An algebraic least squares reduced basis method for the solution of parametrized Stokes equations

Transcription

1 MATHICSE Institute of Matematics Scool of Basic Sciences MATHICSE Tecnical Report Nr September 2017 An algebraic least squares reduced basis metod for te solution of parametrized Stokes equations Niccolo Dal Santo, Simone Deparis, Andrea Manzoni, Alfio Quarteroni Address: ttp://maticse.epfl.c EPFL - SB INSTITUTE of MATHEMATICS Maticse (Bâtiment MA) Station 8 - CH Lausanne - Switzerland

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3 An algebraic least squares reduced basis metod for te solution of parametrized Stokes equations N. DAL SANTO 1, S. DEPARIS 1, A. MANZONI 1, A. QUARTERONI 1,2 September 29, CMCS, École Polytecnique Fédérale de Lausanne (EPFL), Station 8, 1015 Lausanne, Switzerland. 2 Mox-Laboratory for Modeling and Scientic Computing, Department of Matematics, Politecnico di Milano, P.za Leonardo da Vinci 32, Milano, Italy (on leave). Abstract In tis paper we propose a new, purely algebraic, Petrov-Galerkin reduced basis (RB) metod to solve te parametrized Stokes equations, were parameters serve to identify te (variable) domain geometry. Our metod is obtained as an algebraic least squares reduced basis (als-rb) metod, and improves te existing RB metods for Stokes equations in several directions. First of all, it does not require to enric te velocity space, as often done wen dealing wit a velocity-pressure formulation. Tis because it relies on a Petrov-Galerkin RB metod rater tan on a Galerkin RB (G-RB) metod, and exploits a suitable approximation of te matrix-norm in te denition of te (global) supremizing operator. Te knowledge of an analytical map between a reference domain and te pysical, parameter-dependent domain, is not requested. For tis reason, possible deformations of te domain wic result from te solution of a computational problem can be accounted for. Te new metod also provides a fully automated procedure to assemble and solve te RB problem, able to treat any kind of parametrization. We prove te stability of te resulting als-rb problem (in te sense of a suitable inf-sup condition) and provide a numerical comparison between te proposed metod and te current state-of-art G-RB metods relying on te enricment of te velocity space. In particular, our approac results in a ceaper, more convenient option bot during te oine and te online stage of computation, compared to te existing G-RB metods; numerical results are provided on a problem featuring a parameter-dependent geometry and more tan 10 6 degrees of freedom regarding its ig-delity nite element approximation. 1 Introduction Solving saddle-point problems depending on a set of input parameters is a relevant task in several engineering contexts. In tis work, as a special instance of suc a problem, we consider te parametrized Stokes equations, wic describe a viscous incompressible uid wen te nonlinear convective terms are neglected. Te numerical solution of te Stokes equations is in general a callenging task, owever a remarkable family of ecient metods exists, see e.g. [21, 32, 36]. An even more callenging task occurs wen devising an ecient solver wen a parametrized version of te Stokes equations is considered. Tis is precisely te scope of tis investigation. In tis paper we deal wit te ecient solution by means of te reduced basis (RB) metod of parametrized steady Stokes equations of te form ν u + p = f in Ω u = 0 in Ω (1) + b.c. wic can be seen as a prototypical example of parametrized saddle-point problem, modeling te velocity u and te pressure p of a viscous incompressible uid wit viscosity ν in a domain Ω R d, d = 2, 3. In (1), te vector of parameters describes pysical and/or geometrical properties of te system, wereas f is te known rigt and side collecting all te data of te problem; system (1) must be supplied wit proper boundary conditions, wic may also depend on. In particular, we are interested in te ecient solution of (1) for several (say, tousands) instances of. Tis requirement arising, e.g., wen dealing wit uncertainty quantication, sensitivity analysis or PDEconstrained optimization makes usual ig-delity approximation tecniques, suc as te nite element 1

4 (FE) metod, extremely expensive, if not computationally unaordable. In fact, te FE approximation of problem (1) entails te solution of a parametrized, saddle-point (linear) system wose dimension N is usually very large, were is related to te mes size; in tree-dimensional applications of real-life interest, N is typically of order of millions. Assembling from scratc, and ten solving, te corresponding linear system for any new value of is tus unfeasible. Suc a task can be more easily tackled by means of reduced-order modeling tecniques. Among all te available metodologies, we focus on te reduced basis (RB) metod to deal wit (1). Te main idea of te RB metod for parameter dependent partial dierential equations (PDEs) is to approximate its solution by a linear combination of few global basis functions, obtained from a set of FE solutions (or snapsots) corresponding to dierent parameter values [33, 24]. Tis strategy is pursued in two stages: an oine pase and an online pase. In te former, we construct a RB space V N of dimension N N wose basis is obtained by (properly ortonormalized) linear combinations of FE solutions of te parametrized PDE. In te latter, we require te projection of te residual of te FE problem onto te RB test space to vanis, obtaining a small problem wic replaces te original ig-delity FE problem. Initially applied to linear elliptic PDEs, te RB metod as been extended to saddle-point problems suc as te Stokes equations, and extensively investigated in te past decade. In particular, several works ave been devoted to te analysis and te implementation of te RB metod for addressing problems involving Stokes-like systems; a non-exaustive list includes, among oters: Stokes ows featuring ane [39, 23, 38] and nonane parameter dependence [37]; Navier-Stokes ows depending on pysical and/or geometrical parameters [17, 34, 18, 26]; parametrized optimal control problems [31] or sape optimization problems [28] involving Stokes ows. In all tese cases, te RB metod relies on: 1. a (weak) greedy algoritm for te incremental construction of te RB space, performed by selecting a new basis for te velocity and te pressure upon te use of a residual-based a posteriori error estimator. Tis latter is a -dependent quantity related wit te FE approximation and is not always easily available or computable; 2. a Galerkin projection onto te RB space to generate te RB problem (G-RB metod). Of course, tis is not te only available coice. Regarding point 1., proper ortogonal decomposition (POD), rater tan greedy algoritms, can be used to build te RB space. Wen suc a strategy is employed, a set of FE solutions, called snapsots, are computed and te RB spaces for te velocity and te pressure are constructed, eiter jointly or separately, by performing POD, [3, 8, 12, 20, 25, 41]. Tis option as been considered, e.g., in [4] were two-dimensional Navier-Stokes ows on simple geometries anely parametrized ave been treated. Moreover, we remark tat oter possibilities ave been investigated, e.g. in [19], were proper generalized decomposition (PGD) is applied to te Stokes equations in two-dimensional parametrized geometries. Concerning point 2., a more general Petrov-Galerkin (rater tan Galerkin) projection suc as in te case of a least-squares (LS) metod can be performed, coosing a test space dierent from te trial space, see e.g. [14, 16]. Tis option as been rst explored in te case of two-dimensional, anely parametrized Stokes problems on simple geometries in [1]. Moreover, in bot tese cases parameter-dependent domains Ω were obtained as images of a reference domain Ω 0 troug a parameter-dependent map wose expression was known analytically. Tis is a relevant limitation toward te application of RB metods to more general domains wit varying sape, not necessarily obtained in an explicit way from a priori known, parametrized deformations 1. Wat makes te RB approximation of parametrized Stokes equations ard (and, more generally speaking, parametrized saddle-point problems) is ensuring te stability of te resulting RB problem. Tis is te main reason wy, for instance, reduced-order models for uid dynamics problems ave sometimes focused on approximations for te velocity eld uniquely, recovering ten te pressure in a dierent way, rater tan building a reduced-order approximation based on a mixed velocity-pressure formulation, [12]. Indeed, it is well-known tat in te FE case an inf-sup condition must be satised at te nite dimensional level to ensure te well-posedness of te numerical problem. Tis condition is fullled if eiter P 2 P 1 (Taylor-Hood) couples of FE spaces are used for discretizing te velocity and pressure elds, respectively or a stabilized FE formulation is employed, e.g. by relying on te streamline-upwind Petrov-Galerkin (SUPG) metod. Hereon we rely on te former option for te ig-delity FE problem, see e.g. [33, 29] for furter details about te latter. Concerning te stability of te RB approximation, a stable couple of reduced subspaces for velocity and pressure, satisfying an equivalent inf-sup condition at te reduced level, ensures tat te RB Stokes 1 For example, we can regard te proposed strategy as a step towards te ecient reduction of parametrized uid-structure interaction problems, were te deformation of te domain is computed troug a structural problem coupled wit te uid equations. 2

