Efficient reduced basis treatment of non
➤ Gửi thông báo lỗi ⚠️ Báo cáo tài liệu vi phạmNội dung chi tiết: Efficient reduced basis treatment of non
Efficient reduced basis treatment of non
Mathematical Modelling and Numerical Analysis Modclisation Mathcrnatique et Analyse NiimeriqucWill be ect by the publisherEFFICIENT REDUCED-BASIS TREA Efficient reduced basis treatment of non ATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONSM.A. Grepl* 1, Y. Maday2, N.c. Nguyen3 and A.T. Patera1Abstract. Ill this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter depende Efficient reduced basis treatment of non nce to problems involving (a) nonaffine dependence on the parameter, and (Ờ) nonlinear dependence on the field variable. The method replaces the nonafEfficient reduced basis treatment of non
finc and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We firsMathematical Modelling and Numerical Analysis Modclisation Mathcrnatique et Analyse NiimeriqucWill be ect by the publisherEFFICIENT REDUCED-BASIS TREA Efficient reduced basis treatment of non nd (tt) a stable and inexpensive Interpolation procedure. Wo then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximat ion and the associated offline-online computational procedures. Numerical results arc pr Efficient reduced basis treatment of non esented to assess our approach.1991 Mathematics Subject Classification. 3-'i.l25^5JG0,35K15,35K55.March 21). 2006.1. IntroductionThe design, optimizatEfficient reduced basis treatment of non
ion, control, and characterization of engineering components or systems often requires repeated, reliable, and real-time prediction of selected perforMathematical Modelling and Numerical Analysis Modclisation Mathcrnatique et Analyse NiimeriqucWill be ect by the publisherEFFICIENT REDUCED-BASIS TREA Efficient reduced basis treatment of non online procedures. elliptic PDEs. parabolic PDEs. nonlinear PDEs1 Massachusetts Institute of Technology, Room 3-264, Cambridge, MA USA- University Pierre et Marie Curie-Parish, Ư.MII 7568 Laborntoirc Jacques.Louts Lions. B.c. 1S7. Paris. F-75005 Prance, E-mail: madayU.-inn.jussicu.fr: and Division o Efficient reduced basis treatment of non f Applied Mathematics. Brown University3 Massachusetts Institute of Technology. Room 37-4.tr>. Cambridge, MA USA1 Massachusetts Institute of TechnologEfficient reduced basis treatment of non
y, Room 3-266, Cambridge, MA USA'Here superscript "e" shall refer to "exact.’' We shall Later introduce a "truth approximation" which will bear no supMathematical Modelling and Numerical Analysis Modclisation Mathcrnatique et Analyse NiimeriqucWill be ect by the publisherEFFICIENT REDUCED-BASIS TREA Efficient reduced basis treatment of non ionals of a field variable, u6(ft) such as temperatures or velocities associated with a parametrized partial differential equation that describes the underlying physics; the parameters, or “inputs," ft, serve to identify a particular configuration of the component or system geometry, material proper Efficient reduced basis treatment of non ties, boundary conditions, ami loads. The relevant system l>ehavior is thus described by an implicit input-output relationship. s'ifi). evaluation ofEfficient reduced basis treatment of non
which demands solution of the underlying ]>artial differential equation (PDE).The abstract formulation for an elliptic problem can be stated its folloMathematical Modelling and Numerical Analysis Modclisation Mathcrnatique et Analyse NiimeriqucWill be ect by the publisherEFFICIENT REDUCED-BASIS TREA Efficient reduced basis treatment of non r domain in which our P-tuple (input) parameter /X resides; X‘:(ii) is an appropriate Hilbert space; ỈỈ is a bounded domain in lỉ,f with Lipschitz continuous boundary ỚÍĨ; /(•://), /■(■) are x°-continuoua linear functionals: and «(•,•;p) is a A'^-continuous bilinear form.in actual practice, of cours Efficient reduced basis treatment of non e, we do not. have access to the exact solution; we thus replace uc(fi) with a “truth" approximation, m(p), which resides in (say) a suitably fine pieEfficient reduced basis treatment of non
