NONLINEAR MODELREDUCTION VIA DISCRETE EMPIRICAL … · SIAM J. SCI. COMPUT. c 2010 Society for...

SIAM J. SCI. COMPUT. c© 2010 Society for Industrial and Applied MathematicsVol. 32, No. 5, pp. 2737–2764

NONLINEAR MODEL REDUCTION VIA DISCRETEEMPIRICAL INTERPOLATION∗

SAIFON CHATURANTABUT† AND DANNY C. SORENSEN†

Abstract. A dimension reduction method called discrete empirical interpolation is proposedand shown to dramatically reduce the computational complexity of the popular proper orthogonaldecomposition (POD) method for constructing reduced-order models for time dependent and/orparametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinear-ity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables arepresent, but the complexity of evaluating the nonlinear term remains that of the original problem.The original empirical interpolation method (EIM) is a modification of POD that reduces the com-plexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number ofreduced variables obtained by POD. We propose a discrete empirical interpolation method (DEIM),a variant that is suitable for reducing the dimension of systems of ordinary differential equations(ODEs) of a certain type. As presented here, it is applicable to ODEs arising from finite differ-ence discretization of time dependent PDEs and/or parametrically dependent steady state problems.However, the approach extends to arbitrary systems of nonlinear ODEs with minor modification. Ourcontribution is a greatly simplified description of the EIM in a finite-dimensional setting that pos-sesses an error bound on the quality of approximation. An application of DEIM to a finite differencediscretization of the one-dimensional FitzHugh–Nagumo equations is shown to reduce the dimensionfrom 1024 to order 5 variables with negligible error over a long-time integration that fully capturesnonlinear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions withsimilar state space dimension reduction and accuracy results.

Key words. nonlinear model reduction, proper orthogonal decomposition, empirical interpola-tion methods, nonlinear partial differential equations

AMS subject classifications. 65L02, 65M02

DOI. 10.1137/090766498

1. Introduction. Model order reduction (MOR) seeks to reduce the compu-tational complexity and computational time of large-scale dynamical systems by ap-proximations of much lower dimension that can produce nearly the same input/outputresponse characteristics. The method proposed here is concerned with dimension re-duction of high-dimensional nonlinear ordinary differential equation (ODE) systems.Our approach applies to virtually any system of ODEs. However, systems arising fromdiscretization of partial differential equations (PDEs) are primary examples. Dimen-sion reduction of discretized time dependent and/or parametrized nonlinear PDEsis of great value in reducing computational times in many applications, includingthe neuron modeling and steady state flow problems presented here as illustrations.These discrete systems often must become very high dimensional to achieve the de-sired accuracy in the numerical solutions. We introduce a discrete empirical interpola-tion method (DEIM) to greatly improve the dimension reduction efficiency of properorthogonal decomposition (POD) with Galerkin projection, a popular approach forconstructing reduced-order models of these discrete systems.

∗Received by the editors July 28, 2009; accepted for publication (in revised form) May 31, 2010;published electronically September 7, 2010. This work was supported in part by NSF grant CCF-0634902 and AFOSR grant FA9550-06-1-0245. A brief synopsis of some of the results in this paperappeared in Proceedings of the 48th IEEE Conference on Decision and Control and the 28th ChineseControl Conference (CDC/CCC 2009), 2009, pp. 4316–4321.

http://www.siam.org/journals/sisc/32-5/76649.html†Department of Computational and Applied Mathematics, MS-134, Rice University, 6100 Main

Street, Houston, TX 77005-1892 ([email protected], [email protected]).

2737

2738 SAIFON CHATURANTABUT AND DANNY C. SORENSEN

The POD-Galerkin approach has provided reduced-order models of systems innumerous applications such as compressible flow [38], fluid dynamics [23], aerody-namics [9], and optimal control [22]. However, effective dimension reduction for thePOD-Galerkin approach is usually limited to problems with linear or bilinear termsas in [20, 22, 18, 33]. In fact, this limitation occurs when Galerkin projection is ap-plied with any type of reduced basis, as noted, for example, in [28] for the case ofthe reduced basis proposed in [30]. Its success is limited to the problems of linearelliptic parabolic PDEs with affine parameters or low-order polynomial nonlinearities[32, 25, 26, 41, 29]. When a general nonlinearity is present, the cost to evaluate theprojected nonlinear function still depends on the dimension of the original system,resulting in simulation times that hardly improve over the original system.

