Spectral methods for nonlinear functionals and functional differential equations
We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: First, we prove that continuous nonlinear functionals, functional derivatives, and FDEs c...
Gespeichert in:
| Veröffentlicht in: | Research in the mathematical sciences Jg. 8; H. 2 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Cham
Springer International Publishing
01.06.2021
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 2522-0144, 2197-9847 |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Abstract | We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: First, we prove that continuous nonlinear functionals, functional derivatives, and FDEs can be approximated uniformly on any compact subset of a real Banach space admitting a basis by high-dimensional multivariate functions and high-dimensional partial differential equations (PDEs), respectively. Second, we show that the convergence rate of such functional approximations can be exponential, depending on the regularity of the functional (in particular its Fréchet differentiability), and its domain. We also provide necessary and sufficient conditions for consistency, stability and convergence of cylindrical approximations to linear FDEs. These results open the possibility to utilize numerical techniques for high-dimensional systems such as deep neural networks and numerical tensor methods to approximate nonlinear functionals in terms of high-dimensional functions, and compute approximate solutions to FDEs by solving high-dimensional PDEs. Numerical examples are presented and discussed for prototype nonlinear functionals and for an initial value problem involving a linear FDE. |
|---|---|
| AbstractList | We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: First, we prove that continuous nonlinear functionals, functional derivatives, and FDEs can be approximated uniformly on any compact subset of a real Banach space admitting a basis by high-dimensional multivariate functions and high-dimensional partial differential equations (PDEs), respectively. Second, we show that the convergence rate of such functional approximations can be exponential, depending on the regularity of the functional (in particular its Fréchet differentiability), and its domain. We also provide necessary and sufficient conditions for consistency, stability and convergence of cylindrical approximations to linear FDEs. These results open the possibility to utilize numerical techniques for high-dimensional systems such as deep neural networks and numerical tensor methods to approximate nonlinear functionals in terms of high-dimensional functions, and compute approximate solutions to FDEs by solving high-dimensional PDEs. Numerical examples are presented and discussed for prototype nonlinear functionals and for an initial value problem involving a linear FDE. |
| ArticleNumber | 27 |
| Author | Venturi, Daniele Dektor, Alec |
| Author_xml | – sequence: 1 givenname: Daniele orcidid: 0000-0001-8831-8547 surname: Venturi fullname: Venturi, Daniele email: venturi@ucsc.edu organization: Department of Applied Mathematics, University of California Santa Cruz – sequence: 2 givenname: Alec surname: Dektor fullname: Dektor, Alec organization: Department of Applied Mathematics, University of California Santa Cruz |
| BookMark | eNp9kMtKAzEUhoNUsNa-gKsB19GTyyQzSyneoKCgrkMmF53SZtpkZuHbm3YExUVXyTn8X87Jd44moQsOoUsC1wRA3iQOopIYKMEAVJSYn6ApJbXEdcXlJN9LSjEQzs_QPKUVABApGGcwRS-vW2f6qNfFxvWfnU2F72KRB6zb4HQs_BBM33ZBr1Ohg_1TF7b13kUX-jYXbjfofT9doFOfw27-c87Q-_3d2-IRL58fnha3S2yYYD0mDZXMA7XAS21FY7kGq3ndEC-Z0bRhVoBzVmpRVcJxZohnXDBbclZbI9kMXY3vbmO3G1zq1aob4n5PRUtSMyFoCTlFx5SJXUrRebWN7UbHL0VA7eWpUZ7K8tRBnuIZqv5Bpu0Pv8ui2vVxlI1oynPCh4u_Wx2hvgFga4aR |
| CitedBy_id | crossref_primary_10_1038_s41598_025_03093_6 crossref_primary_10_1007_s10915_021_01539_3 |
| Cites_doi | 10.1016/j.jcp.2019.108940 10.1017/S0022112008001821 10.1007/s10208-016-9317-9 10.4064/sm-76-1-1-58 10.1007/BF01023868 10.1137/S1064827501387826 10.1016/j.jcp.2018.08.057 10.1007/s00220-011-1261-6 10.1017/CBO9780511897818.011 10.1016/0001-8708(65)90041-1 10.1007/BF02392270 10.1016/j.jcp.2009.10.043 10.1142/5715 10.1216/RMJ-1976-6-1-1 10.1016/j.jcp.2019.05.024 10.1007/978-3-642-51633-7 10.1142/4319 10.1016/j.jcp.2019.109125 10.1016/0021-9045(81)90023-X 10.1007/BF01578539 10.1512/iumj.1981.30.30011 10.1007/BF01022182 10.1016/0022-1236(90)90147-D 10.1007/978-3-319-67110-9_3 10.4064/sm-57-2-147-190 10.1007/978-3-319-56436-4 10.1007/s00365-008-9017-z 10.1103/PhysRevE.101.013104 10.1016/0021-9045(70)90039-0 10.1098/rspa.2013.0001 10.1016/j.exmath.2018.03.002 10.1088/0305-4470/10/5/011 10.1090/proc/13249 10.1063/1.4827679 10.1137/18M1202670 10.1016/j.jcp.2020.109744 10.1080/00029890.1982.11995506 10.1007/978-3-642-69894-1 10.1016/0034-4877(78)90055-1 10.1016/j.jcp.2017.11.039 10.1063/1.1665649 10.1016/j.jco.2013.10.001 10.1051/m2an/2011045 10.4064/sm-45-1-15-29 10.1090/S0002-9939-1961-0120342-2 10.1515/9781400835348 10.1093/acprof:oso/9780198509233.001.0001 10.1017/S0022112009993685 10.1090/S0002-9904-1972-13048-9 10.4007/annals.2003.157.257 10.1007/978-3-540-30726-6 10.1017/CBO9780511618352 10.1103/PhysRevA.8.423 10.1137/07070111X 10.1016/j.jcp.2018.06.038 10.1073/pnas.43.4.336 10.1515/9783110550962 10.1098/rspa.2011.0186 10.1098/rspa.2013.0754 10.1007/s10915-019-00972-9 10.1007/s00365-013-9186-2 10.1103/PhysRevA.33.467 10.1007/978-3-642-61943-4 10.1006/jfan.1997.3108 10.1016/j.jcp.2013.03.001 10.1016/j.physrep.2017.12.003 10.1023/A:1020503505540 10.1023/B:JOTH.0000029696.94590.94 10.1016/j.jcp.2020.109341 10.1016/j.jcp.2011.01.002 10.1016/j.jfa.2008.03.015 10.1016/j.jcp.2018.10.045 10.1073/pnas.1922204117 10.1090/S0002-9947-1981-0607110-7 10.1016/j.exmath.2010.03.001 10.2140/pjm.1958.8.887 10.1063/1.1694652 10.1201/9780429503559 10.1103/PhysRev.136.B864 |
| ContentType | Journal Article |
| Copyright | The Author(s) 2021 The Author(s) 2021. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
| Copyright_xml | – notice: The Author(s) 2021 – notice: The Author(s) 2021. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
| DBID | C6C AAYXX CITATION |
| DOI | 10.1007/s40687-021-00265-4 |
| DatabaseName | Springer Nature OA Free Journals CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | CrossRef |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 2197-9847 |
