{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T07:37:22Z","timestamp":1771486642541,"version":"3.50.1"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,4,6]],"date-time":"2022-04-06T00:00:00Z","timestamp":1649203200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,4,6]],"date-time":"2022-04-06T00:00:00Z","timestamp":1649203200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2022,6]]},"DOI":"10.1007\/s10444-022-09928-4","type":"journal-article","created":{"date-parts":[[2022,4,6]],"date-time":"2022-04-06T10:21:12Z","timestamp":1649240472000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations"],"prefix":"10.1007","volume":"48","author":[{"given":"Jun","family":"Liu","sequence":"first","affiliation":[]},{"given":"Xiang-Sheng","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Shu-Lin","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Tao","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,4,6]]},"reference":[{"key":"9928_CR1","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1007\/s11075-013-9818-4","volume":"67","author":"R Andreev","year":"2014","unstructured":"Andreev, R.: Space-time discretization of the heat equation. Numer. Algor. 67, 713\u2013731 (2014)","journal-title":"Numer. Algor."},{"key":"9928_CR2","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1002\/nla.1951","volume":"22","author":"R Andreev","year":"2014","unstructured":"Andreev, R., Tobler, C.: Multilevel preconditioning and low-rank tensor iteration for space-time simultaneous discretizations of parabolic PDEs. Numer. Linear Algebra Applic. 22, 317\u2013337 (2014)","journal-title":"Numer. Linear Algebra Applic."},{"key":"9928_CR3","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1090\/S0025-5718-1985-0790649-9","volume":"45","author":"AOH Axelsson","year":"1985","unstructured":"Axelsson, A.O.H., Verwer, J.G.: Boundary value techniques for initial value problems in ordinary differential equations. Math. Comp. 45, 153\u2013171 (1985)","journal-title":"Math. Comp."},{"key":"9928_CR4","doi-asserted-by":"crossref","unstructured":"Balay, S., Abhyankar, S., Adams, M.F., Benson, S., Brown, J., Brune, P., Buschelman, K., Constantinescu, E., Dalcin, L., Dener, A., Eijkhout, V., Gropp, W.D., Hapla, V., Isaac, T., Jolivet, P., Karpeev, D., Kaushik, D., Knepley, M.G., Kong, F., Kruger, S., May, D.A., McInnes, L.C., Mills, R.T., Mitchell, L., Munson, T., Roman, J.E., Rupp, K., Sanan, P., Sarich, J., Smith, B.F., Zampini, S., Zhang, H., Zhang, H., Zhang, J.: PETSc\/TAO users manual, Tech. Report ANL-21\/39 - Revision 3.16 Argonne National Laboratory (2021)","DOI":"10.2172\/1893326"},{"key":"9928_CR5","unstructured":"Balay, S., Abhyankar, S., Adams, M.F., Benson, S., Brown, J., Brune, P., Buschelman, K., Constantinescu, E.M., Dalcin, L., Dener, A., Eijkhout, V., Gropp, W.D., Hapla, V., Isaac, T., Jolivet, P., Karpeev, D., Kaushik, D., Knepley, M.G., Kong, F., Kruger, S., May, D.A., McInnes, L.C., Mills, R.T., Mitchell, L., Munson, T., Roman, J.E., Rupp, K., Sanan, P., Sarich, J., Smith, B.F., Zampini, S., Zhang, H., Zhang, H., Zhang, J: PETSc Web page https:\/\/petsc.org\/ (2021)"},{"key":"9928_CR6","doi-asserted-by":"publisher","first-page":"136","DOI":"10.1016\/j.cpc.2015.08.025","volume":"197","author":"AT Barker","year":"2015","unstructured":"Barker, A.T., Stoll, M.: Domain decomposition in time for PDE-constrained optimization. Comput. Phys. Commun. 197, 136\u2013143 (2015)","journal-title":"Comput. Phys. Commun."