In the finite element (FE) context, this inefficiency arises from the high computa-tional complexity in repeatedly calculating the inner products required to evaluate theweak form of the nonlinearities as discussed in [7, 19, 28]. In particular, in [28], Nguyenand Peraire discuss the limitations of such approaches and give a number of examplesof equations involving nonpolynomial nonlinearities. Specifically, they study linearelliptic equations with nonaffine parameter dependence, nonlinear elliptic equations,and nonlinear time dependent convection-diffusion equations. They demonstrate forthese examples that the standard POD-Galerkin approach does not admit the sortof precomputation that is possible with polynomial nonlinearities. They propose areduced basis method with a best-points interpolation method (BPIM; see [27]) toselecting interpolation points.

Several approaches have been proposed to address the problem of reducing thecomplexity of evaluating the nonlinear term of the POD reduced model in the contextof finite difference (FD) and finite volume (FV) discretization as well as differential-algebraic equations (e.g., in circuit simulation). Missing point estimation (MPE) wasoriginally proposed in [2] to improve the complexity of the POD-Galerkin reducedsystem from FV discretization, essentially, by solving only a subset of equations ofthe original model. A reduced system is obtained by first extracting certain equationscorresponding to specially chosen spatial grid points and then projecting the extractedsystem onto the space spanned by the restricted POD with components/rows corre-sponding to only these selected grid points. This procedure can be viewed as perform-ing the Galerkin projection onto the truncated POD basis via a specially constructedinner product as defined in [5] which evaluates only at selected grid points insteadof computing the usual L2 inner product. Two heuristic methods for selecting thesespatial grid points are introduced in the thesis [2] (also in subsequent publications;see, e.g., [1, 4, 3]) by aiming to minimize aliasing effects in using only partial spatialpoints. This was shown to be equivalent to a criterion for preserving the orthogo-nality of the restricted POD basis vectors which is further translated into a criterionfor controlling condition number growth. These grid point selection procedures arelater improved by incorporating a greedy algorithm from [42]. The applications ofthe MPE method are primarily in the context of a linear time varying system arisingfrom FV discretization of a nonlinear computational fluid dynamic model for a glassmelting furnace [2, 1, 4, 3]. It has also been used in modeling heat transfer in electricalcircuits [40] and in subsurface flow simulation [11].

Alternatively, techniques for approximating a nonlinear function can be used inconjunction with the POD-Galerkin projection method to overcome this computa-tional inefficiency. There are a number of examples that use MOR approaches withnonlinear approximation based on precomputation of coefficients defining multilinear

DISCRETE EMPIRICAL INTERPOLATION METHOD 2739

forms of polynomial nonlinearities followed by POD-Galerkin projection [14, 15, 31, 6,16, 10]. One of these approaches is found in the trajectory piecewise-linear (TPWL)approximation [34, 37], which is based on approximating a nonlinear function by aweighted sum of linearized models at selected points along a state trajectory. Theselinearization points are selected using prior knowledge from a training trajectory (orits approximation) of the full-order nonlinear system [36]. The TPWL approach wassuccessfully applied to several practical nonlinear systems, especially in the circuitsimulations [35, 36, 37, 40, 8]. However, there are still many nonlinear functions thatmay not be approximated well by using low degree piecewise polynomials unless thereare very many constituent polynomials.

The DEIM approach proposed here approximates a nonlinear function by combin-ing projection with interpolation. DEIM constructs specially selected interpolationindices that specify an interpolation-based projection to provide a nearly L2 opti-mal subspace approximation to the nonlinear term without the expense of orthogonalprojection. This approach is a discrete variant of the empirical interpolation method(EIM) introduced by Barrault, Maday, Nguyen, and Patera [7], which was originallyposed in an empirically derived finite-dimensional function space. We were motivatedto develop this DEIM variant to apply to arbitrary systems of ODEs regardless oftheir origin. For illustration purposes, we shall focus on FD discretized systems oftime dependent and/or parametrized nonlinear PDEs. The procedure presented inthis paper can also be applied to general nonlinear ODEs, including a FV discretizedsystem and the system of coefficients from FE discretization.