5 problem is well-posed. Tis property is not automatically fullled if te RB problem is constructed troug a Galerkin projection employing RB spaces made solely of ortonormalized solutions of (1) obtained for dierent values of parameters. Two overcome tis sortcoming, two strategies ave been designed. A. Te velocity space can be augmented by means of a set of enricing basis functions computed troug te pressure supremizing operator, wic depends on te divergence term. Tis yields a RB problem wit additional degrees of freedom for te velocity eld (as many as te pressure variable), see [38] for te details. In presence of parameter-dependent domains, te supremizing operator is -dependent, so tat to recover computational eciency (and avoid te construction of te pressure supremizing operator for any value of online), an oine enricment is employed. Tis strategy leads to a RB problem wic is inf-sup stable in practice, but its well-posedness is not proven rigorously. Suc a framework as been originally introduced in conjunction wit a (weak) greedy algoritm [39, 23, 38], and later for te POD case. In te former case, for eac pressure basis selected by te greedy algoritm, a supremizing function is used to augment te velocity space. In te latter case, owever, te basis functions are not directly related to any precise instance of te parameter, so tat a set of enricing functions for te velocity space must be computed in advance starting from te pressure snapsots, ten POD is applied to build te enricing basis [4]. Tis tecnique allows to build a stable RB problem, owever it is not clear, a priori, ow many supremizing functions are needed to properly stabilize te problem. Taking as many enricing functions as te number of velocity and pressure basis is a working rule of tumb, owever very likely tis leads to an excessive number of basis functions. B. We can exploit a Petrov-Galerkin (PG) metod [1, 33] to build an automatically stable RB problem; we coose te least squares (LS) metod. Te resulting LS-RB metod relies on a test space wic is obtained as te image of te RB space troug a global supremizing operator involving bot velocity and pressure elds. Te corresponding algebraic construction of tis operator substantially relies on te coice of te matrix-norm to be used for te velocity and pressure spaces. By tis approac te resulting RB problem is automatically stable tat is, it satises te required inf-sup condition as usually appens wen dealing wit PG-RB metods for weakly coercive problems, see [33] for furter details. However, te existing formulation of te LS-RB metod for Stokes equations proposed in [1] presumes te existence of an explicit dependent function wic enables to recast te problem on a reference domain. Witout tis function available, as in te case were te deformation results from te solution of a FE problem, te computational work to build te RB problem is unbearable. In tis work we propose a new, purely algebraic, PG-RB metod to address large-scale parametrized Stokes equations in domains wit varying geometry. Our metod can be seen as an algebraic LS-RB metod; for tis reason we refer to it as als-rb metod. Te als-rb metod extends and improves te existing RB metods for Stokes equations in several directions, potentially becoming a paradigm for te ecient construction of a stable and accurate RB metod for Stokes equations and, more generally speaking, weakly coercive problems. Indeed: 1. it relies on POD rater tan on a greedy algoritm for te construction of te RB spaces for velocity and pressure, tus avoiding te evaluation of potentially expensive a posteriori error bounds; owever, like in te greedy case, an exponential decay of te residuals wit respect to N is obtained; 2. it does not need an enricment of te velocity space, by relying on a Petrov-Galerkin metod; 3. it exploits suitable approximations of te matrix-norm in te denition of te supremizing operator; 4. te resulting als-rb problem rigorously is inf-sup stable; 5. it does not require any analytical map between a reference domain and te pysical domain Ω ; 6. it provides a fully automated procedure to assemble and solve te RB problem, able to treat any kind of parametrization. To enance te eciency of te resulting RB metods, we employ te discrete empirical interpolation metod (DEIM) [5, 15], and te Matrix DEIM (M-DEIM) [30] to nd an ane approximation of te (generally nonane) rigt and sides and matrices of te ig-delity system. We analyze te well-posedness of te als-rb metod and prove tat under suitable conditions it satises an inf-sup stable RB approximation. A numerical comparison between te proposed als-rb metod and te current state-of-art G-RB metod relying on te enriced velocity space, sows tat te former approac is ceaper, bot during te oine pase, since te velocity enricing snapsots must not be computed, and te online pase, due to te generally smaller dimension of te resulting RB problem. 3

6 We apply te als-rb metod to solve te tree-dimensional Stokes system dened over parameterdependent domains, wit up to millions of degrees of freedom for te ig delity solution, for wic te mapping from a reference domain is not necessarily known analytically. For te case at and, we consider a geometry wic is parametrized wit respect to te deformation obtained by solving an elliptic PDE problem wic armonically extends some Diriclet data, playing te role of imposed sape deformation. Te paper breaks down as follows. In Section 2 we briey recall te Stokes equations and teir FE approximation. In Section 3 we introduce te POD-RB metod, wit particular empasis on ow to construct a stable RB metod eiter using a Galerkin or Petrov Galerkin projection. Ten, we present te als-rb and its analysis. In Section 4 we present numerical results obtained wit te dierent RB approximations and in Section 5 we draw some conclusions. Concerning notation, ereon we denote scalar elds by lower case letters, as a R, vector elds wit an arrow, as a R d, for d = 2, 3, vectors (like nite elements vectors) by bold lower case letters, as a R n, and matrices by bold capital letters, as A R n n. We denote by (, ) 2 te Euclidean scalar product and by K 2 (A) te condition number of te matrix A wit respect te Euclidean norm. Moreover, given a symmetric and positive denite matrix Y R n n, we denote by (, ) Y te scalar product and by Y te norm dened as (a, b) Y = a T Yb a, b R n and a Y = (a, a) Y a R n, respectively. Finally, we denote by K Y (A) te condition number of A wit respect to te norm Y. 2 Parametrized Stokes equations: setting and preliminaries In tis section we introduce te Stokes equations in parametrized domains, togeter wit teir weak formulation and a corresponding FE approximation. Denote by D R l, l N te parameter space and by D a vector of parameters encoding pysical and/or geometrical properties. Heron, te apex means tat a variable depends on te specic coice of te parameter. Given a dependent domain Ω R d, d = 2, 3, suc tat, for any D, Ω = Γ out Γ in Γ w and Γ out Γ in = Γ w Γ in = Γ out Γ w =, te Stokes equations read ν u + p = f in Ω u = 0 in Ω u = g D on Γ in u = 0 on Γ w p n + ν u n = g N on Γ out, were ( u, p ) are te velocity and pressure elds of a viscous incompressible Newtonian uid wit viscosity ν, respectively. We introduce a regular enoug lifting function r g D ( H 1 (Ω ) ) d and te following - dependent spaces V = { v ( H 1 (Ω ) ) d : v Γ = v w Γ = 0 }, in Q = L 2 (Ω ) or Q = L 2 0(Ω ) if Γ out =, equipped wit scalar products (and corresponding induced norms) (, ) V = (, ) (H 1 0 (Ω )) and (, ) d Q = (, ) L2 (Ω ). For a given D, te weak formulation of problem (2) reads: nd ( u, p ) V Q suc tat { a ( u, v) + b ( v, p ) = f ( v) v V b ( u, q) = b ( r (3) g D, q) q Q, were we dene te forms in (3) for u, v V, q Q a ( u, v) = ν u : vdω, Ω b ( v, q) = q vdω Ω f ( v) = f vdω + Ω Γ out g N vdγ out a ( r g D, v). (2) 4

