{"title":"Some Comments on Preconditioning for Normal Equations and Least Squares","authors":"A. Wathen","doi":"10.1137/20m1387948","DOIUrl":"https://doi.org/10.1137/20m1387948","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77377388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Delayed Gradient Methods for Symmetric and Positive Definite Linear Systems","authors":"Qinmeng Zou, F. Magoulès","doi":"10.1137/20m1321140","DOIUrl":"https://doi.org/10.1137/20m1321140","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80829303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Deforming $||.||_{1}$ into $||.||_{infty}$ via Polyhedral Norms: A Pedestrian Approach","authors":"M. Gaudioso, J. Hiriart-Urruty","doi":"10.1137/21m1391481","DOIUrl":"https://doi.org/10.1137/21m1391481","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73701075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Analyzing Pattern Formation in the Gray-Scott Model: An XPPAUT Tutorial","authors":"Demi L. Gandy, M. R. Nelson","doi":"10.1137/21m1402868","DOIUrl":"https://doi.org/10.1137/21m1402868","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85204860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Federal student loans are fixed-rate debt contracts with three main special features: (i) borrowers can use income-driven schemes to make payments proportional to their income above subsistence, (ii) after several years of good standing, the remaining balance is forgiven but taxed as ordinary income, and (iii) accrued interest is simple, i.e., not capitalized. For a very small loan, the cost-minimizing repayment strategy dictates maximum payments until full repayment, forgoing both income-driven schemes and forgiveness. For a very large loan, the minimal payments allowed by income-driven schemes are optimal. For intermediate balances, the optimal repayment strategy may entail an initial period of minimum payments to exploit the non-capitalization of accrued interest, but when the principal is being reimbursed maximal payments always precede minimum payments. Income-driven schemes and simple accrued interest mostly benefit borrowers with very large balances. MSC (2010): 91G20, 91G80.
{"title":"Minimizing the Repayment Cost of Federal Student Loans","authors":"P. Guasoni, Yu‐Jui Huang","doi":"10.1137/22M1505840","DOIUrl":"https://doi.org/10.1137/22M1505840","url":null,"abstract":". Federal student loans are fixed-rate debt contracts with three main special features: (i) borrowers can use income-driven schemes to make payments proportional to their income above subsistence, (ii) after several years of good standing, the remaining balance is forgiven but taxed as ordinary income, and (iii) accrued interest is simple, i.e., not capitalized. For a very small loan, the cost-minimizing repayment strategy dictates maximum payments until full repayment, forgoing both income-driven schemes and forgiveness. For a very large loan, the minimal payments allowed by income-driven schemes are optimal. For intermediate balances, the optimal repayment strategy may entail an initial period of minimum payments to exploit the non-capitalization of accrued interest, but when the principal is being reimbursed maximal payments always precede minimum payments. Income-driven schemes and simple accrued interest mostly benefit borrowers with very large balances. MSC (2010): 91G20, 91G80.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79540102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generating Resonant and Repeated Root Solutions to Ordinary Differential Equations Using Perturbation Methods","authors":"Bernardo Gouveia, H. Stone","doi":"10.1137/21m1395922","DOIUrl":"https://doi.org/10.1137/21m1395922","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72475552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hamiltonicity of Cubic Planar Graphs with Bounded Face Sizes","authors":"Frantisek Kardos","doi":"10.1137/22m1476915","DOIUrl":"https://doi.org/10.1137/22m1476915","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78429685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dimensionality Reduction via Dynamical Systems: The Case of t-SNE","authors":"G. Linderman, S. Steinerberger","doi":"10.1137/21m1446769","DOIUrl":"https://doi.org/10.1137/21m1446769","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88818900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We investigate the use of reduced precision arithmetic to solve the linear equation for the Newton step. If one 3 neglects the backward error in the linear solve, then well-known convergence theory implies that using single precision in the 4 linear solve has very little negative e(cid:11)ect on the nonlinear convergence rate. 5 However, if one considers the e(cid:11)ects of backward error, then the usual textbook estimates are very pessimistic and even the 6 state-of-the-art estimates using probabilistic rounding analysis do not fully conform to experiments. We report on experiments 7 with a speci(cid:12)c example. We store and factor Jacobians in double, single, and half precision. In the single precision case we 8 observe that the convergence rates for the nonlinear iteration do not degrade as the dimension increases and that the nonlinear 9 iteration statistics are essentially identical to the double precision computation. In half precision we see that the nonlinear 10 convergence rates, while poor, do not degrade as the dimension increases. 11 Audience. This paper is intended for students who have completed or are taking an entry-level graduate course in 12 numerical analysis and for faculty who teach numerical analysis. The important ideas in the paper are O notation, (cid:13)oating 13 point precision, backward error in linear solvers, and Newton’s method.
{"title":"Newton's Method in Mixed Precision","authors":"C. Kelley","doi":"10.1137/20m1342902","DOIUrl":"https://doi.org/10.1137/20m1342902","url":null,"abstract":". We investigate the use of reduced precision arithmetic to solve the linear equation for the Newton step. If one 3 neglects the backward error in the linear solve, then well-known convergence theory implies that using single precision in the 4 linear solve has very little negative e(cid:11)ect on the nonlinear convergence rate. 5 However, if one considers the e(cid:11)ects of backward error, then the usual textbook estimates are very pessimistic and even the 6 state-of-the-art estimates using probabilistic rounding analysis do not fully conform to experiments. We report on experiments 7 with a speci(cid:12)c example. We store and factor Jacobians in double, single, and half precision. In the single precision case we 8 observe that the convergence rates for the nonlinear iteration do not degrade as the dimension increases and that the nonlinear 9 iteration statistics are essentially identical to the double precision computation. In half precision we see that the nonlinear 10 convergence rates, while poor, do not degrade as the dimension increases. 11 Audience. This paper is intended for students who have completed or are taking an entry-level graduate course in 12 numerical analysis and for faculty who teach numerical analysis. The important ideas in the paper are O notation, (cid:13)oating 13 point precision, backward error in linear solvers, and Newton’s method.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":null,"pages":null},"PeriodicalIF":10.2,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86218883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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