Our DEIM approach is closely related to MPE in the sense that both methodsemploy a small selected set of spatial grid points to avoid evaluation of the expensiveL2 inner products at every time step that are required to evaluate the nonlinearities.However, the fundamental procedures for constructing a reduced system and the al-gorithms for selecting a set of spatial grid points are different. While MPE focuseson reducing the number of equations and using a restricted inner product on thePOD basis vectors, DEIM focuses on approximating each nonlinear function so thata certain coefficient matrix can be precomputed and, as a result, the complexity inevaluating the nonlinear term becomes proportional to the small number of selectedspatial indices. Hence, the reduced system from the MPE procedure considers onlya POD basis for the state variables, but the one from the DEIM procedure considersboth a POD basis for the state variables and a POD basis related to each nonlinearterm.

The POD-DEIM approach is also closely related to the approach called interpola-tion of function snapshots suggested in [40] as an alternative to MPE for constructinga reduced system for a nonlinear circuit model. The main steps of both approachesare the same. The nonlinear approximation is computed by using some selected spa-tial points, and then Galerkin projection is applied to the system. However, a keydifference is that in [40] the basis matrices used for spanning the unknowns (statevariables) and the nonlinear function in the reduced system are obtained from a least-squares solution of the snapshot matrices in such a way that the unknown coefficientsof the resulting reduced system still have the original interpretations of state variablesinstead of using basis matrices from SVD truncation as done in our POD-DEIM ap-proach. No concrete algorithm was proposed in [40] for selecting indices (besides theones used in MPE). However, it was suggested in [40] to select them to minimize anupper bound of the approximation error, which is an idea similar to the one leadingto our error bound for DEIM approximation (see (3.8) and (3.9) in section 3.2).


More recently, Galbally et al. [17] applied the techniques of gappy POD, EIM, andBPIM to develop an approach to uncertainty quantification in a nonlinear combus-tion problem governed by an advection-diffusion-reaction PDE. The nonlinear terminvolved an exponential nonlinearity of Arrhenius type. In [17], there is a detailedexplanation of why POD-Galerkin does not reduce the complexity of evaluating thenonlinear term. They also developed a masked projection framework that is very sim-ilar to the projection methodology developed in this paper that shows the similarityof the gappy POD, EIM, and BPIM approaches.

Our discussion is organized as follows. Section 2 describes the problem setup.Dimension reduction via POD is reviewed in section 2.1 followed by a discussion ofthe fundamental complexity issue in section 2.2. The DEIM approximation is intro-duced in section 3. The key to complexity reduction is to replace the orthogonalprojection of POD with the interpolation projection of DEIM in the same POD basis.An algorithm for selecting the interpolation indices used in the DEIM approximationis presented in section 3.1. Section 3.2 provides an error bound on this interpolatoryapproximation indicating that it is nearly as good as orthogonal projection. Sec-tion 3.3 illustrates the validity of this error bound and the high quality of the DEIMapproximations with selected numerical examples. In section 3.4, we explain how toapply the DEIM approximation to nonlinear terms in POD reduced-order models ofFD discretized systems, and then we extend this to general nonlinear ODEs in sec-tion 3.5. Finally, in section 4, we give computational evidence of DEIM effectivenessin two specific problem settings. DEIM applied to a 1024 variable discretization ofthe FitzHugh–Nagumo equations produced a 5 variable reduced-order model whichwas able to capture limit cycle behavior over a long-time integration. Similar effec-tiveness is demonstrated for a two-dimensional steady state parametrically dependentflow problem. Throughout the discussion, we shall refer to a reduced-order systemobtained directly from POD-Galerkin projection as POD reduced system and the oneobtained from the POD-Galerkin approach with the DEIM approximation as POD-DEIM reduced system.

The numerical examples given here were selected because they are simple and yetstill present a challenge to model reduction. The purpose is to illustrate the main ideaswithout the complexity of the equations potentially obscuring them. In two recentpapers, we have demonstrated the effectiveness of DEIM in two very different and farmore complex applications. One is in neural modeling, where we reduce numerousexamples of full Hodgkin–Huxley models of realistic spiking neurons [21]. The otheris in two-phase miscible flow in porous media with varying Peclet number, both withand without chemistry at the interface of the different fluids [12].

