{"id":786,"date":"2017-11-03T14:54:51","date_gmt":"2017-11-03T12:54:51","guid":{"rendered":"http:\/\/www.math.sk\/mpmnew\/?page_id=786"},"modified":"2024-04-28T20:07:48","modified_gmt":"2024-04-28T18:07:48","slug":"inzinierske-statnice","status":"publish","type":"page","link":"https:\/\/www.math.sk\/mpm\/inzinierske-statnice\/","title":{"rendered":"In\u017einierske \u0161t\u00e1tnice"},"content":{"rendered":"

T\u00e9my ot\u00e1zok in\u017einierskych \u0161t\u00e1tnic pre \u0161tudijn\u00fd program Matematicko-po\u010d\u00edta\u010dov\u00e9 modelovanie:<\/h2>\n

1. Rie\u0161enie hyperbolick\u00e9ho syst\u00e9mu s kon\u0161tantn\u00fdmi koeficientami<\/h3>\n

Pop\u00ed\u0161te rie\u0161enie line\u00e1rneho hyperbolick\u00e9ho syst\u00e9mu parci\u00e1lnych diferenci\u00e1lnych rovn\u00edc s kon\u0161tantn\u00fdmi koeficientami v jednorozmernom pr\u00edpade pomocou charakteristick\u00fdch r\u00fdchlost\u00ed a premenn\u00fdch. Formulujte predpoklady, kedy je tento postup mo\u017en\u00fd. (LeVeque, str. 31-32)<\/p>\n

2. Met\u00f3da kone\u010dn\u00fdch objemov pre hyperbolick\u00e9 rovnice<\/h3>\n

Pop\u00ed\u0161te v\u0161eobecn\u00fd tvar met\u00f3dy kone\u010dn\u00fdch objemov na rie\u0161enie skal\u00e1rnej neline\u00e1rnej hyperbolickej rovnice pre jednorozmern\u00fd pr\u00edpad. Pou\u017eite upwind met\u00f3du pre toky cez hranicu kontroln\u00e9ho objemu pre pr\u00edpad line\u00e1rnej rovnice s kon\u0161tantn\u00fdmi koeficientmi (LeVeque, str.73-74).<\/p>\n

3. Neline\u00e1rny model hustoty prem\u00e1vky<\/h3>\n

Pop\u00ed\u0161te neline\u00e1rnu hyperbolick\u00fa rovnicu pre matematick\u00fd model hustoty prem\u00e1vky v konzervat\u00edvnom a nekonzervat\u00edvnom tvare. Pop\u00ed\u0161te rie\u0161enie rovnice v nekonzervat\u00edvnom tvare pre spojit\u00fa po\u010diato\u010dn\u00fa hustotu v tvare vlny pomocou charakteristick\u00fdch kriviek (LeVeque, str.203-208).<\/p>\n

4.\u00a0Rie\u0161enie Riemannovej \u00falohy pre model hustoty prem\u00e1vky<\/h3>\n

Pop\u00ed\u0161te rie\u0161enie Riemanovej \u00falohy pre model hustoty prem\u00e1vky s pr\u00edpustn\u00fdm a nepr\u00edpustn\u00fdm \u0161okom, definujte r\u00fdchlos\u0165 pr\u00edpustn\u00e9ho \u0161oku, nakreslite pr\u00edslu\u0161n\u00e9 charakteristick\u00e9 krivky.<\/p>\n

5. Architekt\u00fara paraleln\u00fdch po\u010d\u00edta\u010dov, modely paraleln\u00e9ho programovania, charakteristika \u0161tandardu MPI (Message Passing Interface)<\/h3>\n

6. Z\u00e1klady paraleln\u00e9ho programovania v prostred\u00ed MPI, pr\u00edkazy hromadnej a jednoduchej (point-to-point) komunik\u00e1cie<\/h3>\n

7. Z\u00e1kladn\u00e9 princ\u00edpy met\u00f3dy okrajov\u00fdch prvkov<\/h3>\n

Charakterizujte met\u00f3du hrani\u010dn\u00fdch elementov. Na\u010drtnite z\u00e1kladn\u00fd postup rie\u0161enia okrajov\u00fdch \u00faloh pomocou BEM a ilustrujte ju na pr\u00edklade okrajov\u00fdch \u00faloh pre stacion\u00e1rne vedenie tepla v homog\u00e9nnom, izotropnom kontinuu. Porovnajte z\u00e1kladn\u00e9 atrib\u00faty BEM a\u00a0FEM.<\/p>\n