},{"key":"9928_CR7","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1016\/0168-9274(93)90043-Q","volume":"11","author":"L Brugnano","year":"1993","unstructured":"Brugnano, L., Mazzia, F., Trigiante, D.: Parallel implementation of BVM methods. Appl. Numer. Math. 11, 115\u2013124 (1993)","journal-title":"Appl. Numer. Math."},{"key":"9928_CR8","volume-title":"Solving Differential Problems by Multistep Initial and Boundary Value Methods","author":"L Brugnano","year":"2003","unstructured":"Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publ., Amsterdam (2003)"},{"key":"9928_CR9","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1007\/s10589-013-9550-y","volume":"56","author":"R Buchholz","year":"2013","unstructured":"Buchholz, R., Engel, H., Kammann, E., Tr\u00f6ltzsch, F.: On the optimal control of the Schl\u00f6gl-model. Comput. Optim. Appl. 56, 153\u2013185 (2013)","journal-title":"Comput. Optim. Appl."},{"key":"9928_CR10","doi-asserted-by":"crossref","unstructured":"Bueler, E: PETSc for Partial Differential Equations: Numerical Solutions in C and Python. SIAM (2020)","DOI":"10.1137\/1.9781611976311"},{"key":"9928_CR11","unstructured":"Caklovic, G., Speck, R., Frank, M.: A parallel implementation of a diagonalization-based parallel-in-time integrator, arXiv:2103.12571 (2021)"},{"key":"9928_CR12","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1007\/BF01990529","volume":"33","author":"D Calvetti","year":"1993","unstructured":"Calvetti, D., Reichel, L.: Fast inversion of Vandermonde-like matrices involving orthogonal polynomials. BIT Numer. Math. 33, 473\u2013484 (1993)","journal-title":"BIT Numer. Math."},{"key":"9928_CR13","doi-asserted-by":"crossref","unstructured":"Chen, F., Hesthaven, J.S., Zhu, X.: On the use of reduced basis methods to accelerate and stabilize the Parareal method. In: Reduced Order Methods for Modeling and Computational Reduction, vol. 9, pp 187\u2013214. Springer, Berlin (2014)","DOI":"10.1007\/978-3-319-02090-7_7"},{"key":"9928_CR14","doi-asserted-by":"crossref","unstructured":"Chopp, D.: Introduction to High Performance Scientific Computing. SIAM (2019)","DOI":"10.1137\/1.9781611975642"},{"key":"9928_CR15","doi-asserted-by":"publisher","first-page":"A52","DOI":"10.1137\/110861002","volume":"35","author":"X Dai","year":"2013","unstructured":"Dai, X., Maday, Y.: Stable parareal in time method for first- and second-order hyperbolic systems. SIAM J. Sci. Comput. 35, A52\u2013A78 (2013)","journal-title":"SIAM J. Sci. Comput."},{"key":"9928_CR16","doi-asserted-by":"crossref","unstructured":"Danieli, F., Southworth, B.S., Wathen, A.J.: Space-time block preconditioning for incompressible flow, arXiv:2101.07003 (2021)","DOI":"10.1137\/21M1390773"},{"key":"9928_CR17","volume-title":"Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms","author":"P Deuflhard","year":"2004","unstructured":"Deuflhard, P.: Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms. Springer, Berlin (2004)"},{"key":"9928_CR18","doi-asserted-by":"publisher","first-page":"105","DOI":"10.2140\/camcos.2012.7.105","volume":"7","author":"M Emmett","year":"2012","unstructured":"Emmett, M., Minion, M.L.: Toward an efficient parallel in time method for partial differential equations. Comm. App. Math. Comp. Sci. 7, 105\u2013132 (2012)","journal-title":"Comm. App. Math. Comp. Sci."},{"key":"9928_CR19","doi-asserted-by":"publisher","first-page":"C635","DOI":"10.1137\/130944230","volume":"36","author":"RD Falgout","year":"2014","unstructured":"Falgout, R.D., Friedhoff, S., Kolev, T.V., MacLachlan, S.P., Schroder, J.B.: Parallel time integration with multigrid. SIAM J. Sci. Comput. 36, C635\u2013C661 (2014)","journal-title":"SIAM J. Sci. Comput."