7 Problem (3) can be written as a symmetric non-coercive problem, provided we dene te space = V Q, equipped wit te scalar product and te norm (( u, p), ( v, q)) = ( u, v) V + (p, q) Q, ( u, p), ( v, q), ( v, q) = (( v, q), ( v, q)), ( v, q). Next, we introduce te forms A : R, F : R given by A (( u, p ), ( v, q)) = a ( u, v) + b ( v, p ) + b ( u, q) F (( v, q)) = f ( v) b ( r g D, q). System (3) can tus be equivalently written as: nd z suc tat A ( z, w) = F ( w) w. (4) Te well-posedness of problem (4) is ensured according to te general teory of saddle-point problems [10, 11]. 2.1 Finite element approximation of te Stokes equations Numerical metods based on (Petrov-)Galerkin projection onto a nite dimensional subspace, as te nite element (FE) or spectral element metods, represent a successful tecnique to andle te numerical approximation of (2), see e.g. [13, 21, 35]. However, wen tey are employed, a discrete inf-sup condition must be satised to ensure te well-posedness of te numerical problem. All te reduced order models considered in tis paper for te ecient solution of te parametrized problem (4) inge upon a ig-delity nite element approximation, wic we introduce in tis section. We consider a domain deformation dependent on ; te corresponding meses are also taken as a deformation of a reference mes, ence not aecting te topology of te degrees of freedom. In tis work, we do not focus on renement, terefore we consider a xed regular mes wic is ne enoug for te problem at and. Let us denote by V and Q two nite dimensional FE spaces of dimension N u and N p, respectively, wit V V and Q Q. Moreover, set = V Q wit dimension N = N u + N p. Te FE approximation of problem (4) reads: nd z suc tat A ( z, w ) = F ( w ) w. (5) We furter assume tat te following uniform inf-sup condition olds: tere exists a positive constant β min > 0, independent of, suc tat β = inf z sup w A ( z, w ) z w β min D. (6) A couple of FE spaces wic fullls condition (6) is given by P 2 P 1 (Taylor-Hood) nite elements, for velocity and pressure, respectively. Condition (6) ensures te stability of problem (5). Problem (5) can be equivalently written as a parametrized linear system A z = g, (7) featuring a saddle-point structure, were [ A D = (B ] [ )T B, z u ] 0 = p [ and g f ] = r, (8) were A RN N and z, g RN. More precisely D RN u N u, B RN p N u, f RN u and nally r RN p. Te solution of system (7) exploits suitable iterative metods properly preconditioned. Several tecniques relying on domain decomposition, multilevel metods and block factorizations ave been proposed as preconditioners, see e.g. [21, 32, 40, 36] and references terein. See also [6, 7] for an extensive review on numerical metods for saddle-point systems. 5

8 Condition (6) can be algebraically expressed as follows: tere exists β min > 0 suc tat β = inf z R N sup w R N w T A z z w β min D, (9) were te symmetric and positive denite matrix RN N encodes te scalar product (, ) on te space and is built as a block diagonal matrix of te form [ ] = u 0 0 ; (10) p u R N u N u and p R N p N p encode te scalar products on te spaces V and Q, respectively. Notice tat since te computational domain is dependent, also te matrix depends on te parameter. 3 POD-based RB metods for te parametrized Stokes equations Te RB metod represents a convenient framework for te reduction of parametrized PDEs [33, 24]. Suc a metod essentially combines a tecnique to generate a low-dimensional subspace of were te RB solution is sougt, and a Galerkin-type projection (eiter Galerkin or Petrov-Galerkin) onto te subspace to obtain te corresponding RB problem to be solved. Here we rely upon proper ortogonal decomposition (POD) because we do not ave a ceaply computable a-posteriori error bound, wic would instead be required if a greedy algoritm were performed. Ten, a new algebraic PG-RB metod is investigated for te sake of te construction of te RB problem, and compared to te (indeed, more classical) Galerkin-RB metod wic relies on properly enriced RB spaces. In te following, we recall te essential elements of tis tecnique and ow it is used to build a RB approximation. 3.1 Constructing te RB space: proper ortogonal decomposition Let us consider a set of n s FE vectors {s i } ns i=1 RN (called snapsots) collected as columns of a matrix S = [s 1... s ns ], S R N n s. For any prescribed dimension N, te POD allows to nd an ortonormal basis {ξ i } ns i=1 and te corresponding N-dimensional subspace, spanned by te columns of te matrix V = [ξ 1... ξ N ], V R N N wic best approximates {s i } ns i=1 up to a tolerance ε POD wit respect to a prescribed norm induced by a symmetric positive denite matrix R N N. Te POD metod takes advantage of te singular value decomposition (SVD) of te matrix S S = UΣZ T, wit U R N N and Z R n s n s ortogonal matrices and Σ = diag(σ 1,... σ ns ), Σ R N n s, containing te singular values σ 1 σ 2 σ ns 0. Ten, V is provided by te rst N columns of U, wic form by construction an ortonormal basis for te best N-dimensional approximation subspace. More in particular, denoting I N R N N te N-dimensional identity matrix, te following proposition olds [33]. Proposition 3.1. Let V N = {W R N N : W T W = I N } be te set of all N-dimensional - ortonormal bases. Ten n s i=1 s i VV T s i 2 = min W V N n s i=1 s i WW T s i 2 = n s i=n+1 Moreover, notice tat te relative error on all te snapsots is related to {σ i } ns i relation n s i=1 s i VV T s i 2 n s = s i 2 i=1 n s σi 2 i=n+1 n s σi 2 i=1 σ 2 i. troug te following. (11) From a practical perspective, te POD basis construction is done following Algoritm 1. At rst te correlation matrix C ns = S T S is formed and te corresponding eigenvalue problem is solved. Ten, for a given tolerance ε POD, (11) is employed to control te relative error on te approximation of te snapsots and select N basis functions. Alternatively, one could directly provide a dimension N instead of ε POD, leading to a similar algoritm POD(S,, N). 6