2. Problem formulation. The method we are about to develop is really amethod for reducing the dimension of general large-scale ODE systems regardless oftheir origin. However, a considerable source of such systems is the semidiscretizationof time dependent or parameter dependent PDEs. Thus, we shall develop this methodin the context of FD discretized systems arising from two types of nonlinear PDEswhich are used for our numerical examples in section 4. One is time dependent, andthe other is a parametrized steady state problem. We explain how to handle generalnonlinearities in section 3.5.

An FD discretization of a scalar nonlinear PDE in one spatial variable results ina system of nonlinear ODEs of the form

d

dty(t) = Ay(t) + F(y(t)),(2.1)


with appropriate initial conditions. Here t ∈ [0, T ] denotes time, y(t) = [y1(t), . . . ,yn(t)]

T ∈ Rn, A ∈ R

n×n is a constant matrix, and F is a nonlinear function evaluatedat y(t) componentwise, i.e., F = [F (y1(t)), . . . , F (yn(t))]

T , F : I �→ R for I ⊂ R.The matrix A is the discrete approximation of the linear spatial differential operator,and F is a nonlinear function of a scalar variable.

Steady nonlinear PDEs (in several spatial dimensions) might give rise similarlyto a corresponding FD discretized system of the form

Ay(μ) + F(y(μ)) = 0,(2.2)

with the corresponding Jacobian

J(y(μ)) := A+ JF(y(μ)),(2.3)

where y(μ) = [y1(μ), . . . ,yn(μ)]T ∈ R

n, with A and F defined as for (2.1). Notethat, from (2.3), the Jacobian of the nonlinear function is a diagonal matrix given by

JF(y(μ)) = diag{F ′(y1(μ)), . . . , F′(yn(μ))} ∈ R

n×n,(2.4)

where F ′ denotes the first derivative of F . The parameter μ ∈ D ⊂ Rd, d = 1, 2, . . . ,

generally represents the system’s configuration in terms of its geometry, materialproperties, etc.

To simplify exposition, we have considered time dependence and parametric de-pendence separately. Note, however, that the two may be merged to address timedependent parametrized systems. For example, if one wished to allow for a variety ofinitial conditions in a time dependent problem, including them as parameters wouldbe a possibility.

The dimension n of (2.1) and (2.2) reflects the number of spatial grid points usedin the FD discretization. As noted, the dimension n can become extremely large whenhigh accuracy is required. Hence, solving these systems becomes computationallyintensive or possibly infeasible. Approximate models with much smaller dimensionsare needed to recover the efficiency.

Projection-based techniques are commonly used for constructing a reduced-ordersystem. They construct a reduced-order system of order k � n that approximatesthe original system from a subspace spanned by a reduced basis of dimension k in R

n.We use Galerkin projection as the means for dimension reduction. In particular, letVk ∈ R

n×k be a matrix whose orthonormal columns are the vectors in the reducedbasis. Then, by replacing y(t) in (2.1) by Vky(t), y(t) ∈ R

k and projecting thesystem (2.1) onto Vk, the reduced system of (2.1) is of the form

d

dty(t) = VT

k AVk︸︷︷︸A

y(t) +VTk F(Vky(t)).(2.5)

Similarly, the reduced-order system of (2.2) is of the form

VTk AVk︸︷︷︸A

y(μ) +VTk F(Vky(μ)) = 0,(2.6)

with corresponding Jacobian

J(y(μ)) := A+VTk JF(Vky(μ))Vk,(2.7)


where A = VTk AVk ∈ R

k×k. The choice of the reduced basis clearly affects the qual-ity of the approximation. The techniques for constructing a set of reduced basis usea common observation that, for a particular system, the solution space is often at-tracted to a low-dimensional manifold. POD constructs a set of global basis functionsfrom a singular value decomposition (SVD) of snapshots, which are discrete samplesof trajectories associated with a particular set of boundary conditions and inputs. Itis expected that the samples will be on or near the attractive manifold. Once thereduced model has been constructed from this reduced basis, it may be used to ob-tain approximate solutions for a variety of initial conditions and parameter settings,provided the set of samples is rich enough. This empirically derived basis is clearlydependent on the sampling procedure.

Among the various techniques for obtaining a reduced basis, POD constructs a re-duced basis that is optimal in the sense that a certain approximation error concerningthe snapshots is minimized. Thus, the space spanned by the basis from POD oftengives an excellent low-dimensional approximation. The POD approach is thereforeused here as a starting point.