cewise-linear finite clement approximation space A' c X” of very large dimension JV. Our “truth” approximation is thus: given any ft e T>, we evaluateMathematical Modelling and Numerical Analysis Modclisation Mathcrnatique et Analyse NiimeriqucWill be ect by the publisherEFFICIENT REDUCED-BASIS TREA Efficient reduced basis treatment of non tion is sufficiently rich such that ti(p) and «*■(//) and hence s(p) and .<(//) are indistinguishable at the accuracy level of interest. The reduced-basis approximation shall be built upon this reference (or "truth”) finite element, approximation, and the reduced-basis error will thus l>e evaluated Efficient reduced basis treatment of non with respect to ix(p) E X. Our formulation must l>e stable and efficient as Af -» 00.We now t urn to the abstract formulation for the controlled parabEfficient reduced basis treatment of non
olic case. For simplicity, in this paper we will direct ly consider a time-discrete framework associated to t he time interval / =)0, t/]. We divide /Mathematical Modelling and Numerical Analysis Modclisation Mathcrnatique et Analyse NiimeriqucWill be ect by the publisherEFFICIENT REDUCED-BASIS TREA Efficient reduced basis treatment of non lso introduce K - {I,...,A'}. We shall consider Euler-Backwanl for the time integration; we can also readily treat higher-order schemes such as Crank-Nicolson [12], The “truth" approximation is thus: given any /< € 'D. we evaluate the output s(/i,/fc) = /(«(/*, f*)), Vk ệ K, where u(fi,tk) (. X sati Efficient reduced basis treatment of non sfies+ Atrt(u(//.tx),«.■;//.) =*),«) -t- At f(tt;ji)b(f.k), v» E X, V£ € K, (3)with initial condition (say) !/(//,t°) = «u(/x) = Í). Here, /(-,/() andEfficient reduced basis treatment of non
f(-) are Y" continuous (A” c Y") linear functionals, /«(•,•) is a yc-eontinuous bilinear form, and b(tk) is the control input. We note that the outpuMathematical Modelling and Numerical Analysis Modclisation Mathcrnatique et Analyse NiimeriqucWill be ect by the publisherEFFICIENT REDUCED-BASIS TREA Efficient reduced basis treatment of non permit the mptdyer accwmte and reliable prediction of these PDE-indnced input-output relationships in real time, or in the limit oj many queries relevant, for example, in the design. optimization, control, and characterization contexts. To achieve this goal we will pursue the reduced-basis method. T Efficient reduced basis treatment of non he reduced-basis method was first introduced in the late 1970s for the nonlinear analysis of structures (1,25) and subsequently attracted and analyzedEfficient reduced basis treatment of non
(5,11,28,33) and extended [16,18.26) to a much larger class of parametrized partial differential equations. The foundation of the reduced basis methoMathematical Modelling and Numerical Analysis Modclisation Mathcrnatique et Analyse NiimeriqucWill be ect by the publisherEFFICIENT REDUCED-BASIS TREA Efficient reduced basis treatment of non its projection on a finite and low dimensional vector space: for sufficiently well chosen Hi, then: exist coefficients Cj cjv(/i) such that the finite sum iSi-i «'|1'(/'|) *s very close to w(/<) for any I>.More recently, the reduced-basis approach and also associated 0 posteriori error estimation p Efficient reduced basis treatment of non rocedures have been successfully developed for (») linear elliptic and parabolic PDEs that are affine in the parameter [13,20,21,29,40] the bilinear fEfficient reduced basis treatment of non
orm a(w.v,/ỉ) can be expressed asQ u(w,v;n) O7(/r)v),-40—1where the f-)'1' : T> — H and rt’(tti,w), 1 < q < Q. are parameter dependent, functions and Mathematical Modelling and Numerical Analysis Modclisation Mathcrnatique et Analyse NiimeriqucWill be ect by the publisherEFFICIENT REDUCED-BASIS TREA Efficient reduced basis treatment of non n particular. a{w. v:/i) satisfies (4) and is at most quadratic in w (but 3https://khothuvien.cori!of course linear in t>). In these cases a very efficient offiine-online computational st rategy relevant in the many-query and real-time contexts can be developed. The operation count for the online st Efficient reduced basis treatment of non age in which, given a new parameter value, we calculate the reduced-basis output and associated error bound depends on a low power of the dimension ofEfficient reduced basis treatment of non
t he reduced-basis space N (typically small) and Q-. but it is independent of Ai t he dimension of the underlying “truth’' finite element approximatiMathematical Modelling and Numerical Analysis Modclisation Mathcrnatique et Analyse NiimeriqucWill be ect by the publisherEFFICIENT REDUCED-BASIS TREAMathematical Modelling and Numerical Analysis Modclisation Mathcrnatique et Analyse NiimeriqucWill be ect by the publisherEFFICIENT REDUCED-BASIS TREAGọi ngay
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