| ExternalDocumentID | 10_1007_s40687_021_00265_4 |
| GrantInformation_xml | – fundername: Directorate for Mathematical and Physical Sciences grantid: NSF-DMS 1439786 funderid: http://dx.doi.org/10.13039/100000086 – fundername: Army Research Office (US) grantid: W911NF1810309 |
| GroupedDBID | -EM 0R~ 406 5VS AAAVM AACDK AAHNG AAJBT AASML AATNV AATVU AAUYE AAYZH ABAKF ABDZT ABECU ABFTV ABJNI ABJOX ABKCH ABMQK ABSXP ABTEG ABTKH ABTMW ABXPI ACAOD ACDTI ACGFS ACHSB ACMLO ACOKC ACPIV ACZOJ ADHHG ADINQ ADKNI ADKPE ADTPH ADURQ ADYFF ADZKW AEFQL AEJRE AEMSY AESKC AFBBN AFQWF AGDGC AGJBK AGMZJ AGQEE AGRTI AHBYD AHSBF AHYZX AIAKS AIGIU AILAN AITGF AJZVZ ALMA_UNASSIGNED_HOLDINGS AMKLP AMXSW AMYLF AMYQR ASPBG AXYYD BGNMA C24 C6C CSCUP DPUIP EBLON EBS EJD FIGPU FINBP FNLPD FSGXE GGCAI GROUPED_DOAJ IKXTQ IWAJR J-C JZLTJ KOV LLZTM M4Y M~E NPVJJ NQJWS NU0 O9J OK1 PT4 ROL RSV SISQX SJYHP SNE SNPRN SOHCF SOJ SPISZ SRMVM SSLCW STPWE TSG UOJIU UTJUX UZXMN VFIZW Z88 ZMTXR AAYXX ABBRH ABDBE ABFSG ABRTQ ACSTC AEZWR AFDZB AFHIU AFOHR AHPBZ AHWEU AIXLP AMVHM ATHPR AYFIA CITATION |
| ID | FETCH-LOGICAL-c363t-1b273f02d045ad6bd4a0da49b1f73ca2b3d60eed7a6886e43c1f3463d5439dc73 |
| IEDL.DBID | RSV |
| ISICitedReferencesCount | 9 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000645248100002&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 2522-0144 |
| IngestDate | Thu Sep 25 00:55:16 EDT 2025 Tue Nov 18 22:04:22 EST 2025 Sat Nov 29 06:11:21 EST 2025 Fri Feb 21 02:49:04 EST 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 2 |
| Keywords | 47J05 35R15 46G05 65J15 46N40 |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c363t-1b273f02d045ad6bd4a0da49b1f73ca2b3d60eed7a6886e43c1f3463d5439dc73 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ORCID | 0000-0001-8831-8547 |
| OpenAccessLink | https://link.springer.com/10.1007/s40687-021-00265-4 |
| PQID | 2519366250 |
| PQPubID | 2044439 |
| ParticipantIDs | proquest_journals_2519366250 crossref_primary_10_1007_s40687_021_00265_4 crossref_citationtrail_10_1007_s40687_021_00265_4 springer_journals_10_1007_s40687_021_00265_4 |
| PublicationCentury | 2000 |
| PublicationDate | 2021-06-01 |
| PublicationDateYYYYMMDD | 2021-06-01 |
| PublicationDate_xml | – month: 06 year: 2021 text: 2021-06-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | Cham |
| PublicationPlace_xml | – name: Cham – name: Heidelberg |
| PublicationTitle | Research in the mathematical sciences |
| PublicationTitleAbbrev | Res Math Sci |
| PublicationYear | 2021 |
| Publisher | Springer International Publishing Springer Nature B.V |
| Publisher_xml | – name: Springer International Publishing – name: Springer Nature B.V |
| References | MartinPCSiggiaEDRoseHAStatistical dynamics of classical systemsPhys. Rev. A19738423437 VenturiDWanXKarniadakisGEStochastic low-dimensional modelling of a random laminar wake past a circular cylinderJ. Fluid Mech.200860633936724286741146.76018 VenturiDTartakovskyDMTartakovskyAMKarniadakisGEExact PDF equations and closure approximations for advective–reactive transportJ. Comput. Phys.201324332334330641711349.35068 AronszajnNDifferentiability of Lipschitz mapping between Banach spacesStudia Math.1976571471904256080342.46034 EngelK-JNagelROne-Parameter Semigroups for Linear Evolution Equations1999BerlinSpringer0952.47036 VenturiDWanXKarniadakisGEStochastic bifurcation analysis of Rayleigh–Bénard convectionJ. Fluid Mech.201065039141326390691189.76213 BartleRGJoichiJTThe preservation of convergence of measurable functions under compositionProc. Am. Math. Soc.1961121221261203420097.04401 HanW. E, JLiQA mean-field optimal control formulation of deep learningRes. Math. Sci20191063891852 PeskinMESchroedeDVAn Introduction to Quantum Field Theory2018Boca RatonCRC Press DektorAVenturiDDynamically orthogonal tensor methods for high-dimensional nonlinear PDEsJ. Comput. Phys.202040410912540447281453.65280 RheeH-KArisRAmundsonNRFirst-Order Partial Differential Equations2001MineolaDover0982.35002 CanutoCHussainiMYQuarteroniAZangASpectral Methods: Fundamentals in Single Domains2006BerlinSpringer1093.76002 RodgersAVenturiDStability analysis of hierarchical tensors methods for time-dependent PDEsJ. Comput. Phys.202040910934140703801435.65149 BoelensAMPVenturiDTartakovskyDMParallel tensor methods for high-dimensional linear PDEsJ. Comput. Phys.201837551953938745481416.65387 Hanche-OlsenHHoldenHThe Kolmogorov–Riesz compactness theoremExpo. Math.20102838539527344541208.46027 Hanche-OlsenHHoldenHAn improvement of the Kolmogorov–Riesz compactness theoremExpo. Math.201937849139642171425.46018 CampitiMTacelliCRate of convergence in Trotter’s approximation theoremConstr. Approx.200828233334124533701181.41024 SchwartzJTNonlinear Functional Analysis1969LondonGordon and Breach Science Publishers0203.14501 SchepARCompactness properties of an operator which imply that it is an integral operatorTrans. Am. Math. Soc.198126511111196071100538.47018 BaezJCSawinSFunctional integration on spaces of connectionsJ. Funct. Anal.1997150112614736230891.46040 JensenRVFunctional integral approach to classical statistical dynamicsJ. Stat. Phys.1981252183210624745 VenturiDThe numerical approximation of nonlinear functionals and functional differential equationsPhys. Rep.20187321102377496306852746 SchachermayerWIntegral operators on lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^p$$\end{document} spacesIndiana Uni. Math. J.19813011231400422.47012 ParrRGWeitaoYDensity-Functional Theory of Atoms and Molecules1994OxfordOxford University Press KatoTRemarks on pseudo-resolvents and infinitesimal generators of semigroupsProc. Jpn. Acad.1959354674680095.10502 BertuzziAGandolfiAGermaniAA Weierstrass-like theorem for real separable Hilbert spacesJ. Approx. Theory19813276816295830493.41043 GikhmanIISkorokhodAVThe Theory of Stochastic Processes I2004BerlinSpringer1068.60004 BrennanCVenturiDData-driven closures for stochastic dynamical systemsJ. Comput. Phys.201837228129838474331415.65014 MatveevOVBases in Sobolev spaces on bounded domains with Lipschitzian boundaryMath. Notes20027237338219631681037.46033 BelloutHOn a special Schouder basis for the Sobolev spaces w01,p(ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{0}^{1, p}(\omega )$$\end{document}Ill. J. Math.1995392187195 Dektor, A., Venturi, D.: Dynamic tensor approximation for high-dimensional nonlinear PDEs pp. 1–23 (2020). arXiv: 2007.09538 VenturiDSapsisTPChoHKarniadakisGEA computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systemsProc. R. Soc. A2012468213975978328923111365.92023 VenturiDKarniadakisGEConvolutionless Nakajima–Zwanzig equations for stochastic analysis in nonlinear dynamical systemsProc. R. Soc. A2014470216612031902221371.60119 XiuDKarniadakisGEThe Wiener–Askey polynomial chaos for stochastic differential equationsSIAM J. Sci. Comput.200224261964419510581014.65004 MelroseRMIT Mathematics 18.102/18.1022020BerlinSpringer Skorohod, A.V.: Integration in Hilbert Space. Springer. Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge (1974) CarmonaRDelarueFProbabilistic Theory of Mean Field Games with Applications I-II2018BerlinSpringer1422.91014 AdamsRAFournierJJFSobolev Spaces2003AmsterdamElsevier1098.46001 DoostanAOwhadiHA non-adapted sparse approximation of PDEs with stochastic inputsJ. Comput. Phys.201123083015303427743281218.65008 VainbergMMVariational Methods for the Study of Nonlinear Operators1964TorontoHolden-Day0122.35501 WienerNNonliner Problems in Random Theory1966CambridgeMIT Press GuidettiDKarasozenBPiskarevSApproximation of abstract differential equationsJ. Math. Sci.20041223013305420841851111.47063 MankiewiczPOn the differentiability of Lipschitz mappings in Fréchet spacesStudia Math.19734515293310550219.46006 LinLZepeda-NunezLProjection-based embedding theory for solving Kohn-Sham density functional theorySIAM Multiscale Model. Simul.20191741274130040417051434.65325 RudinWPrinciples of Mathematical Analysis19763New YorkMcGraw-Hill0346.26002 Zinn-JustinJQuantum Field Theory and Critical Phenomena20024OxfordOxford University Press0865.00014 CombePRodriguezRRideauGSirugue-CollinMOn the cylindrical approximation of the Feynman path integralRep. Math. Phys.1978312792945079440421.28014 ErnstOGMuglerAStarkloffH-JUllmannEOn the convergence of generalized polynomial chaos expansionsESAIM: Math. Model. Numer. Anal.201246231733928556451273.65012 CiesielskiZFigielTSpline bases in classical function spaces on compact c∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c^{\infty }$$\end{document} manifoldsPart I. Studia Mathematica1983761587281950599.46041 CiliaRGutiérrezJMOperators with an integral representationProc. Am. Math. Soc.20161445275529035562711353.47040 NashedMZRallLBDifferentiability and related properties of non-linear operators: some aspects of the role of differentials in non-linear functional analysisNonlinear Functional Analysis and Applications1971CambridgeAcademic Press0236.46050 RaissiMKarniadakisGEHidden physics models: machine learning of nonlinear partial differential equationsJ. Comput. Phys.201835712514137594151381.68248 FoxRFFunctional-calculus approach to stochastic differential equationsPhys. Rev. A1986331467476822180 HesthavenJSGottliebSGottliebDSpectral Methods for Time-Dependent Problems2007CambridgeCambridge University Press1111.65093 DiestelJUhlJJThe Radon–Nikodym theorem for Banach space valued measuresRocky Mt. J. Math.1976611463998520339.46031 MoninASYaglomAMStatistical Fluid Mechanics2007MineolaDover1140.76004 PinkusAN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document}-Widths in Approximation Theory1985BerlinSpringer0551.41001 XiuDNumerical Methods for Stochastic Computations: A Spectral Approach2010PrincetonPrinceton University Press1210.65002 Dektor, A., Rodgers, B., Venturi, D.: Rank-adaptive tensor methods for high-dimensional nonlinear PDEs, pp. 1–24 (2020). arXiv:2012.05962 ZakharovVKMikhalevAVRodionovTVFundamentals of Functions and Measure Theory2018BerlinDe Gruyter06817012 HopfEStatistical hydromechanics and functional calculusJ. Rat. Mech. Anal.19521187123591190049.41704 DeVoreRPetrovaGWojtaszczykPGreedy algorithms for reduced bases in Banach spacesConstruct. Approx.20133745546630546111276.41021 EnfloPA counterexample to the approximation problem in Banach spacesActa Math.19731303093174024680267.46012 van Neerven, J.: Stochastic evolution equations. ISEM Lecture Notes (2008) TrotterHFApproximation of semi-groups of operatorsPacific J. Math.195888879191034200099.10302 KlyatskinVIDynamics of Stochastic Systems2005AmsterdamElsevier Publishing Company1213.93002 SingerIBases in Banach Spaces I1970BerlinSpringer0198.16601 ChoHVenturiDKarniadakisGEStatistical analysis and simulation of random shocks in Burgers equationProc. R. Soc. A201421714701211371.76081 McArthurCWDevelopment in Schauder basis theoryBull. Am. Math. Soc.1972788779083137660257.46012 BoelensAMPVenturiDTartakovskyDMTensor methods for the Boltzmann-BGK equationJ. Comput. Phys.20204211097444136198 HohenbergPKohnWInhomogeneous electron gasPhys. Rev.1964136B864B871180312 SchneiderRUschmajewAApproximation rates for the hierarchical tensor format in periodic Sobolev spacesJ. Complex.2014302567131665211329.41033 ZhuYZabarasNKoutsourelakisP-SPerdikarisPPhysics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled dataJ. Comput. Phys.2019394568139574521452.68172 VenturiDConjugate flow action functionalsJ. Math. Phys.20135411350231370401302.35340 AmitDJMartín-MayorVField Theory, the Renormalization Group and Critical Phenomena2005New YorkWorld Scientific Publishing1088.81001 KoldaTBaderBWTensor decompositions and applicationsSIREV20095145550025350561173.65029 PhythianRThe functional formalism of classical statistical dynamicsJ. Phys. A: Math. Gen.1977105777788471820 BogoliubovNNOn the theory of superfluidityJ. Phys. (USSR)194711233222177 GangboWLiWOsherSPuthawalaMUnnormalized optimal transportJ. Comput. Phys.201939910894 H Hanche-Olsen (265_CR42) 2010; 28 RG Bartle (265_CR7) 1961; 12 M Campiti (265_CR15) 2008; 28 R Carmona (265_CR17) 2018 H Hanche-Olsen (265_CR43) 2019; 37 JS Hesthaven (265_CR46) 2007 D Venturi (265_CR99) 2013; 469 AR Schep (265_CR81) 1981; 265 A Rodgers (265_CR76) 2020; 409 D Preiss (265_CR71) 1990; 91 Z Semadeni (265_CR85) 1965; 1 NN Bogoliubov (265_CR12) 1947; 11 T Kato (265_CR53) 1959; 35 RG Parr (265_CR67) 1994 DJ Amit (265_CR3) 2005 RF Fox (265_CR37) 1986; 33 VK Zakharov (265_CR103) 2018 A Bertuzzi (265_CR9) 1981; 32 A Doostan (265_CR29) 2011; 230 RA Adams (265_CR1) 2003 VI Klyatskin (265_CR54) 2005 T Alankus (265_CR2) 1988; 53 A Pinkus (265_CR70) 1985 C Canuto (265_CR16) 2006 P Hänggi (265_CR45) 1989 AMP Boelens (265_CR11) 2020; 421 H-K Rhee (265_CR75) 2001 Y Zhu (265_CR104) 2019; 394 D Venturi (265_CR98) 2010; 650 H Bellout (265_CR8) 1995; 39 W Gangbo (265_CR40) 2019; 399 RV Jensen (265_CR52) 1981; 25 MM Vainberg (265_CR90) 1964 OV Matveev (265_CR60) 2002; 72 CW McArthur (265_CR61) 1972; 78 E Hopf (265_CR48) 1952; 1 265_CR25 P Hohenberg (265_CR47) 1964; 136 JK Hunter (265_CR49) 2001 P Combe (265_CR23) 1978; 31 W Schachermayer (265_CR80) 1981; 30 GB Folland (265_CR35) 2013 R Melrose (265_CR62) 2020 D Venturi (265_CR94) 2014; 470 R Cilia (265_CR22) 2016; 144 N Wiener (265_CR100) 1966 PC Martin (265_CR59) 1973; 8 G Rosen (265_CR77) 1971; 12 H Cho (265_CR19) 2017 D Jackson (265_CR50) 2004 P Mankiewicz (265_CR58) 1973; 45 D Venturi (265_CR97) 2008; 606 D Venturi (265_CR93) 2018; 732 265_CR38 M Raissi (265_CR74) 2019; 378 P Enflo (265_CR32) 1973; 130 C Dopazo (265_CR30) 1998; 17 R Seiringer (265_CR84) 2011; 306 D Venturi (265_CR96) 2013; 243 K-J Engel (265_CR33) 1999 D Xiu (265_CR102) 2002; 24 M Raissi (265_CR73) 2018; 357 I Singer (265_CR86) 1970 265_CR91 M Bachmayr (265_CR5) 2016; 16 Z Ciesielski (265_CR21) 1983; 76 R DeVore (265_CR27) 2013; 37 A Dektor (265_CR26) 2020; 404 J Diestel (265_CR28) 1976; 6 D Guidetti (265_CR41) 2004; 122 KO Friedrichs (265_CR39) 1957; 43 OG Ernst (265_CR34) 2012; 46 JT Schwartz (265_CR83) 1969 J Foo (265_CR36) 2010; 229 TJ Morrison (265_CR64) 2001 P Hänggi (265_CR44) 1985 R Phythian (265_CR69) 1977; 10 D Venturi (265_CR95) 2012; 468 RC James (265_CR51) 1982; 89 JC Baez (265_CR6) 1997; 150 265_CR24 ME Peskin (265_CR68) 2018 YT Chow (265_CR20) 2019; 80 W Rudin (265_CR78) 1976 L Ruthotto (265_CR79) 2020; 117 II Gikhman (265_CR88) 2004 HF Trotter (265_CR89) 1958; 8 D Xiu (265_CR101) 2010 R Schneider (265_CR82) 2014; 30 AV Bukhvalov (265_CR14) 1978; 9 D Venturi (265_CR92) 2013; 54 N Aronszajn (265_CR4) 1976; 57 L Lin (265_CR56) 2019; 17 AS Monin (265_CR63) 2007 C Brennan (265_CR13) 2018; 372 W. E, J Han (265_CR31) 2019; 10 H Cho (265_CR18) 2014; 2171 J Lindenstrauss (265_CR57) 2003; 157 K Ohkitani (265_CR66) 2020; 101 265_CR87 J Zinn-Justin (265_CR105) 2002 PM Prenter (265_CR72) 1970; 3 T Kolda (265_CR55) 2009; 51 MZ Nashed (265_CR65) 1971 AMP Boelens (265_CR10) 2018; 375 |
| References_xml | – reference: WienerNNonliner Problems in Random Theory1966CambridgeMIT Press – reference: VenturiDThe numerical approximation of nonlinear functionals and functional differential equationsPhys. Rep.20187321102377496306852746 – reference: HohenbergPKohnWInhomogeneous electron gasPhys. Rev.1964136B864B871180312 – reference: OhkitaniKStudy of the Hopf functional equation for turbulence: Duhamel principle and dynamical scalingPhys. Rev. E20201010131044065599 – reference: SchachermayerWIntegral operators on lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^p$$\end{document} spacesIndiana Uni. Math. J.19813011231400422.47012 – reference: ChoHVenturiDKarniadakisGEStatistical analysis and simulation of random shocks in Burgers equationProc. R. Soc. A201421714701211371.76081 – reference: ErnstOGMuglerAStarkloffH-JUllmannEOn the convergence of generalized polynomial chaos expansionsESAIM: Math. Model. Numer. Anal.201246231733928556451273.65012 – reference: TrotterHFApproximation of semi-groups of operatorsPacific J. Math.195888879191034200099.10302 – reference: BrennanCVenturiDData-driven closures for stochastic dynamical systemsJ. Comput. Phys.201837228129838474331415.65014 – reference: SchneiderRUschmajewAApproximation rates for the hierarchical tensor format in periodic Sobolev spacesJ. Complex.2014302567131665211329.41033 – reference: HanW. E, JLiQA mean-field optimal control formulation of deep learningRes. Math. Sci20191063891852 – reference: AmitDJMartín-MayorVField Theory, the Renormalization Group and Critical Phenomena2005New YorkWorld Scientific Publishing1088.81001 – reference: HesthavenJSGottliebSGottliebDSpectral Methods for Time-Dependent Problems2007CambridgeCambridge University Press1111.65093 – reference: DiestelJUhlJJThe Radon–Nikodym theorem for Banach space valued measuresRocky Mt. J. Math.1976611463998520339.46031 – reference: DeVoreRPetrovaGWojtaszczykPGreedy algorithms for reduced bases in Banach spacesConstruct. Approx.20133745546630546111276.41021 – reference: CiliaRGutiérrezJMOperators with an integral representationProc. Am. Math. Soc.20161445275529035562711353.47040 – reference: HopfEStatistical hydromechanics and functional calculusJ. Rat. Mech. Anal.19521187123591190049.41704 – reference: VenturiDWanXKarniadakisGEStochastic bifurcation analysis of Rayleigh–Bénard convectionJ. Fluid Mech.201065039141326390691189.76213 – reference: AronszajnNDifferentiability of Lipschitz mapping between Banach spacesStudia Math.1976571471904256080342.46034 – reference: DopazoCO’BrienEEFunctional formulation of nonisothermal turbulent reactive flowPhys. Fluids19981711196819750303.76027 – reference: ZhuYZabarasNKoutsourelakisP-SPerdikarisPPhysics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled dataJ. Comput. Phys.2019394568139574521452.68172 – reference: VenturiDKarniadakisGEConvolutionless Nakajima–Zwanzig equations for stochastic analysis in nonlinear dynamical systemsProc. R. Soc. A2014470216612031902221371.60119 – reference: CombePRodriguezRRideauGSirugue-CollinMOn the cylindrical approximation of the Feynman path integralRep. Math. Phys.1978312792945079440421.28014 – reference: HunterJKNachtergaeleBApplied Analysis2001New YorkWorld Scientific0981.46002 – reference: BoelensAMPVenturiDTartakovskyDMParallel tensor methods for high-dimensional linear PDEsJ. Comput. Phys.201837551953938745481416.65387 – reference: Skorohod, A.V.: Integration in Hilbert Space. Springer. Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge (1974) – reference: GangboWLiWOsherSPuthawalaMUnnormalized optimal transportJ. Comput. Phys.201939910894040131501453.49016 – reference: SingerIBases in Banach Spaces I1970BerlinSpringer0198.16601 – reference: VenturiDSapsisTPChoHKarniadakisGEA computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systemsProc. R. Soc. A2012468213975978328923111365.92023 – reference: EngelK-JNagelROne-Parameter Semigroups for Linear Evolution Equations1999BerlinSpringer0952.47036 – reference: RheeH-KArisRAmundsonNRFirst-Order Partial Differential Equations2001MineolaDover0982.35002 – reference: HänggiPPesqueraLRodriguezMThe functional derivative and its use in the description of noisy dynamical systemsStochastic Processes Applied to Physics1985New YorkWorld Scientific6995 – reference: PeskinMESchroedeDVAn Introduction to Quantum Field Theory2018Boca RatonCRC Press – reference: GuidettiDKarasozenBPiskarevSApproximation of abstract differential equationsJ. Math. Sci.20041223013305420841851111.47063 – reference: VenturiDWanXMikuleviciusRRozovskiiBLKarniadakisGEWick–Malliavin approximation to nonlinear stochastic partial differential equations: analysis and simulationsProc. R. Soc. A2013469215812031050161371.60108 – reference: BachmayrMSchneiderRUschmajewATensor networks and hierarchical tensors for the solution of high-dimensional partial differential equationsFound. Comput. Math.20161661423147235797141357.65153 – reference: EnfloPA counterexample to the approximation problem in Banach spacesActa Math.19731303093174024680267.46012 – reference: BoelensAMPVenturiDTartakovskyDMTensor methods for the Boltzmann-BGK equationJ. Comput. Phys.20204211097444136198 – reference: Hanche-OlsenHHoldenHThe Kolmogorov–Riesz compactness theoremExpo. Math.20102838539527344541208.46027 – reference: ZakharovVKMikhalevAVRodionovTVFundamentals of Functions and Measure Theory2018BerlinDe Gruyter06817012 – reference: DektorAVenturiDDynamically orthogonal tensor methods for high-dimensional nonlinear PDEsJ. Comput. Phys.202040410912540447281453.65280 – reference: Hanche-OlsenHHoldenHAn improvement of the Kolmogorov–Riesz compactness theoremExpo. Math.201937849139642171425.46018 – reference: FollandGBReal Analysis: Modern Techniques and Their Applications2013HobokenWiley0549.28001 – reference: MoninASYaglomAMStatistical Fluid Mechanics2007MineolaDover1140.76004 – reference: FoxRFFunctional-calculus approach to stochastic differential equationsPhys. Rev. A1986331467476822180 – reference: DoostanAOwhadiHA non-adapted sparse approximation of PDEs with stochastic inputsJ. Comput. Phys.201123083015303427743281218.65008 – reference: McArthurCWDevelopment in Schauder basis theoryBull. Am. Math. Soc.1972788779083137660257.46012 – reference: MatveevOVBases in Sobolev spaces on bounded domains with Lipschitzian boundaryMath. Notes20027237338219631681037.46033 – reference: CarmonaRDelarueFProbabilistic Theory of Mean Field Games with Applications I-II2018BerlinSpringer1422.91014 – reference: PrenterPMA Weierstrass theorem for real, separable Hilbert spacesJ. Approx. Theory197033413514332140206.42202 – reference: JensenRVFunctional integral approach to classical statistical dynamicsJ. Stat. Phys.1981252183210624745 – reference: MartinPCSiggiaEDRoseHAStatistical dynamics of classical systemsPhys. Rev. A19738423437 – reference: CiesielskiZFigielTSpline bases in classical function spaces on compact c∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c^{\infty }$$\end{document} manifoldsPart I. Studia Mathematica1983761587281950599.46041 – reference: Dektor, A., Venturi, D.: Dynamic tensor approximation for high-dimensional nonlinear PDEs pp. 1–23 (2020). arXiv: 2007.09538 – reference: PinkusAN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document}-Widths in Approximation Theory1985BerlinSpringer0551.41001 – reference: VenturiDTartakovskyDMTartakovskyAMKarniadakisGEExact PDF equations and closure approximations for advective–reactive transportJ. Comput. Phys.201324332334330641711349.35068 – reference: VenturiDWanXKarniadakisGEStochastic low-dimensional modelling of a random laminar wake past a circular cylinderJ. Fluid Mech.200860633936724286741146.76018 – reference: BogoliubovNNOn the theory of superfluidityJ. Phys. (USSR)194711233222177 – reference: GikhmanIISkorokhodAVThe Theory of Stochastic Processes I2004BerlinSpringer1068.60004 – reference: ChowYTLiWOsherSYinWAlgorithm for Hamilton–Jacobi equations in density space via a generalized Hopf formulaJ. Sci. Comput.2019801195123939772031422.91102 – reference: MankiewiczPOn the differentiability of Lipschitz mappings in Fréchet spacesStudia Math.19734515293310550219.46006 – reference: PhythianRThe functional formalism of classical statistical dynamicsJ. Phys. A: Math. Gen.1977105777788471820 – reference: SemadeniZSpaces of continuous functions on compact setsAdv. Math.196513193821854700135.34802 – reference: VainbergMMVariational Methods for the Study of Nonlinear Operators1964TorontoHolden-Day0122.35501 – reference: XiuDKarniadakisGEThe Wiener–Askey polynomial chaos for stochastic differential equationsSIAM J. Sci. Comput.200224261964419510581014.65004 – reference: LinLZepeda-NunezLProjection-based embedding theory for solving Kohn-Sham density functional theorySIAM Multiscale Model. Simul.20191741274130040417051434.65325 – reference: ChoHVenturiDKarniadakisGEJinSPareschiLNumerical methods for high-dimensional kinetic equationsUncertainty Quantification for Kinetic and Hyperbolic Equations2017BerlinSpringer931251404.65117 – reference: RuthottoLOsherSLiWNurbekyanLFungSWA machine learning framework for solving high-dimensional mean field game and mean field control problemsPNAS202011717918391934236167 – reference: CanutoCHussainiMYQuarteroniAZangASpectral Methods: Fundamentals in Single Domains2006BerlinSpringer1093.76002 – reference: BukhvalovAVIntegral representation of linear operatorsJ. Math. Sci.197891291370397.47017 – reference: SchepARCompactness properties of an operator which imply that it is an integral operatorTrans. Am. Math. Soc.198126511111196071100538.47018 – reference: KatoTRemarks on pseudo-resolvents and infinitesimal generators of semigroupsProc. Jpn. Acad.1959354674680095.10502 – reference: BertuzziAGandolfiAGermaniAA Weierstrass-like theorem for real separable Hilbert spacesJ. Approx. Theory19813276816295830493.41043 – reference: SeiringerRhe excitation spectrum for weakly interacting bosonsCommun. Math. Phys.20113065655781226.82039 – reference: RaissiMPerdikarisPKarniadakisGEPhysics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equationsJ. Comput. Phys.201937860670738816951415.68175 – reference: PreissDDifferentiability of Lipschitz functionsJ. Funct. Anal.19909131234510589750711.46036 – reference: BartleRGJoichiJTThe preservation of convergence of measurable functions under compositionProc. Am. Math. Soc.1961121221261203420097.04401 – reference: SchwartzJTNonlinear Functional Analysis1969LondonGordon and Breach Science Publishers0203.14501 – reference: RosenGFunctional calculus theory for incompressible fluid turbulenceJ. Math. Phys.19711258128202814080218.76064 – reference: RudinWPrinciples of Mathematical Analysis19763New YorkMcGraw-Hill0346.26002 – reference: ParrRGWeitaoYDensity-Functional Theory of Atoms and Molecules1994OxfordOxford University Press – reference: Friedrichs, K.O., Shapiro, H.N.: Integration of functionals. New York University, Institute of Mathematical Sciences (1957) – reference: HänggiPMossFMcClintockPVEColored noise in continuous dynamical systemNoise in Nonlinear Dynamical Systems1989CambridgeCambridge University Press307347 – reference: RaissiMKarniadakisGEHidden physics models: machine learning of nonlinear partial differential equationsJ. Comput. Phys.201835712514137594151381.68248 – reference: KlyatskinVIDynamics of Stochastic Systems2005AmsterdamElsevier Publishing Company1213.93002 – reference: FriedrichsKOShapiroHNIntegration over a Hilbert space and outer extensionsProc. Natl. Acad. Sci.19574343363381087110077.31303 – reference: JamesRCBases in Banach spacesAm. Math. Monthly1982896256406788080506.46006 – reference: LindenstraussJPreissDOn Fréchet differentiability of Lipschitz maps between Banach spacesAnn. Math.200315725728819542671171.46313 – reference: XiuDNumerical Methods for Stochastic Computations: A Spectral Approach2010PrincetonPrinceton University Press1210.65002 – reference: BaezJCSawinSFunctional integration on spaces of connectionsJ. Funct. Anal.1997150112614736230891.46040 – reference: BelloutHOn a special Schouder basis for the Sobolev spaces w01,p(ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{0}^{1, p}(\omega )$$\end{document}Ill. J. Math.1995392187195 – reference: van Neerven, J.: Stochastic evolution equations. ISEM Lecture Notes (2008) – reference: Zinn-JustinJQuantum Field Theory and Critical Phenomena20024OxfordOxford University Press0865.00014 – reference: FooJKarniadakisGEMulti-element probabilistic collocation method in high dimensionsJ. Comput. Phys.20102291536155725782381181.65014 – reference: KoldaTBaderBWTensor decompositions and applicationsSIREV20095145550025350561173.65029 – reference: JacksonDFourier Series and Orthogonal Polynomials2004MineolaDover1084.42001 – reference: NashedMZRallLBDifferentiability and related properties of non-linear operators: some aspects of the role of differentials in non-linear functional analysisNonlinear Functional Analysis and Applications1971CambridgeAcademic Press0236.46050 – reference: CampitiMTacelliCRate of convergence in Trotter’s approximation theoremConstr. Approx.200828233334124533701181.41024 – reference: VenturiDConjugate flow action functionalsJ. Math. Phys.20135411350231370401302.35340 – reference: MorrisonTJFunctional Analysis: An Introduction to Banach Space Theory2001HobokenWiley1005.46004 – reference: RodgersAVenturiDStability analysis of hierarchical tensors methods for time-dependent PDEsJ. Comput. Phys.202040910934140703801435.65149 – reference: Dektor, A., Rodgers, B., Venturi, D.: Rank-adaptive tensor methods for high-dimensional nonlinear PDEs, pp. 1–24 (2020). arXiv:2012.05962 – reference: AdamsRAFournierJJFSobolev Spaces2003AmsterdamElsevier1098.46001 – reference: AlankusTThe generating functional for the probability density functions of Navier–Stokes turbulenceJ. Stat. Phys.1988535–6126112710677.76052 – reference: MelroseRMIT Mathematics 18.102/18.1022020BerlinSpringer – volume: 399 start-page: 108940 year: 2019 ident: 265_CR40 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.108940 – volume: 606 start-page: 339 year: 2008 ident: 265_CR97 publication-title: J. Fluid Mech. doi: 10.1017/S0022112008001821 – volume: 16 start-page: 1423 issue: 6 year: 2016 ident: 265_CR5 publication-title: Found. Comput. Math. doi: 10.1007/s10208-016-9317-9 – volume: 76 start-page: 1 year: 1983 ident: 265_CR21 publication-title: Part I. Studia Mathematica doi: 10.4064/sm-76-1-1-58 – ident: 265_CR38 – volume-title: MIT Mathematics 18.102/18.102 year: 2020 ident: 265_CR62 – volume: 53 start-page: 1261 issue: 5–6 year: 1988 ident: 265_CR2 publication-title: J. Stat. Phys. doi: 10.1007/BF01023868 – volume: 24 start-page: 619 issue: 2 year: 2002 ident: 265_CR102 publication-title: SIAM J. Sci. Comput. doi: 10.1137/S1064827501387826 – volume: 375 start-page: 519 year: 2018 ident: 265_CR10 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.08.057 – volume: 306 start-page: 565 year: 2011 ident: 265_CR84 publication-title: Commun. Math. Phys. doi: 10.1007/s00220-011-1261-6 – start-page: 307 volume-title: Noise in Nonlinear Dynamical Systems year: 1989 ident: 265_CR45 doi: 10.1017/CBO9780511897818.011 – volume: 1 start-page: 319 year: 1965 ident: 265_CR85 publication-title: Adv. Math. doi: 10.1016/0001-8708(65)90041-1 – volume: 130 start-page: 309 year: 1973 ident: 265_CR32 publication-title: Acta Math. doi: 10.1007/BF02392270 – volume: 2171 start-page: 1 issue: 470 year: 2014 ident: 265_CR18 publication-title: Proc. R. Soc. A – volume: 229 start-page: 1536 year: 2010 ident: 265_CR36 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2009.10.043 – volume-title: Field Theory, the Renormalization Group and Critical Phenomena year: 2005 ident: 265_CR3 doi: 10.1142/5715 – volume: 6 start-page: 1 issue: 1 year: 1976 ident: 265_CR28 publication-title: Rocky Mt. J. Math. doi: 10.1216/RMJ-1976-6-1-1 – volume: 394 start-page: 56 year: 2019 ident: 265_CR104 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.05.024 – volume-title: Bases in Banach Spaces I year: 1970 ident: 265_CR86 doi: 10.1007/978-3-642-51633-7 – volume: 39 start-page: 187 issue: 2 year: 1995 ident: 265_CR8 publication-title: Ill. J. Math. – volume-title: Applied Analysis year: 2001 ident: 265_CR49 doi: 10.1142/4319 – volume: 10 start-page: 6 year: 2019 ident: 265_CR31 publication-title: Res. Math. Sci – ident: 265_CR24 doi: 10.1016/j.jcp.2019.109125 – volume: 32 start-page: 76 year: 1981 ident: 265_CR9 publication-title: J. Approx. Theory doi: 10.1016/0021-9045(81)90023-X – volume: 9 start-page: 129 year: 1978 ident: 265_CR14 publication-title: J. Math. Sci. doi: 10.1007/BF01578539 – start-page: 69 volume-title: Stochastic Processes Applied to Physics year: 1985 ident: 265_CR44 – volume: 30 