9 Algoritm 1 POD 1: procedure POD(S,, ε POD ) 2: form te correlation matrix C ns = S T S 3: solve te eigenvalue problem C ns ψ i = σ 2 i ψ i, i = 1,..., n s 4: set ξ i = 1 σ i Sψ i 5: dene N as te minimum integer suc tat 6: dene V = [ξ 1... ξ N ] 7: end procedure N i=1 σ2 i ns i=1 σ2 i > 1 ε 2 POD 3.2 Projection-based RB metods Te RB metod relies on te idea tat te solution of te parametrized system (7), for a certain value of te parameter, can be well approximated by a linear combination of N basis functions { ξ i } N i=1 obtained by ortonormalizing te solutions of te same problem for oter values of te parameter. Te basis functions are collected in te so-called RB space, wic is dened as V N = span{ ξ i, i = 1,..., N} (12) of dimension N N. From an algebraic standpoint V N is represented by te matrix V = [ξ 1... ξ N ] R N N, were ξ i, i = 1,..., N are te FE vector representation of te basis ξ i, i = 1,..., N. From a practical standpoint, te vector basis {ξ i } N i=1 is constructed by exploiting POD were te snapsots are FE solutions of te FE linear system for many instances of te parameter, i.e. s i = z i, i = 1,..., n s Ten, te RB approximation is constructed by introducing a set of (possibly dependent) functions {w i }N i=1 suc tat a test space W N is obtained as W N = span{w i, i = 1,..., N}. Algebraically, W N is represented by a matrix W R N N, wic is generally dierent from V and may be -dependent. If W V we ave te more general Pevtrov Galerkin-RB approximation, oterwise if W = V we come up wit te usual Galerkin case. For te sake of generality, we consider te PG-RB problem, wic reads: nd z N V N suc tat A ( z N, w N) = F ( w N ) w N W N. (13) In order to obtain a well-posed RB approximation, an inf-sup condition at te RB level must also be satised. Specically, tere must exist βn min > 0 independent of suc tat β N = inf z N V N sup w N W N A ( z N, w N ) z N w N β min N > 0 D. (14) Ensuring tis condition, as we will see in te following, essentially depends on te type of projection used to generate te RB problem and te way te RB spaces are built. Problem (13) leads to te following algebraic RB linear system A N z N = g N, (15) were te RB matrix A N RN N and te RB rigt and side g N RN are dened as A N = (W ) T A V, g N = (W ) T g. (16) We igligt tat te PG-RB approximation depends on te coice of te test space W N. As remarked above, te matrix V is built employing te POD metod, wic in te Stokes case turns to [ ] V u V = Nu 0 0 V p = [ ] ξ 1... ξ Nu ξ Nu+1... ξ N, (17) N p were VN u u R N u Nu and V p N p R N p Np are te basis to approximate te velocity u p, respectively. In particular [ ] [ ] ϕi 0 ξ i = i = 1,..., N 0 u, ξ Nu+i = ψ p i = 1,..., N p, i and te pressure 7

10 were { } Nu ϕ i and { } Np ψ i i are te basis functions for te velocity and te pressure RB space, tat is, i V u N u = [ ϕ u 1... ϕ Nu ], V p N p = [ ψ 1... ψ Np ]. } ns, i=1 Te construction of te RB spaces is tus performed by rst collecting a set of FE snapsots { u i { p i } ns, solutions of (7) for dierent instances of te parameters { } ns i=1 i, and ten performing POD i=1 separately on te two spaces ) ) = P OD (S u, u, ε POD, V p N p = P OD (S p, p, ε POD. V u N u Te matrices VN u u and V p N p are constructed by selecting te largest N u and N p eigenmodes respectively, as explained in Algoritm 1, nally obtaining N u + N p = N, see [33]. Notice tat a priori N u N p. Te trial reduced basis spaces are dened by considering te sets of basis { ϕ i } Nu i=1 and {ψ i} Np i=1, wose FE vector representations are given by {ϕ i } Nu i=1 and {ψ i} Np i=1. Ten, we dene V Nu = span{ ϕ i, i = 1,..., N u } Q Np = span{ψ i, i = 1,..., N p }, and V N = (V Nu, Q Np ). We nally remark tat te dimension N = N u + N p of te RB system is smaller of te dimension N of te FE linear system of several orders of magnitude: N N, so tat problem (15) is solved by direct metods. Several ways to produce a well-posed Stokes RB problem, relying eiter on Galerkin or Petrov-Galerkin projection, are available. Te Galerkin RB approximation relies on a velocity enricment strategy, were te velocity RB space V Nu is augmented wit an additional RB space built troug a pressure supremizing operator, as explained e.g. in [4]. Tis metod as been proposed for bot POD and greedy RB space construction, and, even if empirically it works properly, it does not rigorously ensure te well-posedness of te resulting RB problem. Furtermore, wen using POD it is unclear ow large te augmenting space sould be. In te following sections we report bot te Galerkin and Petrov-Galerkin RB approximations as well as alternative ways to construct te RB spaces, to deal wit te parametrized Stokes equations; in te section of numerical experiments, we will report results using bot tese tecniques Galerkin-RB metod wit velocity enricment A Galerkin-RB formulation is obtained by coosing W N = V N (or algebraically W = V) in (16), resulting in a RB approximation wose well-posedness is guaranteed by satisfying te following assumption: tere must exist > 0 suc tat β min N β N = inf q N Q Np b ( v N, q N ) sup v N V Nu v N V q N Q β min N > 0 D. (18) Unfortunately, as explained above, condition (18) is not automatically satised wen te RB spaces V Nu and Q Np are constructed by POD, or by greedy algoritms, by considering basis functions extracted from velocity and pressure snapsots only. Consequently, we consider an "enriced" velocity space formulation, as proposed in [4], were te velocity space V Nu is augmented to guarantee te well-posedness of te resulting RB approximation. Algebraically, tis is pursued by building a matrix VN s s R N u Ns wose columns form a basis for te enricing RB velocity space. Ten, te G-RB approximation is built by considering V = W = Ṽ in (16), were Ṽ is dened as [ V u Ṽ = Nu VN s ] s V p. N p Te enricing strategies are based upon te use of te pressure-supremizing operator T p : Q V suc tat for any given q Q, it yields T p (q ) as te solution of te following problem (T p (q ), v ) V = b ( v, q ) v V. (19) Equation (19) corresponds to a FE problem wose algebraic formulation yields te following linear system u t p (q ) = (B )T q, (20) were q R N p is te FE vector representation of q Q. Two strategies ave been developed to build a well-posed G-RB approximation for a new parameter : 8