},{"key":"9928_CR20","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/s00791-017-0283-9","volume":"18","author":"RD Falgout","year":"2017","unstructured":"Falgout, R.D., Friedhoff, S., Kolev, T.V., MacLachlan, S.P., Schroder, J.B., Vandewalle, S.: Multigrid methods with space\u2013time concurrency. Comput. Vis. Sci. 18, 123\u2013143 (2017)","journal-title":"Comput. Vis. Sci."},{"key":"9928_CR21","doi-asserted-by":"publisher","first-page":"697","DOI":"10.1002\/nme.1653","volume":"67","author":"C Farhat","year":"2006","unstructured":"Farhat, C., Cortial, J., Dastillung, C., Bavestrello, H.: Time-parallel implicit integrators for the near-real-time prediction of linear structural dynamic responses. Int. J. Numer. Methods Eng. 67, 697\u2013724 (2006)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"9928_CR22","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1093\/qjmam\/7.3.367","volume":"3","author":"L Fox","year":"1954","unstructured":"Fox, L.: A note on the numerical integration of first order differential equations. Quart. J. Mech. Appl. Math. 3, 367\u2013378 (1954)","journal-title":"Quart. J. Mech. Appl. Math."},{"key":"9928_CR23","doi-asserted-by":"publisher","first-page":"232","DOI":"10.1093\/qjmam\/10.2.232","volume":"10","author":"L Fox","year":"1957","unstructured":"Fox, L., Mitchell, A.R.: Boundary value techniques for the numerical solution of initial value problems in ordinary differential equations. Quart. J. Mech. Appl. Math. 10, 232\u2013243 (1957)","journal-title":"Quart. J. Mech. Appl. Math."},{"key":"9928_CR24","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1007\/978-3-319-52389-7_15","volume":"116","author":"MJ Gander","year":"2017","unstructured":"Gander, M.J., Halpern, L.: Time parallelization for nonlinear problems based on diagonalization. Lect. Notes Comput. Sci. Eng. 116, 163\u2013170 (2017)","journal-title":"Lect. Notes Comput. Sci. Eng."},{"key":"9928_CR25","doi-asserted-by":"publisher","first-page":"A220","DOI":"10.1137\/17M1148347","volume":"41","author":"MJ Gander","year":"2019","unstructured":"Gander, M.J., Halpern, L., Rannou, J., Ryan, J.: A direct time parallel solver by diagonalization for the wave equation. SIAM J. Sci. Comput. 41, A220\u2013A245 (2019)","journal-title":"SIAM J. Sci. Comput."},{"key":"9928_CR26","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1553\/etna_vol51s135","volume":"51","author":"A Goddard","year":"2019","unstructured":"Goddard, A., Wathen, A.: A note on parallel preconditioning for all-at-once evolutionary PDEs. Electron Trans. Numer. Anal. 51, 135\u2013150 (2019)","journal-title":"Electron Trans. Numer. Anal."},{"key":"9928_CR27","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1007\/s002110050018","volume":"67","author":"I Gohberg","year":"1994","unstructured":"Gohberg, I., Olshevsky, V.: Fast inversion of Chebyshev\u2013Vandermonde matrices. Numer. Math. 67, 71\u201392 (1994)","journal-title":"Numer. Math."},{"key":"9928_CR28","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1006\/jcom.1997.0442","volume":"13","author":"I Gohberg","year":"1997","unstructured":"Gohberg, I., Olshevsky, V.: The fast generalized Parker-Traub algorithm for inversion of Vandermonde and related matrices. J. Complex. 13, 208\u2013234 (1997)","journal-title":"J. Complex."