2.1. Proper orthogonal decomposition (POD). POD is a method for con-structing a low-dimensional approximation representation of a subspace in Hilbertspace. It is essentially the same as the SVD in a finite-dimensional space or in Eu-clidean space. It efficiently extracts the basis elements that contain characteristics ofthe space of expected solutions of the PDE. The POD basis in Euclidean space maybe specified formally as follows.

Given a set of snapshots {y1, . . . ,yns} ⊂ Rn (recall snapshots are samples of

trajectories), let Y = span{y1, . . . ,yns} ⊂ Rn and r = dim(Y ). A POD basis of

dimension k < r is a set of orthonormal vectors {φ}ki=1 ⊂ Rn whose linear span best

approximates the space Y . The basis set {φ}ki=1 solves the minimization problem

min{φi}k

i=1

ns∑j=1

∥∥∥∥∥yj −k∑

i=1

(yTj φi)φi

∥∥∥∥∥2

2

,

with φTi φj = δij =

{1 if i = j ,0 if i �= j,

i, j = 1, . . . , k.

(2.8)

It is well known that the solution to (2.8) is provided by the set of the left singularvectors of the snapshot matrix Y = [y1, . . . ,yns ] ∈ R

n×ns . In particular, suppose thatthe SVD of Y is

Y = VΣWT ,

where V = [v1, . . . ,vr] ∈ Rn×r and W = [w1, . . . ,wr] ∈ R

ns×r are orthogonal andΣ = diag(σ1, . . . , σr) ∈ R

r×r, with σ1 ≥ σ2 ≥ · · · ≥ σr > 0. The rank of Y isr ≤ min(n, ns). Then the POD basis or the optimal solution of (2.8) is {vi}ki=1. Theminimum 2-norm error from approximating the snapshots using the POD basis is thengiven by

ns∑j=1

∥∥∥∥∥yi −k∑

i=1

(yTj vi)vi

∥∥∥∥∥2

2

=

r∑i=k+1

σ2i .

We refer the reader to [23] for more details on the POD basis in general Hilbert space.The choice of the snapshot ensemble is a crucial factor in constructing a POD

basis. This choice can greatly affect the approximation of the original solution space,


but it is a separate issue and will not be discussed here. POD is a popular approachbecause it works well in many applications and often provides an excellent reduced ba-sis. However, as discussed in the introduction, when POD is used in conjunction withthe Galerkin projection, effective dimension reduction is usually limited to the lin-ear terms or low-order polynomial nonlinearities. Systems with general nonlinearitiesneed additional treatment, which will be presented in section 3.

2.2. A problem with complexity of the POD-Galerkin approach. Thissection illustrates the computational inefficiency that occurs in solving the reduced-order system that is directly obtained from the POD-Galerkin approach. Equation(2.5) has the nonlinear term

N(y) := VTk︸︷︷︸

k×n

F(Vky(t))︸︷︷︸n×1

.(2.9)

N(y) has a computational complexity that depends on n, the dimension of the orig-inal full-order system (2.1). It requires on the order of 2nk flops for matrix-vectormultiplications, and it also requires a full evaluation of the nonlinear function F atthe n-dimensional vectorVky(t). In particular, suppose the complexity for evaluatingthe nonlinear function F with q components is O(α(q)), where α is some function ofq. Then the complexity for computing (2.9) is roughly O(α(n) + 4nk). As a result,solving this system might still be as costly as solving the original system. Here, the4nk flops are a result of the two matrix-vector products required to form the argu-ment of f and then to form the projection. We count both the multiplications andadditions as flops.

The same inefficiency occurs when solving the reduced-order system (2.6) for thesteady nonlinear PDEs by Newton iteration. At each iteration, besides the nonlinearterm of the form (2.9), the Jacobian of the nonlinear term (2.7) must also be computedwith a computational cost that still depends on the full-order dimension n:

JF(y(μ)) := VTk︸︷︷︸

k×n

JF(Vky(μ))︸︷︷︸n×n

Vk︸︷︷︸n×k

.(2.10)

The computational complexity for computing (2.10) is roughlyO(α(n)+2n2k+2nk2+2nk) if we treat JF as dense. The 2n2k term becomes O(nk) if JF is sparse ordiagonal.