11 build for eac pressure basis {ξ i } N i=n u+1 te corresponding supremizing functions {t p (ξ i )} N i=n u+1 and dene V s N s = [t p (ξ Nu+1)... t p (ξ Nu+1)], leading to a RB formulation wic by denition satises (18). However, in tis way te construction of te supremizing enricing functions is not computationally feasible, because it entails (online) te solution of N p FE linear system for eac new value of ; compute a set of supremizing snapsots { t i p (p i )} n s corresponding to te pressure snapsots { p i i=1 troug (20), and ten build te matrix VN s s troug POD V s N s = P OD( {t i p (p i )} n s i=1, u, ε POD ). Notice tat tis option does not ensure tat condition (18) (or any equivalent one) is satised. Moreover, te number N s of basis functions for VN s s is cosen, wit a rule of tumb, equal to N u, doubling te size of te RB velocity space. Tis looks like a reliable option wic yields a stable RB problem for te steady Navier-Stokes equations, see [4] LS-RB metod Instead of performing a Galerkin projection onto properly enriced RB spaces, te Petrov-Galerkin (PG)- RB metod uses a dierent test space W and naturally builds an inf-sup stable RB problem. Te PG-RB metod as been rstly analyzed for te anely parametrized Stokes equations in [1] were te RB space is built upon a greedy algoritm. In tis work we deepen te analysis carried out in [1], propose several strategies to make tis metod computationally ecient and use instead te POD metod for te construction of te RB space, wic does not need any error estimator. Moreover, we do not assume to ave an analytical function wic maps te reference domain Ω 0 to te pysical domain Ω ; te main consequence is tat we consider te more general case were recasting te problem on a reference, parameterindependent domain Ω 0 is not possible. We restrict ourselves to te case of PG-RB metod built troug te least-squares (LS) metod, wic automatically guarantees to obtain an inf-sup stable problem. Wit tis aim, we introduce a dierent (global) supremizing operator T :, suc tat (T ( z ), w ) = A ( z, w ) w. (21) Wit respect to te denition of T p provided by (19), bot te velocity and pressure appear, togeter wit te full Stokes operator at te rigt and side. Given z, problem (21) is a dependent FE problem wic needs to be solved for T ( z ). Ten, te RB problem reads as (13), were te test space is cosen as W N = span{ T ( ξ i ), i = 1,..., N }, wile te trial RB space is cosen as in (12) wit te corresponding matrix V as in (17). From an algebraic standpoint, given z R N, te supremizing solution t (z ) is obtained by solving te linear system t (z ) = A z. (22) Te projection matrix W, wose columns are supremizers of type (22) and form a basis for te (parameterdependent) test space, is ten given by } ns W = ( ) 1 A V, (23) were is te dependent norm matrix (10). Finally, te linear system (15) representing te LS-RB problem is recovered wit A N = VT (A )T ( ) 1 A V g N = VT (A )T ( ) 1 g. (24) i=1 Te following results old, see also [1, 33]. Lemma 3.1. Assume tat condition (9) olds and W is taken as in (23). Ten, te LS-RB problem (15) is uniformly inf-sup stable, tat is, tere exists β min > 0 independent of suc tat β N = inf z N R N sup w N R N wn T A N z N Vz N W w N β min D. 9

12 Moreover, it as a unique solution z N RN for any D, wic satises z N 1 β g ( ) 1. N Proof. We report an algebraic variant of te proof of te result sown in [1]. Starting from (22) and te Caucy-Scwarz inequality w T A z = w T t (z ) t (z ) w and te equality is reaced for w = t (z ). Ten, we ave w R N, β N = inf z N R N inf z R N sup w N R N t (z ) z wn T A N z N Vz N W w N β min. = inf z N R N t (Vz N ) Vz N Te proof is concluded by employing te Babu ska teorem for non-coercive problems satisfying an inf-sup stability property, see [2]. Remark 3.1. Te solution z N RN of problem (15) solves te following minimization problem z N = arg min g v N R N A Vv N 2 (, (25) ) 1 i.e. te RB solution minimizes te residual in te norm induced by te symmetric positive denite matrix ( ) 1, see [33] for furter details. 3.3 Algebraic LS-RB metod Te LS-RB metod described in Section requires to build te -dependent matrix ( ) 1 or to solve approximately te N linear systems (22) associated wit te matrix to construct a stable RB problem for any new parameter instances D considered online. If an analytical map is available, one can recast problem (4) over te reference domain Ω 0 by using te Jacobian of te map. In tis way, te LS-RB problem would be built wit respect to te reference domain, and te independence of te norm matrix on would be easily acieved. However, if te displacement of te domain is not analitically available, it is not possible to rely on tis strategy. In tis section we propose a purely algebraic PG-RB metod wic can be viewed as an algebraic LS-RB (als-rb) metod described above for parametrized noncoercive problems as (7). Compared to te approximate enricment of te velocity space described in section (3.2.1), te als-rb metod allows to build a RB problem wic is automatically and rigorously inf-sup stable and encefort it does not require to enric te velocity space doubling te degrees of freedom of te velocity. Te underlying idea is to substitute te matrix appearing in te denition of te test space (23) by a properly cosen surrogate P R N N. To tis aim, we suppose te following assumption to old. Assumption 3.1. Te matrix P R N N is symmetric and positive denite and induces a norm x 2 P = (x, x) P = x T P x for any x R N. Moreover, tere exist two positive constants c and C independent of suc tat c x P x C x P x R N. (26) Next, we introduce a sligtly modied supremizing operator T P : V V V problem dened by te following (T P ( z ), w ) P = A ( z, w ) w V, (27) were te dierence wit respect to (21) is te coice of te scalar product wit respect to wic te operator is built. Reasoning as in te previous section, we introduce a PG problem under te form: nd z N V N suc tat were now te test space is cosen as A ( z N, w N ) = F ( w N ) w N W N,P, (28) W N,P = span { T P ( ξ i ), i = 1,..., N }, 10

13 were { ξ i } N i=1 are te RB functions dening V N in (12). Problem (27) is algebraically equivalent to solving and yields a projection matrix of te following form P t P (z ) = A z, (29) Finally, te corresponding RB system is W P = P 1 A V. (30) A N,P z N = g N,P, (31) were te RB matrix A N,P R N N and te RB rigt and side g N,P R N are dened as A N,P = V T (A )T P 1 A V g N,P = V T (A )T P 1 g. (32) Remark 3.2. Equations (32) are similar to te ones in (24), provided tat is substituted wit P. In te following we provide results sowing te stability of system (31) and te optimality properties satised by te solution z N of (31). Proposition 3.2. Assume tat condition (9) olds, W is taken as in (30). and let assumption 3.1 old. Ten problem (31) is inf-sup stable, more precisely β P,N = inf z N R N sup w N R N wt N A N,P z N Vz N W P w N Moreover, problem (31) as a unique solution z N RN for any D, wic satises Proof. Starting from (29), it olds z N 1 β g ( ) 1. P,N c C βmin D. (33) w T A z = w T P t P (z ) t P (z ) P w P w R N, were te equality is reaced for w = t P (z ). Consequently, using te inequalities in (26) we ave β P,N = inf z N R N sup w N R N = 1 C inf t P (Vz N ) P z N R N Vz N = 1 C inf z R N sup w R N = c C β c C βmin. wt N A N,P z N Vz N W P w N 1 C 1 C inf z N R N inf t P (z ) P z R N z w T A z z w c P C inf z R N sup w R N sup w N R N wt N A N,P z N Vz N W P w N P w T A z z w By applying te Babu ska teorem for non-coercive problems satisfying an inf-sup stability property, see [2], concludes te proof. Proposition 3.3. Let assumption 3.1 old, ten problem (31) corresponds to solving te minimization problem z N = arg min g v N R N A Vv N 2. (34) P 1 Proof. We consider te quadratic functional J(v N ) = g A Vv N 2, v P 1 N R N, 11