start-page: 123 issue: 1 year: 1981 ident: 265_CR80 publication-title: Indiana Uni. Math. J. doi: 10.1512/iumj.1981.30.30011 – volume: 25 start-page: 183 issue: 2 year: 1981 ident: 265_CR52 publication-title: J. Stat. Phys. doi: 10.1007/BF01022182 – volume: 91 start-page: 312 year: 1990 ident: 265_CR71 publication-title: J. Funct. Anal. doi: 10.1016/0022-1236(90)90147-D – start-page: 93 volume-title: Uncertainty Quantification for Kinetic and Hyperbolic Equations year: 2017 ident: 265_CR19 doi: 10.1007/978-3-319-67110-9_3 – volume: 57 start-page: 147 year: 1976 ident: 265_CR4 publication-title: Studia Math. doi: 10.4064/sm-57-2-147-190 – volume-title: Probabilistic Theory of Mean Field Games with Applications I-II year: 2018 ident: 265_CR17 doi: 10.1007/978-3-319-56436-4 – volume: 28 start-page: 333 issue: 2 year: 2008 ident: 265_CR15 publication-title: Constr. Approx. doi: 10.1007/s00365-008-9017-z – volume: 101 start-page: 013104 year: 2020 ident: 265_CR66 publication-title: Phys. Rev. E doi: 10.1103/PhysRevE.101.013104 – volume: 3 start-page: 341 year: 1970 ident: 265_CR72 publication-title: J. Approx. Theory doi: 10.1016/0021-9045(70)90039-0 – volume: 469 start-page: 1 issue: 2158 year: 2013 ident: 265_CR99 publication-title: Proc. R. Soc. A doi: 10.1098/rspa.2013.0001 – volume: 37 start-page: 84 year: 2019 ident: 265_CR43 publication-title: Expo. Math. doi: 10.1016/j.exmath.2018.03.002 – volume: 10 start-page: 777 issue: 5 year: 1977 ident: 265_CR69 publication-title: J. Phys. A: Math. Gen. doi: 10.1088/0305-4470/10/5/011 – volume-title: Density-Functional Theory of Atoms and Molecules year: 1994 ident: 265_CR67 – volume-title: Nonlinear Functional Analysis year: 1969 ident: 265_CR83 – volume: 144 start-page: 5275 year: 2016 ident: 265_CR22 publication-title: Proc. Am. Math. Soc. doi: 10.1090/proc/13249 – volume: 54 start-page: 113502 year: 2013 ident: 265_CR92 publication-title: J. Math. Phys. doi: 10.1063/1.4827679 – volume: 17 start-page: 1274 issue: 4 year: 2019 ident: 265_CR56 publication-title: SIAM Multiscale Model. Simul. doi: 10.1137/18M1202670 – volume: 421 start-page: 109744 year: 2020 ident: 265_CR11 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2020.109744 – volume: 11 start-page: 23 year: 1947 ident: 265_CR12 publication-title: J. Phys. (USSR) – volume: 89 start-page: 625 year: 1982 ident: 265_CR51 publication-title: Am. Math. Monthly doi: 10.1080/00029890.1982.11995506 – volume-title: $$N$$-Widths in Approximation Theory year: 1985 ident: 265_CR70 doi: 10.1007/978-3-642-69894-1 – volume: 31 start-page: 279 year: 1978 ident: 265_CR23 publication-title: Rep. Math. Phys. doi: 10.1016/0034-4877(78)90055-1 – volume-title: Functional Analysis: An Introduction to Banach Space Theory year: 2001 ident: 265_CR64 – volume: 357 start-page: 125 year: 2018 ident: 265_CR73 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2017.11.039 – volume: 12 start-page: 812 issue: 5 year: 1971 ident: 265_CR77 publication-title: J. Math. Phys. doi: 10.1063/1.1665649 – volume: 30 start-page: 56 issue: 2 year: 2014 ident: 265_CR82 publication-title: J. Complex. doi: 10.1016/j.jco.2013.10.001 – volume-title: First-Order Partial Differential Equations year: 2001 ident: 265_CR75 – volume: 46 start-page: 317 issue: 2 year: 2012 ident: 265_CR34 publication-title: ESAIM: Math. Model. Numer. Anal. doi: 10.1051/m2an/2011045 – volume: 45 start-page: 15 year: 1973 ident: 265_CR58 publication-title: Studia Math. doi: 10.4064/sm-45-1-15-29 – volume: 12 start-page: 122 year: 1961 ident: 265_CR7 publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-1961-0120342-2 – volume-title: Sobolev Spaces year: 2003 ident: 265_CR1 – ident: 265_CR25 – volume: 404 start-page: 109125 year: 2020 ident: 265_CR26 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2019.109125 – volume-title: Numerical Methods for Stochastic Computations: A Spectral Approach year: 2010 ident: 265_CR101 doi: 10.1515/9781400835348 – volume-title: Quantum Field Theory and Critical Phenomena year: 2002 ident: 265_CR105 doi: 10.1093/acprof:oso/9780198509233.001.0001 – volume: 650 start-page: 391 year: 2010 ident: 265_CR98 publication-title: J. Fluid Mech. doi: 10.1017/S0022112009993685 – volume: 78 start-page: 877 year: 1972 ident: 265_CR61 publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0002-9904-1972-13048-9 – volume: 157 start-page: 257 year: 2003 ident: 265_CR57 publication-title: Ann. Math. doi: 10.4007/annals.2003.157.257 – volume-title: Spectral Methods: Fundamentals in Single Domains year: 2006 ident: 265_CR16 doi: 10.1007/978-3-540-30726-6 – volume-title: Spectral Methods for Time-Dependent Problems year: 2007 ident: 265_CR46 doi: 10.1017/CBO9780511618352 – volume-title: Nonlinear Functional Analysis and Applications year: 1971 ident: 265_CR65 – volume: 8 start-page: 423 year: 1973 ident: 265_CR59 publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.8.423 – volume-title: Real Analysis: Modern Techniques and Their Applications year: 2013 ident: 265_CR35 – volume: 51 start-page: 455 year: 2009 ident: 265_CR55 publication-title: SIREV doi: 10.1137/07070111X – volume: 372 start-page: 281 year: 2018 ident: 265_CR13 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.06.038 – volume: 43 start-page: 336 issue: 4 year: 1957 ident: 265_CR39 publication-title: Proc. Natl. Acad. Sci. doi: 10.1073/pnas.43.4.336 – volume-title: Fundamentals of Functions and Measure Theory year: 2018 ident: 265_CR103 doi: 10.1515/9783110550962 – volume: 468 start-page: 759 issue: 2139 year: 2012 ident: 265_CR95 publication-title: Proc. R. Soc. A doi: 10.1098/rspa.2011.0186 – volume: 470 start-page: 1 issue: 2166 year: 2014 ident: 265_CR94 publication-title: Proc. R. Soc. A doi: 10.1098/rspa.2013.0754 – volume: 80 start-page: 1195 year: 2019 ident: 265_CR20 publication-title: J. Sci. Comput. doi: 10.1007/s10915-019-00972-9 – volume-title: Variational Methods for the Study of Nonlinear Operators year: 1964 ident: 265_CR90 – volume: 37 start-page: 455 year: 2013 ident: 265_CR27 publication-title: Construct. Approx. doi: 10.1007/s00365-013-9186-2 – volume: 33 start-page: 467 issue: 1 year: 1986 ident: 265_CR37 publication-title: Phys. Rev. A doi: 10.1103/PhysRevA.33.467 – volume-title: The Theory of Stochastic Processes I year: 2004 ident: 265_CR88 doi: 10.1007/978-3-642-61943-4 – volume: 150 start-page: 1 issue: 1 year: 1997 ident: 265_CR6 publication-title: J. Funct. Anal. doi: 10.1006/jfan.1997.3108 – volume: 243 start-page: 323 year: 2013 ident: 265_CR96 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2013.03.001 – volume: 732 start-page: 1 year: 2018 ident: 265_CR93 publication-title: Phys. Rep. doi: 10.1016/j.physrep.2017.12.003 – volume: 72 start-page: 373 year: 2002 ident: 265_CR60 publication-title: Math. Notes doi: 10.1023/A:1020503505540 – volume: 122 start-page: 3013 year: 2004 ident: 265_CR41 publication-title: J. Math. Sci. doi: 10.1023/B:JOTH.0000029696.94590.94 – volume: 409 start-page: 109341 year: 2020 ident: 265_CR76 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2020.109341 – volume-title: Fourier Series and Orthogonal Polynomials year: 2004 ident: 265_CR50 – volume: 35 start-page: 467 year: 1959 ident: 265_CR53 publication-title: Proc. Jpn. Acad. – volume: 1 start-page: 87 issue: 1 year: 1952 ident: 265_CR48 publication-title: J. Rat. Mech. Anal. – volume-title: Statistical Fluid Mechanics year: 2007 ident: 265_CR63 – volume: 230 start-page: 3015 issue: 8 year: 2011 ident: 265_CR29 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2011.01.002 – ident: 265_CR91 doi: 10.1016/j.jfa.2008.03.015 – volume-title: One-Parameter Semigroups for Linear Evolution Equations year: 1999 ident: 265_CR33 – volume-title: Nonliner Problems in Random Theory year: 1966 ident: 265_CR100 – volume: 378 start-page: 606 year: 2019 ident: 265_CR74 publication-title: J. Comput. Phys. doi: 10.1016/j.jcp.2018.10.045 – volume: 117 start-page: 9183 issue: 17 year: 2020 ident: 265_CR79 publication-title: PNAS doi: 10.1073/pnas.1922204117 – volume: 265 start-page: 111 issue: 1 year: 1981 ident: 265_CR81 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1981-0607110-7 – volume: 28 start-page: 385 year: 2010 ident: 265_CR42 publication-title: Expo. Math. doi: 10.1016/j.exmath.2010.03.001 – volume: 8 start-page: 887 year: 1958 ident: 265_CR89 publication-title: Pacific J. Math. doi: 10.2140/pjm.1958.8.887 – volume: 17 start-page: 1968 issue: 11 year: 1998 ident: 265_CR30 publication-title: Phys. Fluids doi: 10.1063/1.1694652 – volume-title: An Introduction to Quantum Field Theory year: 2018 ident: 265_CR68 doi: 10.1201/9780429503559 – volume-title: Principles of Mathematical Analysis year: 1976 ident: 265_CR78 – ident: 265_CR87 – volume: 136 start-page: B864 year: 1964 ident: 265_CR47 publication-title: Phys. Rev. doi: 10.1103/PhysRev.136.B864 – volume-title: Dynamics of Stochastic Systems year: 2005 ident: 265_CR54 |
| SSID | ssj0001763430 ssib053846527 |
| Score | 2.2888281 |
| Snippet | We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential... |
| SourceID | proquest crossref springer |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| SubjectTerms | Applications of Mathematics Approximation Artificial neural networks Banach spaces Boundary value problems Computational Mathematics and Numerical Analysis Continuity (mathematics) Convergence Derivatives Functionals Mathematics Mathematics and Statistics Numerical methods Partial differential equations Spectral methods Tensors |
| Title | Spectral methods for nonlinear functionals and functional differential equations |
| URI | https://link.springer.com/article/10.1007/s40687-021-00265-4 https://www.proquest.com/docview/2519366250 |
| Volume | 8 |
| WOSCitedRecordID | wos000645248100002&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVAVX databaseName: SpringerLINK Contemporary 1997-Present customDbUrl: eissn: 2197-9847 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0001763430 issn: 2522-0144 databaseCode: RSV dateStart: 20141201 isFulltext: true titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22 providerName: Springer Nature |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LSwMxEB60etCDb7FaJQdvGuhuskl6FLF40FJ8lN6WzWNBKEW71d_vJPuoigp6XDYJYTKT-TLJfANwqo2z2gpOhcwN5S7RVOeRppwpjd7ZOhHY-Uc3cjBQ43FvWCWFFfVr9_pKMuzUTbIbuh40CP-kwB8cEsqXYQXdnfIFG-7uR7UWoQVzkVTcliHSgibEQ9GRGMEG9UeIKnvm-2E_e6gF7PxyUxocUH_zf1Pfgo0KcJKLUkO2YclNd2D9tmFrLXZh6IvQ-4gHKetJFwSRLJmWJBrZjHjfV4YMC5JN7YdvUtdXwX1iQtxLyRte7MFj_-rh8ppWlRaoYYLNaaQRxeTd2CLAy6zQlmddm_GejnLJTBZrZkUXvanMhFLCcWainHHBbIJ4xhrJ9qGFs3IHQBCh5YmMFW6bOIoSWvaszBWqis_ZjqI2RLV0U1PRkPtqGJO0IVAO0kpRWmmQVsrbcNb0eS5JOH5t3akXLa0Mskh9gi4TeNjrtuG8XqTF759HO_xb8yNYi8M6-zhNB1rz2as7hlXzNn8qZidBUd8BT2TgjA |
| linkProvider | Springer Nature |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LSwMxEB60CurBt1itmoM3DXQ32ez2KGKp2JaitfS2bB4LQinarf5-J9lHVVTQ47JJCJOZzJdJ5huAc6mMllpwKsJUUW4CSWXqScpZJNE7ayMcO_-oG_b70XjcGhRJYVn52r28knQ7dZXshq4HDcI-KbAHh4DyZVjh6LEsY_79w6jUIrRgLoKC29JFWtCEuCs64iPYoPYIUWTPfD_sZw-1gJ1fbkqdA2pv_W_q27BZAE5ylWvIDiyZ6S5s9Cq21mwPBrYIvY14kLyedEYQyZJpTqKRzIj1fXnIMCPJVH_4JmV9FdwnJsS85Lzh2T48tm-G1x1aVFqgigk2p55EFJM2fY0AL9FCap40dcJb0ktDphJfMi2a6E3DRESRMJwpL2VcMB0gntEqZAdQw1mZQyCI0NIg9CPcNnGUSMiwpcM0QlWxOdueVwevlG6sChpyWw1jElcEyk5aMUordtKKeR0uqj7POQnHr60b5aLFhUFmsU3QZQIPe806XJaLtPj982hHf2t-BmudYa8bd2_7d8ew7rs1tzGbBtTms1dzAqvqbf6UzU6d0r4D3nnjcA |
| linkToPdf | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV3fS8MwED50iuiDv8Xp1Dz4psG2SdPuUdShOMdAHXsrTdOAMMpcq3-_l7TdpqggPpa2ob3c5b675L4DOJVJqqQSnIpAJ5SnvqRSu5JyFkr0zioVlp1_0A16vXA4bPfnqvjtafd6S7KsaTAsTVlxMVb6Ylr4hm4IjcMcLzBBhE_5Iixxc5DexOuPg1qj0Jq58CueS5t1QXPitgGJh8CDmnCiqqT5ftjP3moGQb_smlpn1Nn4_29swnoFRMllqTlbsJBm27D2MGVxzXegb5rTm0wIKftM5wQRLslKco14QoxPLFOJOYkzNXdN6r4ruH6MSPpa8onnu_DcuXm6uqVVBwaaMMEK6kpEN9rxFAK_WAmpeOyomLelqwOWxJ5kSjjoZYNYhKFIOUtczbhgykeco5KA7UEDvyrdB4LITfuBF-JyiqOEQgZtFegQVcjUcrtuE9xa0lFS0ZObLhmjaEqsbKUVobQiK62IN-Fs-s64JOf49elWPYFRZah5ZAp3mcAg0GnCeT1hs9s_j3bwt8dPYKV_3Ym6d737Q1j17JSbVE4LGsXkLT2C5eS9eMknx1Z_PwCy-uxU |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Spectral+methods+for+nonlinear+functionals+and+functional+differential+equations&rft.jtitle=Research+in+the+mathematical+sciences&rft.au=Venturi+Daniele&rft.au=Dektor+Alec&rft.date=2021-06-01&rft.pub=Springer+Nature+B.V&rft.issn=2522-0144&rft.eissn=2197-9847&rft.volume=8&rft.issue=2&rft_id=info:doi/10.1007%2Fs40687-021-00265-4&rft.externalDBID=NO_FULL_TEXT |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2522-0144&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2522-0144&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2522-0144&client=summon |