8. Princ\u00edpy diskretiz\u00e1cie v met\u00f3de okrajov\u00fdch prvkov<\/h3>\n

Hrani\u010dn\u00e9 elementy. Diskretiz\u00e1cia hranice v 2-d a 3-d \u00faloh\u00e1ch. Aproxim\u00e1cia geometrie a fyzik\u00e1lnych pol\u00ed interpol\u00e1ciou. Line\u00e1rne 1 a 2 \u2013 rozmern\u00e9 elementy.<\/p>\n

9. Konvol\u00facia a line\u00e1rna dif\u00fazia\u00a0 v\u00a0spracovan\u00ed obrazu<\/h3>\n

Pop\u00ed\u0161te oper\u00e1ciu konvol\u00facie, v diskr\u00e9tnom a spojitom tvare, v spracovan\u00ed obrazu a jej vz\u0165ah s rovnicou line\u00e1rnej dif\u00fazie (vedenia tepla). Diskutujte ich vlastnosti a vyu\u017eitie vo filtr\u00e1cii a segment\u00e1cii obrazu.<\/p>\n

10. Modely neline\u00e1rnej dif\u00fazie\u00a0 v spracovan\u00ed obrazu<\/h3>\n

Pop\u00ed\u0161te z\u00e1kladn\u00e9 modely v\u00a0tvare parci\u00e1lnych diferenci\u00e1lnych rovn\u00edc pou\u017e\u00edvan\u00e9 vo filtr\u00e1cii a\u00a0segment\u00e1cii obrazu, ako s\u00fa z\u00e1kladn\u00e1 a regularizovan\u00e1 Perona-Malikova rovnica a level-set rovnice riaden\u00e9 krivos\u0165ou. Diskutujte ich vlastnosti a vyu\u017eitie vo filtr\u00e1cii a segment\u00e1cii obrazu.<\/p>\n

11. Diskretiz\u00e1cia neline\u00e1rnych dif\u00faznych rovn\u00edc semi-implicitnou sch\u00e9mou kone\u010dn\u00fdch objemov<\/h3>\n

Odvo\u010fte semi-implicitn\u00fa met\u00f3du kone\u010dn\u00fdch objemov na rie\u0161enie regularizovanej Perona-Malikovej rovnice, pop\u00ed\u0161te SOR algoritmus rie\u0161enia line\u00e1rneho syst\u00e9mu, uk\u00e1\u017ete stabilitu numerick\u00e9ho rie\u0161enia.<\/p>\n

12. Level-set rovnice pre pohyb kriviek a pl\u00f4ch a ich numerick\u00e9 rie\u0161enie<\/h3>\n

Odvo\u010fte \u00farov\u0148ov\u00fa (level-set) formul\u00e1ciu pohybu uzavretej 2D krivky resp. 3D plochy r\u00fdchlostn\u00fdm po\u013eom\u00a0v\u00a0<\/strong>a\u00a0jej \u0161peci\u00e1lne pr\u00edpady, ako je pohyb v\u00a0smere norm\u00e1ly r\u00fdchlos\u0165ou F(x) a\u00a0pohyb z\u00e1visl\u00fd od strednej krivosti. Pre posledn\u00fd pr\u00edpad odvo\u010fte stabiln\u00fa semi-implicitn\u00fa numerick\u00fa sch\u00e9mu kone\u010dn\u00fdch objemov a\u00a0pop\u00ed\u0161te ako sa level-set modely vyu\u017e\u00edvaj\u00fa v\u00a0segment\u00e1cii obrazu.<\/p>\n

13. Diskretiz\u00e1cia level-set rovn\u00edc semi-implicitnou sch\u00e9mou kone\u010dn\u00fdch objemov<\/h3>\n

Odvo\u010fte semi-implicitn\u00fa met\u00f3du kone\u010dn\u00fdch objemov na rie\u0161enie level-set rovnice pre pohyb riaden\u00fd krivos\u0165ou, pop\u00ed\u0161te SOR algoritmus rie\u0161enia line\u00e1rneho syst\u00e9mu, uk\u00e1\u017ete stabilitu numerick\u00e9ho rie\u0161enia.<\/p>\n

