. (1991, January 8\u201312). Hierarchical bases. Proceedings of the ICIAM 91: Second International Conference on Industrial and Applied Mathematics, Washington, DC, USA."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.jedc.2014.03.003","article-title":"Smolyak method for solving dynamic economic models: Lagrange interpolation, anisotropic grid and adaptive domain","volume":"44","author":"Judd","year":"2014","journal-title":"J. Econ. Dyn. Control"},{"key":"ref_28","unstructured":"Bungatrz, H.J. (1998). Finite Elements of Higher Order on Sparse Grids, Shaker Verlag."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Bungartz, H.J., and Dirnstorfer, S. (2004, January 6\u20139). Higher Order Quadrature on Sparse Grids. Proceedings of the Computational Science\u2014ICCS 2004, Krakow, Poland.","DOI":"10.1007\/978-3-540-25944-2_52"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1134\/S1995423908040083","article-title":"Randomized algorithms in interval global optimization","volume":"1","author":"Shary","year":"2008","journal-title":"Numer. Anal. Appl."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/11\/8\/149\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:23:54Z","timestamp":1760127834000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/11\/8\/149"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,1]]},"references-count":30,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["computation11080149"],"URL":"https:\/\/doi.org\/10.3390\/computation11080149","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2023,8,1]]}}}