3. Discrete empirical interpolation method (DEIM). An effective wayto overcome the difficulty described in section 2.2 is to approximate the nonlinearfunction in (2.5) or (2.6) by projecting it onto a subspace that approximates thespace generated by the nonlinear function and that is spanned by a basis of dimensionm � n. This section considers the nonlinear functions F(Vky(t)) and F(Vky(μ))of the reduced-order systems (2.5) and (2.6), respectively, represented by f(τ), whereτ = t or μ. The approximation from projecting f(τ) onto the subspace spanned bythe basis {u1, . . . ,um} ⊂ R

n is of the form

f(τ) ≈ Uc(τ),(3.1)

where U = [u1, . . . ,um] ∈ Rn×m and c(τ) is the corresponding coefficient vector.

To determine c(τ), we select m distinguished rows from the overdetermined systemf(τ) = Uc(τ). In particular, consider a matrix

P = [e℘1 , . . . , e℘m ] ∈ Rn×m,(3.2)


where e℘i = [0, . . . , 0, 1, 0, . . . , 0]T ∈ Rn is the ℘ith column of the identity matrix

In ∈ Rn×n for i = 1, . . . ,m. Suppose PTU is nonsingular. Then the coefficient vector

c(τ) can be determined uniquely from

PT f(τ) = (PTU)c(τ),(3.3)

and the final approximation from (3.1) becomes

f(τ) ≈ Uc(τ) = U(PTU)−1PT f(τ).(3.4)

To obtain the approximation (3.4), we must specify1. the projection basis {u1, . . . ,um} and2. the interpolation indices {℘1, . . . , ℘m} used in (3.2).

The projection basis {u1, . . . ,um} for the nonlinear function f is constructed by apply-ing the POD on the nonlinear snapshots obtained from the original full-order system.These nonlinear snapshots are the sets {F(y(t1)), . . . ,F(y(tns))} and {F(y(μ1)), . . . ,F(y(μns ))} obtained from (2.9) and (2.10), respectively. Note that these values areneeded to generate the trajectory snapshots Y and hence represent no additional costother than the SVD required to obtain U.

The interpolation indices ℘1, . . . , ℘m, used for determining the coefficient vectorc(τ) in the approximation (3.1), are selected inductively from the basis {u1, . . . ,um}by the DEIM algorithm introduced in the next section.

3.1. DEIM: Algorithm for interpolation indices. DEIM is a discrete vari-ant of the empirical interpolation method (EIM) proposed in [7] for constructing anapproximation of a nonaffine parametrized function with spatial variable defined in acontinuous bounded domain Ω. The DEIM algorithm treats the continuous domainΩ as a finite set of discrete points in Ω. The DEIM algorithm selects an index cor-responding to one of these discrete spatial points at each iteration to limit growth ofan error bound. This provides a derivation of a global error bound as presented insection 3.2. For general systems of nonlinear ODEs that are not FD approximationsto PDEs, this spatial connotation of indices will no longer exist. However, the formalprocedure remains unchanged.

Algorithm 1. DEIM

INPUT: {u�}m�=1 ⊂ Rn linearly independent

OUTPUT: �℘ = [℘1, . . . , ℘m]T ∈ Rm

1: [|ρ|, ℘1] = max{|u1|}2: U = [u1], P = [e℘1 ], �℘ = [℘1]

3: for = 2 to m do

4: Solve (PTU)c = PTu� for c

5: r = u� −Uc

6: [|ρ|, ℘�] = max{|r|}

7: U← [U u�], P← [P e℘�], �℘←

[�℘℘�

]8: end for


The notation max in Algorithm 1 is the same as the function max in MATLAB.Thus, [|ρ|, ℘�] = max{|r|} implies |ρ| = |r℘�

| = maxi=1,...,n{|ri|}, with the smallestindex taken in case of a tie. Note that we define ρ = r℘�

in each iteration = 1, . . . ,m.From Algorithm 1, the DEIM procedure constructs a set of indices inductively

on the input basis. The order of the input basis {u�}m�=1 according to the dominantsingular values is important, and an error analysis indicates that the POD basis is asuitable choice for this algorithm. The process starts from selecting the first inter-polation index ℘1 ∈ {1, . . . , n} corresponding to the entry of the first input basis u1