14. Numerick\u00e9 rie\u0161enie eliptickej PDR.<\/h3>\n

Sformulujte okrajov\u00fd probl\u00e9m pre eliptick\u00fa PDR s r\u00f4znymi okrajov\u00fdmi podmienkami v 2D. Definujte slab\u00e9 rie\u0161enie tejto \u00falohy a jeho existenciu. Navrhnite numerick\u00fa sch\u00e9mu zalo\u017een\u00fa\u00a0 na met\u00f3de kone\u010dn\u00fdch objemov a sformulujte z\u00e1kladn\u00e9 predpoklady pre d\u00f4kaz konvergencie numerick\u00e9ho rie\u0161enia ku slab\u00e9mu rie\u0161eniu \u00falohy.<\/p>\n

15. Numerick\u00e9 rie\u0161enie parabolickej PDR.<\/h3>\n

Sformulujte po\u010diato\u010dno-okrajov\u00fd probl\u00e9m. Definujte slab\u00e9 rie\u0161enie tejto \u00falohy. Navrhnite numerick\u00fa sch\u00e9mu zalo\u017een\u00fa na Rotheho met\u00f3de v\u00a0kombin\u00e1cii s met\u00f3dou kone\u010dn\u00fdch objemov a sformulujte z\u00e1kladn\u00e9 predpoklady pre d\u00f4kaz konvergencie numerick\u00e9ho rie\u0161enia ku slab\u00e9mu rie\u0161eniu \u00falohy.<\/p>\n

16. Parametrick\u00e1 krivka a parametriz\u00e1cia.<\/h3>\n

Uve\u010fte defin\u00edciu parametrickej krivky aj s pr\u00edkladom. Definujte dotykov\u00fd vektor, hlavn\u00fa norm\u00e1lu a regul\u00e1rnu krivku. Odvo\u010fte vz\u0165ah pre v\u00fdpo\u010det d\u013a\u017eky krivky. Definujte 1-parametriz\u00e1ciu mno\u017einy. Definujte reparametriz\u00e1ciu a stru\u010dne pop\u00ed\u0161te v\u00fdznam tohto pojmu. Definujte prirodzen\u00fa reparametriz\u00e1ciu, sformulujte tvrdenie o tom, kedy existuje, a pop\u00ed\u0161te, ako by ste ju z\u00edskali.<\/p>\n

17. Zakrivenie parametrickej krivky<\/h3>\n

Definujte krivos\u0165 pre krivku s \u013eubovo\u013en\u00fdm rozmerom a od\u00f4vodnite, pre\u010do sa krivos\u0165 definuje pr\u00e1ve t\u00fdmto sp\u00f4sobom. Pre rovinn\u00fa krivku definujte znamienkov\u00fa krivos\u0165 a uve\u010fte ilustra\u010dn\u00e9 pr\u00edklady. Pre trojrozmern\u00fa krivku definujte Frenetov rep\u00e9r a torziu, sformulujte Frenetove-Serretove vz\u0165ahy a objasnite, \u010do vyjadruj\u00fa. Sformulujte fundament\u00e1lnu vetu o krivk\u00e1ch v 3D a uve\u010fte stru\u010dn\u00fa osnovu d\u00f4kazu.<\/p>\n

18. Regresn\u00e1 \u00faloha v \u0161tatistickom modelovan\u00ed<\/h3>\n

Princ\u00edp a predpoklady line\u00e1rneho regresn\u00e9ho modelu, odhad parametrov. V\u00fdznam kr\u00ed\u017eovej valid\u00e1cie a bootstrap met\u00f3dy. Zov\u0161eobecnenia line\u00e1rneho modelu (GAM, GLM).<\/p>\n

19. Klasifika\u010dn\u00e1 \u00faloha v \u0161tatistickom modelovan\u00ed<\/h3>\n

Dva z\u00e1kladne pr\u00edstupy ku kon\u0161trukcii pravdepodobnostn\u00fdch klasifik\u00e1torov, konkr\u00e9tne met\u00f3dy a ich princ\u00edp. Vyhodnotenie presnosti (celkov\u00e9 a \u0161pecifick\u00e9 chyby, v\u00fdznam ROC a AUC).<\/p>\n

