{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,29]],"date-time":"2025-11-29T16:19:04Z","timestamp":1764433144297,"version":"3.40.5"},"reference-count":10,"publisher":"Walter de Gruyter GmbH","issue":"4","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11331004","11421110002"],"award-info":[{"award-number":["11331004","11421110002"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We describe the minimax reconstruction rates in linear ill-posed\nequations in Hilbert space when smoothness is given in terms of\ngeneral source sets. The underlying fundamental result, the minimax\nrate on ellipsoids, is proved similarly to the seminal study by\nD.\u2009L. Donoho, R.\u2009C. Liu, and B. MacGibbon [4]. These\nauthors highlighted the special role of the truncated series\nestimator, and for such estimators the risk can explicitly be given.\nWe provide several examples, indicating results for\nstatistical estimation in ill-posed problems in Hilbert space.<\/jats:p>","DOI":"10.1515\/cmam-2017-0055","type":"journal-article","created":{"date-parts":[[2017,12,5]],"date-time":"2017-12-05T22:15:51Z","timestamp":1512512151000},"page":"603-608","source":"Crossref","is-referenced-by-count":6,"title":["Minimax Rates for Statistical Inverse Problems Under General Source Conditions"],"prefix":"10.1515","volume":"18","author":[{"given":"Litao","family":"Ding","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences , Fudan University , Shanghai 200433 , P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Math\u00e9","sequence":"additional","affiliation":[{"name":"Weierstrass Institute , Mohrenstr. 39, 10117 Berlin , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,12,5]]},"reference":[{"key":"2025051309580382224_j_cmam-2017-0055_ref_001_w2aab3b7e3739b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"N.  Bissantz, T.  Hohage, A.  Munk and F.  Ruymgaart,\nConvergence rates of general regularization methods for statistical inverse problems and applications,\nSIAM J. Numer. Anal. 45 (2007), no. 6, 2610\u20132636.","DOI":"10.1137\/060651884"},{"key":"2025051309580382224_j_cmam-2017-0055_ref_002_w2aab3b7e3739b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"L.  Cavalier,\nNonparametric statistical inverse problems,\nInverse Problems 24 (2008), no. 3, Article ID 034004.","DOI":"10.1088\/0266-5611\/24\/3\/034004"},{"key":"2025051309580382224_j_cmam-2017-0055_ref_003_w2aab3b7e3739b1b6b1ab2ab3Aa","unstructured":"D. L.  Donoho, R. C.  Liu and B.  MacGibbon,\nMinimax risk for hyperrectangles,\nTechnical Report 123, Department of Statistics, University of California, Berkeley, 1988."},{"key":"2025051309580382224_j_cmam-2017-0055_ref_004_w2aab3b7e3739b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"D. L.  Donoho, R. C.  Liu and B.  MacGibbon,\nMinimax risk over hyperrectangles, and implications,\nAnn. Statist. 18 (1990), no. 3, 1416\u20131437.","DOI":"10.1214\/aos\/1176347758"},{"key":"2025051309580382224_j_cmam-2017-0055_ref_005_w2aab3b7e3739b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"B.  Laurent, J.-M.  Loubes and C.  Marteau,\nNon asymptotic minimax rates of testing in signal detection with heterogeneous variances,\nElectron. J. Stat. 6 (2012), 91\u2013122.","DOI":"10.1214\/12-EJS667"},{"key":"2025051309580382224_j_cmam-2017-0055_ref_006_w2aab3b7e3739b1b6b1ab2ab6Aa","unstructured":"J.-M.  Loubes and V.  Rivoirard,\nReview of rates of convergence and regularity conditions for inverse problems,\nInt. J. Tomogr. Stat. 11 (2009), no. S09, 61\u201382."},{"key":"2025051309580382224_j_cmam-2017-0055_ref_007_w2aab3b7e3739b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"B. A.  Mair and F. H.  Ruymgaart,\nStatistical inverse estimation in Hilbert scales,\nSIAM J. Appl. Math. 56 (1996), no. 5, 1424\u20131444.","DOI":"10.1137\/S0036139994264476"},{"key":"2025051309580382224_j_cmam-2017-0055_ref_008_w2aab3b7e3739b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"P.  Math\u00e9 and S. V.  Pereverzev,\nGeometry of linear ill-posed problems in variable Hilbert scales,\nInverse Problems 19 (2003), no. 3, 789\u2013803.","DOI":"10.1088\/0266-5611\/19\/3\/319"},{"key":"2025051309580382224_j_cmam-2017-0055_ref_009_w2aab3b7e3739b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"P.  Math\u00e9 and S. V.  Pereverzev,\nRegularization of some linear ill-posed problems with discretized random noisy data,\nMath. Comp. 75 (2006), no. 256, 1913\u20131929.","DOI":"10.1090\/S0025-5718-06-01873-4"},{"key":"2025051309580382224_j_cmam-2017-0055_ref_010_w2aab3b7e3739b1b6b1ab2ac10Aa","unstructured":"M. S.  Pinsker,\nOptimal filtration of square-integrable signals in Gaussian noise,\nProbl. Inf. Transm. 16 (1980), no. 2, 52\u201368."}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/18\/4\/article-p603.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0055\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0055\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T10:00:11Z","timestamp":1747130411000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0055\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,5]]},"references-count":10,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,12,5]]},"published-print":{"date-parts":[[2018,10,1]]}},"alternative-id":["10.1515\/cmam-2017-0055"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2017-0055","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"type":"electronic","value":"1609-9389"},{"type":"print","value":"1609-4840"}],"subject":[],"published":{"date-parts":[[2017,12,5]]}}}