with largest magnitude. The remaining interpolation indices, ℘� for = 2, . . . ,m, areselected so that each of them corresponds to the entry with the largest magnitude ofthe residual r = u�−Uc from line 5 of Algorithm 1. The term r can be viewed as theresidual or the error between the input basis u� and its approximation Uc from inter-polating the basis {u1, . . . ,u�−1} at the indices ℘1, . . . , ℘�−1 in line 4 of Algorithm 1.Hence, r℘i = 0 for i = 1, . . . , − 1. However, the linear independence of the inputbasis {u�}m�=1 guarantees that, in each iteration, r is a nonzero vector and hence ρ isalso nonzero. We shall demonstrate in Lemma 3.2 that ρ �= 0 at each step implies thatPTU is always nonsingular and hence that the DEIM procedure is well defined. Thisalso implies that the interpolation indices {℘i}mi=1 are hierarchical and nonrepeated.

Figure 3.1 illustrates the selection procedure in Algorithm 1 for DEIM interpola-tion indices.

To summarize, the DEIM approximation is given formally as follows.Definition 3.1. Let f : D �→ R

n be a nonlinear vector-valued function withD ⊂ R

d for some positive integer d. Let {u�}m�=1 ⊂ Rn be a linearly independent set

for m ∈ {1, . . . , n}. For τ ∈ D , the DEIM approximation of order m for f(τ) in thespace spanned by {u�}m�=1 is given by

f(τ) := U(PTU)−1PT f(τ),(3.5)

where U = [u1, . . . ,um] ∈ Rn×m and P = [e℘1 , . . . , e℘m ] ∈ R

n×m, with {℘1, . . . , ℘m}being the output from Algorithm 1 with the input basis {ui}mi=1.

Notice that f in (3.5) is indeed an interpolation approximation for the original

function f , since f is exact at the interpolation indices; i.e, for τ ∈ D ,

PT f (τ) = PT(U(PTU)−1PT f(τ)

)= (PTU)(PTU)−1PT f(τ) = PT f(τ).

Notice also that the DEIM approximation is uniquely determined by the projec-tion basis {ui}mi=1. This basis not only specifies the projection subspace used in theapproximation, but it also determines the interpolation indices used for computingthe coefficient of the approximation. Hence, the choice of projection basis can greatlyaffect the accuracy of the approximation in (3.5) as shown also in the error boundof the DEIM approximation (3.8) in the next section. As noted, POD introducedin section 2.1 is an effective method for constructing this projection basis, since itprovides an optimal global basis that captures the dynamics of the space generatedfrom snapshots of the nonlinear function.

The selection of the interpolation points is basis dependent. However, once theset of DEIM interpolation indices {℘�}m�=1 is determined from {ui}mi=1, the DEIMapproximation is independent of the choice of basis spanning the space Range(U). Inparticular, let {q�}m�=1 be any basis for Range(U). Then

U(PTU)−1PT f(τ) = Q(PTQ)−1PT f(τ),(3.6)


0 20 40 60 80 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16 DEIM# 1

ucurrent point

0 20 40 60 80 100−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15DEIM# 2

uUc r = u−Uccurrent pointprevious points

0 20 40 60 80 100−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3DEIM# 3


0 20 40 60 80 100−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2DEIM# 4


0 20 40 60 80 100−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3 DEIM# 5


0 20 40 60 80 100−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4DEIM# 6


Fig. 3.1. Illustration of the selection process of indices in Algorithm 1 for the DEIM approx-imation. The input basis vectors are the first six eigenvectors of the discrete Laplacian. From theplots, u = u�, Uc, and r = u� −Uc are defined as in the iteration � of Algorithm 1.

where Q = [q1, . . . ,qm] ∈ Rn×m. To verify (3.6), note that Range(U) = Range(Q)

so that U = QR for some nonsingular matrix R ∈ Rm×m. This substitution gives

U(PTU)−1PT f(τ) = (QR)((PTQ)R)−1PT f(τ) = Q(PTQ)−1PT f(τ).

3.2. Error bound for DEIM. This section provides an error bound for theDEIM approximation. The bound is obtained recursively by limiting the local growthof a certain magnification factor of the best 2-norm approximation error. This errorbound is given formally as follows.