20. Met\u00f3dy neasistovan\u00e9ho \u0161tatistick\u00e9ho modelovania.<\/h3>\n

Modelovanie zdru\u017een\u00e9ho rozdelenia pravdepodobnosti (margin\u00e1lne rozdelenie, podmie\u0148ovanie, nez\u00e1vislos\u0165, parametrick\u00e9 a neparametrick\u00e9 modely, rozklad, odhad parametrov), anal\u00fdza hlavn\u00fdch komponentov (v\u00fdznam a princ\u00edp) a zhlukov\u00e1 anal\u00fdza (met\u00f3dy).<\/p>\n

21. N\u00e1vrh regresn\u00e9ho experimentu.<\/h3>\n

Vysvetlite pojem experiment, pokus. \u010co je cie\u013eom optimaliz\u00e1cie experimentu? \u010co je to n\u00e1vrh experimentu? Ur\u010dte z\u00e1kladn\u00fd postup pri ur\u010dovan\u00ed optim\u00e1lneho n\u00e1vrhu experimentu.<\/p>\n

22. Regresn\u00fd model experimentu s nekorelovan\u00fdmi meraniami.<\/h3>\n

Line\u00e1rny regresn\u00fd model. Dok\u00e1\u017ete, \u017ee v pr\u00edpade regul\u00e1rnej informa\u010dnej matice M<\/strong> s\u00fa odhady z\u00edskan\u00e9 line\u00e1rnou regresiou nevych\u00fdlen\u00e9 a efekt\u00edvne s\u00a0kovarian\u010dnou maticou M<\/strong>-1<\/sup>. Lineariz\u00e1cia neline\u00e1rnej regresie, pos\u00fadenie presnosti lineariz\u00e1cie.<\/p>\n

23. Krit\u00e9ri\u00e1 optimality regresn\u00e9ho experimentu<\/h3>\n

Vysvetlite pojem krit\u00e9ria optimality n\u00e1vrhu experimentu. Uve\u010fte z\u00e1kladn\u00e9 skupiny krit\u00e9ri\u00ed optimality
\na zara\u010fte do nich Vami zn\u00e1me krit\u00e9ri\u00e1 optimality. Odvo\u010fte gradient pre aspo\u0148 jedno vami vybran\u00e9 krit\u00e9rium. Ako ohodnocujeme stupe\u0148 optimality n\u00e1vrhu?.<\/p>\n

24. Met\u00f3dy v\u00fdpo\u010dtu optim\u00e1lneho n\u00e1vrhu experimentu.<\/h3>\n

Zhr\u0148te a pop\u00ed\u0161te s\u00fahrnn\u00e9 a \u010diastkov\u00e9 krit\u00e9ria optimality. Pop\u00ed\u0161te itera\u010dn\u00fa met\u00f3du pre s\u00fahrnn\u00e9 krit\u00e9rium D-optimality a univerz\u00e1lnu itera\u010dn\u00fa met\u00f3du.<\/p>\n

25. Line\u00e1rne programovanie a met\u00f3da vn\u00fatorn\u00e9ho bodu.<\/h3>\n

Formul\u00e1cia \u00falohy line\u00e1rneho programovania pre met\u00f3du vn\u00fatorn\u00e9ho bodu, du\u00e1lna \u00faloha, podmienky komplementarity, centr\u00e1lna cesta, met\u00f3da vn\u00fatorn\u00e9ho bodu a Newtonova met\u00f3da pre centr\u00e1lnu cestu.<\/p>\n

26. Matematick\u00e9 programovanie \u2013 z\u00e1kladn\u00e9 pojmy a Karush-Kuhn-Tuckerove podmienky.<\/h3>\n

Formul\u00e1cia \u00falohy, \u00fa\u010delov\u00e1 funkcia, ohrani\u010denia v tvare rovn\u00edc a nerovn\u00edc, pr\u00edpustn\u00e1 mno\u017eina, optimum. Formul\u00e1cia du\u00e1lnej \u00falohy, Karush-Kuhn-Tuckerove podmienky a ich odvodenie.<\/p>\n

