Many of the most fundamental laws of nature can be formulated as partial differential equations (PDEs). However, since the general solution of many PDEs is unknown, the efficient approximate solution of these equations is one of humanity's greatest challenges. While multigrid represents one of the most effective methods for solving PDEs numerically, in many cases, the design of an efficient or at least working multigrid solver is an open problem. This thesis demonstrates that grammar-guided genetic programming, an evolutionary program synthesis technique, can discover multigrid methods of unprecedented structure that achieve a high degree of efficiency and generalization. For this purpose, we develop a novel context-free grammar that enables the automated generation of multigrid methods in a symbolically-manipulable formal language, based on which we can apply the same multigrid-based solver to problems of different sizes without having to adapt its internal structure. Treating the automated design of an efficient multigrid method as a program synthesis task allows us to find novel sequences of multigrid operations, including the combination of different smoothing and coarse-grid correction steps on each level of the discretization hierarchy.<\/p>","protected":false},"excerpt":{"rendered":"
Viele der grundlegendsten Naturgesetze k\u00f6nnen als partielle Differentialgleichungen (PDGs) formuliert werden. Da jedoch die allgemeine L\u00f6sung vieler PDGs unbekannt ist, stellt die effiziente N\u00e4herungsl\u00f6sung dieser Gleichungen eine der gr\u00f6\u00dften Herausforderungen der Menschheit dar. Obwohl Mehrgitterverfahren eine der effektivsten Methoden zur numerischen L\u00f6sung von PDGs darstellen, ist der Entwurf eines effizienten oder zumindest funktionierenden Mehrgitterl\u00f6sers in […]<\/p>\n","protected":false},"featured_media":5258,"comment_status":"open","ping_status":"closed","template":"","meta":{"_acf_changed":false},"product_brand":[],"product_cat":[54],"product_tag":[],"class_list":{"0":"post-5257","1":"product","2":"type-product","3":"status-publish","4":"has-post-thumbnail","6":"product_cat-fau-studien-aus-der-informatik","7":"has-post-title","8":"has-post-date","9":"has-post-category","10":"has-post-tag","11":"has-post-comment","12":"has-post-author","13":"","15":"first","16":"instock","17":"taxable","18":"shipping-taxable","19":"purchasable","20":"product-type-simple"},"acf":{"autor":"Schmitt, Jonas","orcid":"0000-0002-8891-0046","ror":"00f7hpc57","herausgeber":"","orcid_hrsg":"","rorhrsg":"","bearbeiter":"","sonstige_beteiligte_personen":"","untertitel":"","zusatz":"","jahr":"2024","zeitschrift":"","serie":"","serie_name":false,"serie_band":"","serie_jahr":"","serie_titel_band":"","vortrage":"","issn":"","isbn":"","isbn13":"978-3-96147-731-9","seiten":"xvi, 222","sonstiges":"","band":19,"auflage":"","opus":"","dspace":"31086","e-isbn_pdf":"978-3-96147-732-6","doi":"https:\/\/doi.org\/10.25593\/978-3-96147-732-6","epub":"","e-isbn_epub":"","doi_epub":"","html":"","e-isbn_html":"","doi_html":"","e-isbn_reader":"","doi_reader":"","digibib":"","urn":"urn:nbn:de:101:1-2405170502061.240988996563","supplementary_material":"","sprache":["Englisch"],"ausgabe":"Softcover","suche":" 9783961477319 9783961477326 ","lizenz":"BY-4","bemerkung":"1\tFAU Press\r\n1\tHB\r\n4\tBSB \/ DNB\r\n1\tTNZB\r\n1\tTIB"},"builder_content":"","_links":{"self":[{"href":"https:\/\/www.university-press.fau.de\/en\/wp-json\/wp\/v2\/product\/5257","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.university-press.fau.de\/en\/wp-json\/wp\/v2\/product"}],"about":[{"href":"https:\/\/www.university-press.fau.de\/en\/wp-json\/wp\/v2\/types\/product"}],"replies":[{"embeddable":true,"href":"https:\/\/www.university-press.fau.de\/en\/wp-json\/wp\/v2\/comments?post=5257"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.university-press.fau.de\/en\/wp-json\/wp\/v2\/media\/5258"}],"wp:attachment":[{"href":"https:\/\/www.university-press.fau.de\/en\/wp-json\/wp\/v2\/media?parent=5257"}],"wp:term":[{"taxonomy":"product_brand","embeddable":true,"href":"https:\/\/www.university-press.fau.de\/en\/wp-json\/wp\/v2\/product_brand?post=5257"},{"taxonomy":"product_cat","embeddable":true,"href":"https:\/\/www.university-press.fau.de\/en\/wp-json\/wp\/v2\/product_cat?post=5257"},{"taxonomy":"product_tag","embeddable":true,"href":"https:\/\/www.university-press.fau.de\/en\/wp-json\/wp\/v2\/product_tag?post=5257"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}