},{"key":"9928_CR29","doi-asserted-by":"crossref","unstructured":"Gohberg, I., Olshevsky, V.: Fast inversion of Vandermonde and Vandermonde-like matrices. In: Communications, Computation, Control, and Signal Processing. Springer, pp. 205\u2013221 (1997)","DOI":"10.1007\/978-1-4615-6281-8_10"},{"key":"9928_CR30","doi-asserted-by":"publisher","unstructured":"G\u00fcttel, S., Pearson, J.W.: A spectral-in-time Newton-Krylov method for nonlinear PDE-constrained optimization. https:\/\/doi.org\/10.1093\/imanum\/drab011","DOI":"10.1093\/imanum\/drab011"},{"key":"9928_CR31","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1093\/imanum\/8.4.473","volume":"8","author":"NJ Higham","year":"1988","unstructured":"Higham, N.J.: Fast solution of Vandermonde-like systems involving orthogonal polynomials. IMA J. Numer. Anal. 8, 473\u2013486 (1988)","journal-title":"IMA J. Numer. Anal."},{"key":"9928_CR32","doi-asserted-by":"crossref","unstructured":"Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM (2002)","DOI":"10.1137\/1.9780898718027"},{"key":"9928_CR33","doi-asserted-by":"publisher","first-page":"1288","DOI":"10.1137\/100799010","volume":"32","author":"D Kressner","year":"2011","unstructured":"Kressner, D., Tobler, C.: Low-rank tensor Krylov subspace methods for parametrized linear systems. SIAM J. Matrix Anal. Applic. 32, 1288\u20131316 (2011)","journal-title":"SIAM J. Matrix Anal. Applic."},{"key":"9928_CR34","doi-asserted-by":"publisher","first-page":"827","DOI":"10.4208\/nmtma.2018.s09","volume":"11","author":"X Lin","year":"2018","unstructured":"Lin, X., Ng, M., Sun, H.: A separable preconditioner for time-space fractional Caputo-Riesz diffusion equations. Numer. Math. Theor. Meth. Appl. 11, 827\u2013853 (2018)","journal-title":"Numer. Math. Theor. Meth. Appl."},{"key":"9928_CR35","doi-asserted-by":"publisher","first-page":"661","DOI":"10.1016\/S0764-4442(00)01793-6","volume":"332","author":"JL Lions","year":"2001","unstructured":"Lions, J.L., Maday, Y., Turinici, G.: A \u201cparareal\u201d in time discretization of PDE\u2019s. C. R. Acad. Sci. Paris S\u00e9r. I Math. 332, 661\u2013668 (2001)","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math."},{"key":"9928_CR36","doi-asserted-by":"publisher","first-page":"1912","DOI":"10.1137\/19M1309869","volume":"41","author":"J Liu","year":"2020","unstructured":"Liu, J., Wu, S.L.: A fast block \u03b1-circulant preconditoner for all-at-once systems from wave equations. SIAM J. Matrix Anal. Appl. 41, 1912\u20131943 (2020)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"9928_CR37","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1016\/j.crma.2007.09.012","volume":"346","author":"Y Maday","year":"2008","unstructured":"Maday, Y., R\u00f8nquist, E.M.: Parallelization in time through tensor-product space-time solvers. C. R. Acad. Sci Paris S\u00e9r. I Math. 346, 113\u2013118 (2008)","journal-title":"C. R. Acad. Sci Paris S\u00e9r. I Math."},{"key":"9928_CR38","doi-asserted-by":"publisher","first-page":"A1012","DOI":"10.1137\/16M1062016","volume":"40","author":"E McDonald","year":"2018","unstructured":"McDonald, E., Pestana, J., Wathen, A.: Preconditioning and iterative solution of all-at-once systems for evolutionary partial differential equations. SIAM J. Sci. Comput. 40, A1012\u2013A1033 (2018)","journal-title":"SIAM J. Sci. Comput."},{"key":"9928_CR39","doi-asserted-by":"publisher","first-page":"C28","DOI":"10.1137\/18M1172466","volume":"41","author":"M Neum\u00fcller","year":"2019","unstructured":"Neum\u00fcller, M., Smears, I.: Time-parallel iterative solvers for parabolic evolution equations. SIAM J. Sci. Comput. 41, C28\u2013C51 (2019)","journal-title":"SIAM J. Sci. Comput."},{"key":"9928_CR40","doi-asserted-by":"publisher","first-page":"109156","DOI":"10.1016\/j.jcp.2019.109156","volume":"405","author":"H Nguyen","year":"2020","unstructured":"Nguyen, H., Tsai, R.: A stable parareal-like method for the second order wave equation. J. Comput. Phys. 405, 109156 (2020)","journal-title":"J. Comput. Phys."