14 wic as a unique minimum in u N R N tanks to te nonsingularity of te matrices P and A. We impose its gradient wit respect to v N and evaluated at u N to vanis. By employing te denition of te norm P 1 we obtain 0 = J{v N} v N (u N ) Terefore, u N is suc tat = v N { (g )T P 1 g + vt N V T (A )T P 1 A Vv N 2(g )T P 1 A Vv N = 2V T (A )T P 1 A Vu N 2(g )T P 1 A Vu N = 2A N,P u N 2g N,P. A N,P u N = g N,P, } (u N ) ence it coincides wit te RB solution z N, since te matrix A N,P is invertible Assembling te RB problem Wen building a RB approximation, it is essential to assume te ane dependence on in te FE arrays (7), tat is Q a A = Θ q a()a q, q=1 g = Q g Θ q g()g q, (35) q=1 were Θ q a : D R, q = 1,..., Q a and Θ q g : D R, q = 1,..., Q g are -dependent functions, wile te matrices A q RN N and te vectors g q RN are -independent. If assumption (35) is veried, ten te RB algebraic structures can be written, for te als-rb case, as A N,P = = g N,P = = Q a q 1,q 2=1 Q a q 1,q 2=1 Q a Q g q 1=1 q 2=1 Q a Q g q 1=1 q 2=1 Θ q1 a ()Θ q2 a ()V T (A q1 )T P 1 Aq2 V Θ q1 a ()Θ q2 a ()A q1,q2 N (36) Θ q1 a ()Θ q g()v T (A q1 )T P 1 gq2 In te G-RB case, te algebraic RB structures can be instead obtained as Θ q1 a ()Θ q2 g ()g q1,q2 N. (37) Q a A N = Q a Θ q a()v T A q V = Θ q a()a q N (38) q=1 q=1 Q g Q g N = g Θ q g()v T g q = Θ q g()g q N. (39) q=1 Te matrices A q N, q = 1,..., Q a, A q1,q2 N R N N, q 1, q 2 = 1,..., Q a, and te vectors g q N RN, q = 1,..., Q g, g q1,q2 N R N, q 1 = 1,..., Q a, q 2 = 1,..., Q g can be precomputed and stored during te oine pase. During te online pase, only te sums in (36)(37) and (38)(39) must be calculated out to assemble te RB problem. Notice tat te construction of A N and g N in (38)(39) depends linearly on te number of ane terms Q a and Q g for te G-RB metod. On te oter and, te corresponding als-rb structures A N,P and g N,P in (36)(37) depend quadratically Q a and Q g. Practically, by employing te G-RB metod softens te dependence on te number of ane terms, since less RB structures must be assembled and stored wit respect to te als-rb metod. Tis advantage is also visible in te online pase, since te construction of te RB matrix and rigt and side scale linearly wit respect to Q a and Q g. However, te als-rb matrices and rigt and sides ave a smaller dimension, since te velocity basis is not augmented, q=1 12

15 entailing a lower cost for computing and storing eac array and for computing te solution of te RB system. Finally, notice tat te ane decomposition (35) would not be exploitable in te case of standard LS-RB metod, due to te dependence of te matrix. Indeed, one would need also an ane decomposition of ( ) 1, wic is generally not available since it is never explicitly assembled and its application is performed by solving a linear system were is at te left and side. In te numerical examples considered in tis work, as well as in almost every problem of applied interest, te geometrical dependence of te computational domain on te parameter is generally nonane, terefore an ane representation of A and g cannot be computed. To circumvent tis problem bot te empirical interpolation metod (EIM) or its discrete variants DEIM and Matrix-DEIM [5, 15, 30] oer te possibility to recover an approximate ane decomposition. Wen suc tecniques are employed, te relations (35) are satised up to a certain tolerance, Q a A q=1 Θ q a()a q, Q g g q=1 Θ q g()g q. (40) Q a and Q g are te number of selected basis computed by te corresponding algoritms. In te case of DEIM (resp. M-DEIM), te basis are again built troug POD on a set of n s vectors (resp. matrices) snapsots and up to a provided tolerance. Ten, for a new value of te parameter, te coecients Θ q g : D R q = 1,..., Q a (resp. Θq a : D R, q = 1,..., Q g ) are computed by solving an interpolation problem On te coice of P A natural question arising in tis context regards te coice of te matrix P, since tis directly aects te values of te constants c and C; from (33). Tese constants play indeed a relevant role in te ill-conditioning of te als-rb approximation. Moreover, it is clear tat by taking P = we would ave te optimal case c/c = 1, ence recovering te standard LS-RB metod. Terefore, P sould be cosen as close as possible to, owever it as to be -independent. Te following results give some insigts on ow to properly coose te matrix P. Teir proofs are reported in te appendix A. Lemma 3.2. Let assumption 3.1 old. Te optimal value for te constants C c satisfying (26) are C = P 1/2 ( )1/2 P, c = 1/ ( ) 1/2 P 1/2. (41) From now, we consider C, c as teir optimal values (41). Lemma 3.3. Let assumption 3.1 old. Te two constants C c > 0 satisfying (26) and (41) are suc tat c [ ] 1/2 [ ] 1/2. C = K (P 1 ) = K 2 (P 1/2 P 1/2 ) (42) It is clear from Lemma 3.3 tat te matrix P sould be cosen in suc a way tat te condition number of te preconditioned matrix P 1 does not depend on te mes size, i.e. P sould be an optimal preconditioner for. If tis is not te case, te value of te stability constant of te RB approximation β P,N may depend on. Furtermore, if we set up our RB approximation in a HPC environment, employing a mes partitioner to divide te computational domain among te processors, it is also advisable to coose P suc tat c C does not depend on te size H of te subdomains, i.e. P sould be a scalable preconditioner for. In our numerical experiments P is cosen eiter as P = 0, i.e. as te norm matrix in te reference domain, or as a block diagonal preconditioner of 0, were te two blocks are generated as symmetric and positive denite preconditioners P u R N u N u of 0 u and P p R N p N p of 0 p. 4 Numerical experiments We sow te results obtained wit te RB metods presented in Section 3 implemented witin te LifeV [9] library. We compare te G-RB metod (wit velocity enricment) and te als-rb metod in te case of large-scale Stokes ows in a cylindrical domain wic is nonanely parametrized. Te deformation is not analitically known, since it is retrieved as te solution of an additional FE problem wic armonically extends in te interior of te domain te datum prescribed on a Diriclet boundary. In te following sections, we present te setup of te problem. 13

