Single spin exact gradients for the optimization of complex pulses and pulse sequences

Single spin exact gradients for the optimization of complex pulses and pulse sequences | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Single spin exact gradients for the optimization of complex pulses and pulse sequences Stella Slad, Burkhard Luy This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7180681/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 17 Feb, 2026 Read the published version in Journal of Biomolecular NMR → Version 1 posted 9 You are reading this latest preprint version Abstract The efficient computer optimization of magnetic resonance pulses and pulse sequences involves the calculation of a problem-adapted cost function as well as its gradients with respect to all controls applied. The gradients generally can be calculated as a finite difference approximation, as a GRAPE approximation, or as an exact function, e.g. by the use of the augmented matrix exponentiation, where the exact gradient should lead to best optimization convergence. The majority of pulse optimizations involve a single spin 1/2, for which propagation is either represented by 3D-rotations or quaternions. For both cases highly efficient analytical solutions for gradients with respect to various possible controls have been derived. Controls are either $x$ and $y$ pulses, but also $z$-controls, as well as gradients with respect to amplitude and phase of a pulse shape. In addition, analytical solutions with respect to pseudo controls, involving holonomic constraints to maximum rf-amplitudes, maximum rf-power, or maximum rf-energy, are introduced. Using the hyperbolic tangent function, maximum values are imposed in a fully continuous and differentiable way. The obtained analytical gradients allow the calculation two orders of magnitude faster than the augmented matrix exponential approach. The use of exact gradients for different controls is finally demonstrated in a number of optimizations involving broadband pulses for $^{15}$N, $^{13}$C, and $^{19}$F applications. shaped pulses optimal control exact gradient optimization analytical Full Text Additional Declarations No competing interests reported. Supplementary Files SUPPINFO.zip Cite Share Download PDF Status: Published Journal Publication published 17 Feb, 2026 Read the published version in Journal of Biomolecular NMR → Version 1 posted Editorial decision: Revision requested 10 Dec, 2025 Reviews received at journal 09 Dec, 2025 Reviewers agreed at journal 16 Nov, 2025 Reviews received at journal 24 Sep, 2025 Reviewers agreed at journal 16 Sep, 2025 Reviewers invited by journal 08 Aug, 2025 Editor assigned by journal 22 Jul, 2025 Submission checks completed at journal 22 Jul, 2025 First submitted to journal 21 Jul, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7180681","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":497812246,"identity":"b30bee80-16ae-40b5-88ba-0afd738bff17","order_by":0,"name":"Stella Slad","email":"","orcid":"","institution":"Karlsruhe Institute of Technology (KIT)","correspondingAuthor":false,"prefix":"","firstName":"Stella","middleName":"","lastName":"Slad","suffix":""},{"id":497812247,"identity":"f163d183-ef41-448d-a9f8-831d3f23653d","order_by":1,"name":"Burkhard Luy","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1klEQVRIiWNgGAWjYBACAwYGZmYgzdjADiQ/HJCAChKlBUgyzjggIUGaFmaeAwyEtZhL5D42Lqi5J9vAzGMmbXPGoo6BvXmbBD4tljPSjZNnHCs2BmvJuQF0GM+xMrxaDG6kMR/mYUtIhGj5ANQikWNGhJZ/UC0WIC3ybwhrSeZtg2phADlMgge/FsueZ8zGvH0Jxm3MbMWWPWckJNt40oot8GkxZ09jlub5liDbz9688caPY3X8/OyHN97ApwUO2BgYWCSgDOIB8wdSVI+CUTAKRsHIAQCp5juozBtcZgAAAABJRU5ErkJggg==","orcid":"","institution":"Karlsruhe Institute of Technology (KIT)","correspondingAuthor":true,"prefix":"","firstName":"Burkhard","middleName":"","lastName":"Luy","suffix":""}],"badges":[],"createdAt":"2025-07-21 20:23:18","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7180681/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7180681/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s10858-025-00486-7","type":"published","date":"2026-02-17T15:59:26+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":103252493,"identity":"59fdb145-4a10-4dce-92e9-8849c3e2b66d","added_by":"auto","created_at":"2026-02-23 16:15:00","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":13297456,"visible":true,"origin":"","legend":"","description":"","filename":"MANUSCRIPT.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7180681/v1_covered_d91a1033-da65-4300-8410-f45a70e5af5a.pdf"},{"id":89012642,"identity":"45568f7b-629c-42ba-a81a-98d97cafcf6c","added_by":"auto","created_at":"2025-08-13 17:45:20","extension":"zip","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":823373,"visible":true,"origin":"","legend":"","description":"","filename":"SUPPINFO.zip","url":"https://assets-eu.researchsquare.com/files/rs-7180681/v1/d2a66a3d12d8d62d4d9f273e.zip"}],"financialInterests":"No competing interests reported.","formattedTitle":"Single spin exact gradients for the optimization of complex pulses and pulse sequences","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-biomolecular-nmr","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jnmr","sideBox":"Learn more about [Journal of Biomolecular NMR](http://link.springer.com/journal/10858)","snPcode":"10858","submissionUrl":"https://submission.nature.com/new-submission/10858/3","title":"Journal of Biomolecular NMR","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"shaped pulses, optimal control, exact gradient, optimization, analytical","lastPublishedDoi":"10.21203/rs.3.rs-7180681/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7180681/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The efficient computer optimization of magnetic resonance pulses and pulse sequences involves the calculation of a problem-adapted cost function as well as its gradients with respect to all controls applied. 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