},{"key":"9928_CR41","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719468","volume-title":"Iterative Solution of Nonlinear Equations in Several Variables","author":"JM Ortega","year":"2000","unstructured":"Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM, Philadelphia (2000)"},{"key":"9928_CR42","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/s10915-021-01515-x","volume":"87","author":"D Palitta","year":"2021","unstructured":"Palitta, D.: Matrix equation techniques for certain evolutionary partial differential equations. J. Sci. Comput. 87, 99 (2021)","journal-title":"J. Sci. Comput."},{"key":"9928_CR43","doi-asserted-by":"publisher","first-page":"703","DOI":"10.1090\/S0025-5718-1991-1094957-9","volume":"57","author":"L Reichel","year":"1991","unstructured":"Reichel, L., Opfer, G.: Chebyshev-vandermonde systems. Math. Comput. 57, 703\u2013721 (1991)","journal-title":"Math. Comput."},{"key":"9928_CR44","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00791-018-0296-z","volume":"59","author":"D Ruprecht","year":"2018","unstructured":"Ruprecht, D.: Wave propagation characteristics of Parareal. Comput. Vis. Sci. 59, 1\u201317 (2018)","journal-title":"Comput. Vis. Sci."},{"key":"9928_CR45","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1016\/j.compfluid.2012.02.015","volume":"59","author":"D Ruprecht","year":"2012","unstructured":"Ruprecht, D., Krause, R.: Explicit parallel-in-time integration of a linear acoustic-advection system. Comput. Fluids 59, 72\u201383 (2012)","journal-title":"Comput. Fluids"},{"key":"9928_CR46","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/978-3-319-10705-9_19","volume":"103","author":"J Steiner","year":"2015","unstructured":"Steiner, J., Ruprecht, D., Speck, R., Krause, R.: Convergence of parareal for the Navier-Stokes equations depending on the reynolds number. Lect. Notes Comput. Sci. Eng. 103, 195\u2013202 (2015)","journal-title":"Lect. Notes Comput. Sci. Eng."},{"key":"9928_CR47","doi-asserted-by":"publisher","first-page":"B1","DOI":"10.1137\/130926365","volume":"37","author":"M Stoll","year":"2015","unstructured":"Stoll, M., Breiten, T.: A low-rank in time approach to PDE-constrained optimization. SIAM J. Sci. Comput. 37, B1\u2013B29 (2015)","journal-title":"SIAM J. Sci. Comput."},{"key":"9928_CR48","doi-asserted-by":"publisher","first-page":"644","DOI":"10.1007\/s10915-015-0100-x","volume":"67","author":"SL Wu","year":"2016","unstructured":"Wu, S.L.: Convergence analysis of the Parareal-Euler algorithm for systems of ODEs with complex eigenvalues. J. Sci. Comput. 67, 644\u2013668 (2016)","journal-title":"J. Sci. Comput."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-022-09928-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-022-09928-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-022-09928-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,31]],"date-time":"2023-01-31T20:11:06Z","timestamp":1675195866000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-022-09928-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,6]]},"references-count":48,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,6]]}},"alternative-id":["9928"],"URL":"https:\/\/doi.org\/10.1007\/s10444-022-09928-4","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,4,6]]},"assertion":[{"value":"24 May 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 January 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 April 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}],"article-number":"16"}}