16 4.1 Test case setting: Stokes problem in a parametrized cylinder We consider te Stokes equations in a parameter dependent domain Ω R 3, wic is obtained by deforming a reference domain Ω 0 = { x R 3 : x x 2 1 < 0.25, x 3 (0, 5)} by means of a displacement d obtained as te armonic extension of a boundary deformation. specically, we set More were d solves te following PDE Ω = { x R 3 : x = x + d }, { d = 0 in Ω 0 d = on Ω 0. (43) In our numerical experiments we take = ( 1, 2 ) D = [ 0.3, 0.3] [2, 3] and a Diriclet datum of te form = x 1 1 exp{ 5(x 3 2 ) 2 } x 2 1 exp{ 5(x 3 2 ) 2 }, 0 entailing a deformation of te cylinder by narrowing or enlarging (according to te sign of 2 ) its section in dierent positions along te coordinate x 3 (according to te value of 1 ). Notice tat te solution d of (43) is not known a-priori, terefore we compute its numerical approximation d by writing te variational form of problem (43) and by employing te FE metod. We denote by d RN d te solution of te corresponding FE linear system. Moreover, once te computational domain as been deformed, te lifting function r g D is computed similarly by solving te following problem r g D = 0 in Ω r g D = g D on Γ in r g D = 0 on Γ (44) w r g D n = 0 on Γ out, wic is an armonic extension of te Diriclet data in (2). Here g D is a parabolic prole suc tat te ow rate at te inlet is equal to 1. Problem (44) as well is discretized wit te FE metod wit second order polynomials (P 2 ) basis functions, leading to a parametrized linear system wose solution r RN u is te approximated lifting functions. In Fig. 1, te deformation d is reported for tree dierent values of D. In te numerical experiments we present, Taylor-Hood FE (P 2 P 1 ), wit a mes leading to (a) = (2.7, 0.12) (b) = (2, 0.3) (c) = (3, 0.3) Figure 1: Displacement for dierent values of. N = N u + N p = = degrees of freedom, are employed for te Stokes equations. Te algebraic problem is run on te Piz-Daint cluster wit Cray C40 macines, at te Swiss National Supercomputing Center (CSCS) in Lugano. Te computation as been carried out wit 256 processors. 14

17 4.1.1 FE simulation setup For any parameter considered, we solve te FE problems to approximate te deformation d of problem (43) and te lifting function r g D of problem (44). Te associated algebraic systems are solved by te preconditioned CG metod, wit a tolerance on te Euclidean norm of te residual (rescaled wit te Euclidean norm of te rigt and side) of An algebraic multigrid (AMG) preconditioner from te ML package of Trilinos, see [22], is employed. Once computed te deformation for a new parameter value, we employ a move-mes tool to sape te computational domain and assemble te FE arrays. Tis ensures tat te meses for dierent instances of te parameter are topologically equivalent and tere is a one-to-one correspondence between degrees of freedom. Te linear system (7) resulting from te FE discretization of te Stokes equations is solved wit te preconditioned exible GMRES (FGMRES) metod, were te preconditioner is te Pressure Mass Matrix (PMM) operator, wic exploits te block structure of (8) and employs te mass matrix in pressure to approximate te Scur complement, see [32]. It entails at every iteration te solution of a problem for te velocity (involving te velocity stiness matrix) and one for te pressure (involving te pressure mass matrix). Bot linear systems are solved inexactly wit te preconditioned CG metod, were te preconditioner is still te AMG preconditioner from te ML package of Trilinos, tis time wit a tolerance on te euclidean norm of te residual (rescaled wit te euclidean norm of te rigt and side) of Notice tat we employed te FGMRES (instead of regular GMRES) due to te use of inner iterations for te problems involving te velocity and te pressure, wic, as a matter of fact, yield an iteration dependent preconditioner. Te PMM preconditioner provides satisfactory results in te case of te Stokes equations, cf. [36, 21]. Finally, in order to compute te FE solution wit te exible GMRES metod, up to a nal tolerance of 10 8, our solver requires on average of 38.0 seconds, wic also accounts for te time for deforming te domain, building te lifting function, te PMM preconditioner and te FE solution. In particular, computing te deformation d and te lifting function r requires 2.5 seconds (6.5% of te FE simulation) RB simulation setup During te oine pase, we explore te parameter domain D for building our RB approximation. particular we perform te following steps: we randomly coose a set of n s = 150 parameters { } ns i D; ten we compute te corresponding i=1 velocity snapsots { } u i ns and pressure snapsots { } p i ns i=1 by solving te corresponding linear i=1 system (7). Next, we build te RB space V N by separately computing a basis VN u u for te velocity and V p N p te pressure, by plugging in te POD te same tolerance ε POD = δ RB in bot cases. If te Galerkin-RB metod wit velocity enricment is employed, we also compute n s = 150 supremizer snapsots { t i p (p i )} n s. Since in general we do not take te same number of basis functions for te i=1 velocity and pressure RB spaces, we use a tolerance also for computing te pressure supremizer basis functions. Wit tis aim, we employ POD wit ε POD = δ RB 10 to build te supremizer basis Vs N s, wic numerically conrmed to provide a stable G-RB problem. we compute a basis to anely approximate f, r (wit DEIM) and D, B (wit M-DEIM), by taking n s = 100 snapsots for eac of tese quantities and a tolerance to be plugged in te POD. In te online pase, we perform an analysis of te G-RB and als-rb metods wit respect to te tolerances δ RB (or te number of basis functions N) and, by coosing δ RB, = 10 l, l = 2, 3, 4, 5, 6. We evaluate te accuracy of te RB solutions z N in terms of te rescaled RB residual In r RB = g A Vz N ( ) 1 g ( ) 1, averaging te results obtained for N onl = 100 parameters, dierent from te one employed witin te oine pase. For te als-rb metod, we present te results for two coices of te matrix P : P = 0, i.e. we approximate wit te matrix norm on te reference domain Ω0. Wit tis aim, in te oine pase, we need to solve FE linear systems wit 0 on te left and side to compute te ane terms A q1,q2 N, q 1, q 2 = 1,..., Q a. Tese linear systems are solved wit te CG metod preconditioned wit AMG, up to a tolerance of 10 8 on te euclidean norm of te relative residual; P = P 0, i.e. we take te preconditioner P 0 u of, wic as a block structure P 0 = diag(p 0 u, P 0 p ), were P u R N u N u (resp. Pp R N p N p ) is a symmetric and positive denite AMG preconditioner of 0 u (resp. 0 p). 15