27. Bari\u00e9rov\u00e1 met\u00f3da na rie\u0161enie \u00falohy konvexn\u00e9ho programovania<\/h3>\n

Formul\u00e1cia \u00falohy konvexn\u00e9ho programovania, bari\u00e9rov\u00e1 funkcia, centr\u00e1lna cesta, algoritmus bari\u00e9rovej met\u00f3dy, Newtonova met\u00f3da pre funkciu s bari\u00e9rou a ohrani\u010deniami v tvare rovn\u00edc.<\/p>\n

28. \u00daloha optim\u00e1lneho riadenia pre fixn\u00fd koncov\u00fd \u010das<\/h3>\n

Sformulujte \u00falohu optim\u00e1lneho riadenia pre fixn\u00fd koncov\u00fd \u010das a sp\u00f4sob jej rie\u0161enia. Pou\u017eite a vysvetlite pojmy ako riadiaca a stavov\u00e1 funkcia, stavov\u00e1 diferenci\u00e1lna rovnica, po\u010diato\u010dn\u00fd stav, vo\u013en\u00fd alebo fixn\u00fd koncov\u00fd stav, \u00fa\u010delov\u00e1 funkcia, Hamiltoni\u00e1n, adjungovan\u00e1 \u00faloha, nutn\u00e9 podmienky existencie rie\u0161enia.<\/p>\n

29. Optim\u00e1lne riadenie pohybu hmotn\u00e9ho bodu<\/h3>\n

Sformulujte pr\u00edklad optim\u00e1lneho riadenia s fixn\u00fdm koncov\u00fdm \u010dasom pre pohyb hmotn\u00e9ho bodu v rovine s kon\u0161tantn\u00fdm zr\u00fdchlen\u00edm, ke\u010f riadiacou funkciou je smer zr\u00fdchlenia a ke\u010f maximalizujete hodnotu koncov\u00e9ho stavu pre horizont\u00e1lnu zlo\u017eku r\u00fdchlosti. Na\u010drtnite sp\u00f4sob rie\u0161enia \u00falohy pomocou nutnej podmienky pre jeho existenciu, pri\u010dom zvo\u013ete vo\u013en\u00e9 i fixn\u00e9 koncov\u00e9 stavy.<\/p>\n

30. Stochastick\u00e9 optimaliza\u010dn\u00e9 met\u00f3dy, simulovan\u00e9 \u017e\u00edhanie.<\/h3>\n

Formul\u00e1cia \u00falohy, stru\u010dn\u00fd opis met\u00f3dy Monte Carlo, horolezeck\u00e9ho algoritmu a simulovan\u00e9ho \u017e\u00edhania. Metropolisov algoritmus simulovan\u00e9ho \u017e\u00edhania a jeho aplik\u00e1cia na \u00falohy h\u013eadania najkrat\u0161ej Hamiltonovskej cesty prech\u00e1dzaj\u00facej dan\u00fdmi bodmi roviny.<\/p>\n","protected":false},"excerpt":{"rendered":"

T\u00e9my ot\u00e1zok in\u017einierskych \u0161t\u00e1tnic pre \u0161tudijn\u00fd program Matematicko-po\u010d\u00edta\u010dov\u00e9 modelovanie: 1. Rie\u0161enie hyperbolick\u00e9ho syst\u00e9mu s kon\u0161tantn\u00fdmi koeficientami Pop\u00ed\u0161te rie\u0161enie line\u00e1rneho hyperbolick\u00e9ho syst\u00e9mu parci\u00e1lnych diferenci\u00e1lnych rovn\u00edc s kon\u0161tantn\u00fdmi koeficientami v jednorozmernom pr\u00edpade pomocou charakteristick\u00fdch r\u00fdchlost\u00ed a premenn\u00fdch. Formulujte predpoklady, kedy je tento postup mo\u017en\u00fd. (LeVeque, str. 31-32) 2. Met\u00f3da kone\u010dn\u00fdch objemov pre hyperbolick\u00e9 rovnice Pop\u00ed\u0161te v\u0161eobecn\u00fd tvar […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-786","page","type-page","status-publish","hentry"],"yoast_head":"\nMatematicko-po\u010d\u00edta\u010dov\u00e9 modelovanie<\/title>\n<meta name=\"description\" content=\"Katedra matematiky a deskript\u00edvnej geometrie Stavebnej Fakulty Slovenskej Technickej Univerzity v Bratislave. 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