{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T15:14:06Z","timestamp":1759936446296},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2019,9,10]],"date-time":"2019-09-10T00:00:00Z","timestamp":1568073600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,9,10]],"date-time":"2019-09-10T00:00:00Z","timestamp":1568073600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Max Planck Institute for Developmental Biology"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2019,11]]},"abstract":"<jats:title>Abstract<\/jats:title>\n              <jats:p>Several recent works have developed a new, probabilistic interpretation\nfor numerical algorithms solving linear systems in which the solution is inferred in\na Bayesian framework, either directly or by inferring the unknown action of the\nmatrix inverse. These approaches have typically focused on replicating the behaviour\nof the conjugate gradient method as a prototypical iterative method. In this work,\nsurprisingly general conditions for equivalence of these disparate methods are\npresented. We also describe connections between probabilistic linear solvers and\nprojection methods for linear systems, providing a probabilistic interpretation of a\nfar more general class of iterative methods. In particular, this provides such an\ninterpretation of the generalised minimum residual method. A probabilistic view of\npreconditioning is also introduced. These developments unify the literature on\nprobabilistic linear solvers and provide foundational connections to the literature\non iterative solvers for linear systems. <\/jats:p>","DOI":"10.1007\/s11222-019-09897-7","type":"journal-article","created":{"date-parts":[[2019,9,10]],"date-time":"2019-09-10T08:05:57Z","timestamp":1568102757000},"page":"1249-1263","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Probabilistic linear solvers: a unifying\nview"],"prefix":"10.1007","volume":"29","author":[{"given":"Simon","family":"Bartels","sequence":"first","affiliation":[]},{"given":"Jon","family":"Cockayne","sequence":"additional","affiliation":[]},{"given":"Ilse C. F.","family":"Ipsen","sequence":"additional","affiliation":[]},{"given":"Philipp","family":"Hennig","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,9,10]]},"reference":[{"key":"9897_CR1","unstructured":"Bartels, S., Hennig, P.: Probabilistic approximate least-squares.\nIn: Proceedings of Artificial Intelligence and Statistics (AISTATS)\n(2016)"},{"key":"9897_CR2","unstructured":"Cockayne, J., Oates, C., Sullivan, T.J., Girolami, M.:\nProbabilistic numerical methods for partial differential equations and Bayesian\ninverse problems. \narXiv:1605.07811\n\n (2016)"},{"key":"9897_CR3","unstructured":"Cockayne, J., Oates, C., Sullivan, T.J., Girolami, M.: Bayesian\nprobabilistic numerical methods. \n1702.03673\n\n (2017)"},{"key":"9897_CR4","doi-asserted-by":"crossref","unstructured":"Cockayne, J., Oates, C., Ipsen, I.C.F., Girolami, M.: A Bayesian\nconjugate gradient method. \narXiv:1801.05242\n\n (2018)","DOI":"10.1214\/19-BA1145"},{"issue":"01","key":"9897_CR5","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1017\/s0269964800000255","volume":"1","author":"P Diaconis","year":"1987","unstructured":"Diaconis, P., Shahshahani, M.: The subgroup algorithm for\ngenerating uniform random variables. Probab. Eng. Inf. Sci. 1(01), 15 (1987). \nhttps:\/\/doi.org\/10.1017\/s0269964800000255","journal-title":"Probab. Eng. Inf. Sci."},{"key":"9897_CR6","volume-title":"Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences","author":"GH Golub","year":"2013","unstructured":"Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins\nStudies in the Mathematical Sciences, 4th edn. Johns Hopkins University Press,\nBaltimore (2013)","edition":"4"},{"issue":"1","key":"9897_CR7","doi-asserted-by":"publisher","first-page":"234","DOI":"10.1137\/140955501","volume":"25","author":"P Hennig","year":"2015","unstructured":"Hennig, P.: Probabilistic interpretation of linear solvers. SIAM J.\nOptim. 25(1), 234\u2013260 (2015). \nhttps:\/\/doi.org\/10.1137\/140955501","journal-title":"SIAM J. Optim."},{"key":"9897_CR8","doi-asserted-by":"crossref","first-page":"20150142","DOI":"10.1098\/rspa.2015.0142","volume":"471","author":"P Hennig","year":"2015","unstructured":"Hennig, P., Osborne, M.A., Girolami, M.: Probabilistic numerics and\nuncertainty in computations. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci.471, 20150142 (2015)","journal-title":"Proc. R. Soc. Lond. A Math. Phys. Eng. Sci."},{"key":"9897_CR9","doi-asserted-by":"publisher","unstructured":"Karvonen, T., Sarkka, S.: Classical quadrature rules via gaussian\nprocesses. In: IEEE 27th International Workshop on Machine Learning for Signal\nProcessing (MLSP). IEEE (2017). \nhttps:\/\/doi.org\/10.1109\/mlsp.2017.8168195","DOI":"10.1109\/mlsp.2017.8168195"},{"key":"9897_CR10","unstructured":"Kersting, H., Sullivan, T.J., Hennig, P.: Convergence rates of\nGaussian ODE filters. \narXiv:1807.09737\n\n, 7 (2018)"},{"key":"9897_CR11","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199655410.001.0001","volume-title":"Krylov Subspace Methods. Principles and Analysis","author":"J Liesen","year":"2012","unstructured":"Liesen, J., Strakos, Z.: Krylov Subspace Methods. Principles and\nAnalysis. Oxford University Press, Oxford (2012). \nhttps:\/\/doi.org\/10.1093\/acprof:oso\/9780199655410.001.0001"},{"key":"9897_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/b98874","volume-title":"Numerical Optimization","author":"J Nocedal","year":"1999","unstructured":"Nocedal, J., Wright, S.J.: Numerical Optimization. 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In: Advances in Neural Information Processing Systems,\nvol. 27, pp. 739\u2013747. Curran Associates, Inc. (2014). \nhttp:\/\/papers.nips.cc\/paper\/5451-probabilistic-ode-solvers-with-runge-kutta-means.pdf"},{"issue":"1","key":"9897_CR16","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/s11222-017-9798-7","volume":"29","author":"M Schober","year":"2019","unstructured":"Schober, M., S\u00e4rkk\u00e4, S., Hennig, P.: A probabilistic model for the\nnumerical solution of initial value problems. Stat. Comput. 29(1), 99\u2013122 (2019)","journal-title":"Stat. Comput."},{"key":"9897_CR17","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1016\/j.apnum.2014.02.006","volume":"81","author":"KM Soodhalter","year":"2014","unstructured":"Soodhalter, K.M., Szyld, D.B., Xue, F.: Krylov subspace recycling\nfor sequences of shifted linear systems. Appl. Numer. Math. 81, 105\u2013118 (2014). \nhttps:\/\/doi.org\/10.1016\/j.apnum.2014.02.006","journal-title":"Appl. Numer. Math."},{"key":"9897_CR18","unstructured":"Xi, X., Briol, F.-X., Girolami, M.: Bayesian quadrature for\nmultiple related integrals. 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