18 Bot coices lead to a matrix P wic does not depend on. Notice tat for any new parameter considered online, we solve te FE linear system for computing te deformation d and te lifting function r. Alternatively, we could compute troug te RB metod te RB approximations of d and r, to be exploited during te online pase, similarly to wat as been proposed in [27] for te parametrized Helmoltz scattering problem. However tis goes beyond te scope of tis paper and will be te subject of furter researc. 4.2 Numerical results Oine pase In Tables 1-2 we sow te oine time required to build te structures of te RB approximations wen using δ RB = = 10 6 (comparable results old wen bigger tolerances are used). We recall tat δ RB is used witin POD to build te velocity and pressure RB spaces, wile for building an ane approximation of te FE blocks of A and g in te (M-)DEIM algoritm. In te rst table, we report te computational times to build te (M-)DEIM basis wic provide an ane approximation of te FE matrices and rigt and sides; tese times are sared by bot te G-RB and alsrb metods. In te second table, te total time of te oine computation is reported, togeter wit te details of its tree main stages: snapsots computation, POD and RB ane arrays construction. Snapsots computation is te most demanding pase, and is particularly expensive if te G-RB metod is employed, since it entails te additional computation of n s pressure supremizer snapsots { t i p (p i )} n s. Te second pase, involving te POD to build te RB i=1 spaces, only requires a tiny percentage of te oine time for all te tree metods considered, owever also in tis case, te two variants of te als-rb metod need a sorter time tan te G-RB metod, because tey require only te construction of te velocity and pressure spaces VN u u and V p N p, wile in te G-RB case te pressure supremizer space VN s s must also be constructed. Concerning te construction of te ane RB matrices and vectors, te G-RB metod scales linearly on te number (Q a and Q g ) of ane terms of te FE matrices and rigt and sides, yielding a computational time wic is sorter tan te one obtained wit te als-rb metods for tis pase. However, tere is also a signicant dierence between te two variants of als-rb metod. By employing P = 0, for assembling te ane terms Aq1,q2 N, q 1, q 2 = 1,..., Q a, a FE linear system must be solved for eac combination of te N RB functions {ξ i } N i=1 and Q a ane terms {A q }Qa q=1, leading to N Q a FE linear systems. On te oter and, by employing P = P 0, only N Q a applications of P 1 need to be performed, boosting te computation of te ane RB structures. Finally, 0 te lowest oine time is required by te als-rb metod were P = P 0 is employed, performing te oine pase in about 81% of te time required by te als-rb metod wit P = 0 and 96% of te time required by te G-RB metod. Tis is due to te fact it simultaneously does not require te construction of te pressure supremizing snapsots to augment te velocity RB space and to ceaply construct te RB ane arrays. In Figure 2 te number of RB functions (left) and (M-)DEIM ane terms are reported as function of te tolerances δ RB and, respectively. Te number of pressure RB functions is te same for te G-RB and als-rb metod, owever te number of velocity basis functions doubles in te former case, due to te velocity enricment required to ensure te well-posedness of te resulting G-RB approximation. Table 1: Computational time (seconds) to build RB basis wit δ RB = MDEIM - D MDEIM - B DEIM - f DEIM - r Total (M-)DEIM Table 2: Computational time (seconds) to build (M-)DEIM ane basis wit = G-RB als-rb ( 0 ) als-rb (P 0 ) Snapsots computation POD Ane arrays construction Total (M-)DEIM Total oine pase Online pase In Fig. 3, 4 and 5 te FE solution computed for dierent values of te parameter and te corresponding errors obtained wit te G-RB metod and te als-rb metod wit P = 0 are sown (te als-rb 16

19 computational time online =1e-02 =1e-03 =1e-04 =1e-05 =1e-06 Number of (M-)DEIM functions RHS-velocity RHS-pressure Velocity stiffness Divergence matrix δ rb (a) RB δ DEIM (b) (M-)DEIM. Figure 2: RB and (M-)DEIM functions vs δ RB, = 10 l, l = 2, 3, 4, 5, 6. metod wit P = P 0 provides similar results). (a) FE velocity magnitude. (b) G-RB velocity error magnitude. (c) als-rb velocity error magnitude. (d) FE pressure (e) G-RB norm of pressure error. (f) als-rb norm of pressure error. Figure 3: FE solution and G-RB and als-rb errors for = (2, 0.3). Te proposed als-rb metod, eiter wit P = 0 or P = P 0, allows to obtain an exponential decay of te residual r RB wit respect to te number of RB functions N; te trend, in loglog scale, is reported in Fig. 6. A tolerance = 10 8 as been used for (M-)DEIM algoritms, in order to consider negligible te error induced by anely approximating te FE arrays. An analysis of te convergence of te residual r RB wit varying bot te tolerances δ RB, = 10 l, l = 2, 3, 4, 5, 6 is reported in Fig. 7 for te G-RB and te two variants of te als-rb metods. By using te same tolerances and δ RB, te als-rb metod allows to compute a more accurate solution during te online pase of about 1 order of magnitude. Moreover, notice tat by using te same for te als-rb metods and te G-RB metod, te latter requires a lower tolerance δ RB to reac a solution wit te same accuracy, yielding a muc larger number of RB functions N. Obtaining a more accurate solution wit te als-rb metod is an expected result, since te standard LS-RB metod seeks a RB approximation minimizing te ( ) 1 norm of te residual, and te als-rb metod provides a RB approximation minimizing te P 1 norm, were P 1 ( ) 1, as sown in Proposition 3.3. In Fig. 7, te computational time required to assemble and solve te RB problem is reported for te tree metods by varying bot te tolerances δ RB, = 10 l, l = 2, 3, 4, 5, 6. Depending on te desired level of accuracy and te RB metod employed, te computational time required to solve te RB problem online ranges from 3.75 to 4.3 seconds. Terefore, a solution accurate up to an error of 0.01% on te FE residual r RB is computed in a time ranging from 10% to 12% of te time required by te FE simulation. Notice owever tat te online computational time accounts also for te time employed for assembling 17

20 (a) FE velocity magnitude. (b) G-RB velocity error magnitude. (c) als-rb velocity error magnitude. (d) FE pressure (e) G-RB norm of pressure error. (f) als-rb norm of pressure error. Figure 4: FE solution and G-RB and als-rb errors for = (3, 0.3). (a) FE velocity magnitude. (b) G-RB velocity error magnitude. (c) als-rb velocity error magnitude. (d) FE pressure (e) G-RB norm of pressure error. (f) als-rb norm of pressure error. Figure 5: FE solution and G-RB and als-rb errors for = (2.7, 0.12). and solving te FE problems to compute te deformation d and te lifting function r, wic on average requires 2.5 seconds in total. In our implementation tis is included in te assembly of te RB matrix, wose required computational time is reported in Fig. 9. By substituting in te simulation pipeline te assembly and solution of te FE problems to compute d and r wit a less expensive model, e.g. by using a ceap RB approximation for te computation of d and r, one can compute an accurate solution wit te als-rb metod, wic needs only 5% of te time required by te FE simulation. In Table 3, for te tree metods examined, we compare te minimum computational time to compute a RB approximation wose residual r RB is lower tan a xed target accuracy. Te two versions of te als-rb metod conrm to reac a better accuracy in a lower time. Te 'x' in te te G-RB column states tat te accuracy 10 4 is not reaced wen tis metod wit te given tolerance values δ RB, = 10 l, l = 2, 3, 4, 5, 6. Terefore, one sould furter decrease δ RB and to compute a more accurate solution. Table 3: Computational time (seconds) required by te RB metods to compute a solution satisfying a target accuracy. Accuracy G-RB als-rb ( 0 ) als-rb (P0 ) 1e e e e-04 x