Lemma 3.2. Let f ∈ Rn be an arbitrary vector. Let {u�}m�=1 ⊂ R

n be a givenorthonormal set of vectors. From Definition 3.1, the DEIM approximation of orderm ≤ n for f in the space spanned by {u�}m�=1 is

f = U(PTU)−1PT f ,(3.7)

where U = [u1, . . . ,um] ∈ Rn×m and P = [e℘1 , . . . , e℘m ] ∈ R

n×m, with {℘1, . . . , ℘m}being the output from Algorithm 1 with the input basis {ui}mi=1. An error bound for fis then given by

‖f − f‖2 ≤ C E∗(f) ,(3.8)

where

C = ‖(PTU)−1‖2 and E∗(f) = ‖(I−UUT )f‖2(3.9)

is the error of the best 2-norm approximation for f from the space Range(U). Theconstant C is bounded by

C ≤ (1 +√2n)m−1

|eT℘1u1| = (1 +

√2n)m−1‖u1‖−1

∞ .(3.10)

Proof. This proof provides motivation for the DEIM selection process in terms ofminimizing the local error growth for the approximation.

Consider the DEIM approximation f given by (3.4). We wish to determine a

bound for the error ‖f − f‖2 in terms of the optimal 2-norm approximation for f fromRange(U). This best approximation is given by

f∗ = UUT f ,(3.11)

which minimizes the error ‖f − f‖2 over Range(U). Consider

f = (f − f∗) + f∗ = w + f∗,(3.12)

where w = f − f∗ = (I −UUT )f . Define the projector P = U(PTU)−1PT . From(3.7) and (3.12),

f = Pf = P(w + f∗) = Pw + Pf∗ = Pw+ f∗.(3.13)

Equations (3.12) and (3.13) imply f − f = (I− P)w and

‖f − f‖2 = ‖(I− P)w‖2 ≤ ‖I− P‖2‖w‖2.(3.14)

Note that

‖I− P‖2 = ‖P‖2 = ‖U(PTU)−1PT ‖2(3.15) ≤ ‖U‖2‖(PTU)−1‖2‖PT ‖2 = ‖(PTU)−1‖2.

The first equality in (3.15) follows from the fact that ‖I−P‖2 = ‖P‖2 for any projectorP �= 0 or I (see [39]). The last equality in (3.15) follows from the fact that ‖U‖2 =‖PT ‖2 = 1, since each of the matrices U and P has orthonormal columns.


Note that E∗(f) := ‖w‖2 is the minimum 2-norm error for f∗ defined in (3.11).From (3.15), the bound for the error in (3.14) becomes

‖f − f‖2 ≤ ‖(PTU)−1‖2 E∗(f),(3.16)

and ‖(PTU)−1‖2 is the magnification factor needed to express the DEIM error interms of the optimal approximation error. This establishes the error bound (3.8). An

upper bound for ‖f − f‖2 is now available by giving a bound for the matrix norm.The matrix norm ‖(PTU)−1‖2 depends on the DEIM selection of indices ℘1, . . . , ℘m

through the matrix P. We now show that each iteration of the DEIM algorithm aimsto select an index to limit stepwise growth of ‖(PTU)−1‖2 and hence to limit the size

of the bound for the error ‖f − f‖2.To simplify notation, for = 2, . . . ,m, we denote the relevant quantities at itera-

tion by

U = [u1, . . . ,u�−1] ∈ Rn×(�−1), P = [e℘1 , . . . , e℘�−1

] ∈ Rn×(�−1),

u = u� ∈ Rn, p = e℘�

∈ Rn,

U = [U u] ∈ Rn×�, P = [P p] ∈ R

n×�.(3.17)

Put M = PTU, and consider the matrix norm ‖M−1‖2 from Algorithm 1. At theinitial step of Algorithm 1, P = e℘1 and U = u1. Thus,

M = PTU = eT℘1u1, ‖M−1‖2 =

1

|eT℘1u1| = ‖u1‖−1

∞ ≥ 1.(3.18)

That is, for m = 1, C = ‖M−1‖2 = ‖u1‖−1∞ . Note that the choice of the first

interpolation index ℘1 minimizes the matrix norm ‖M−1‖2 and hence minimizes theerror bound (3.8).

Now consider a general step ≥ 2 with matrices defined in (3.17). With M =

PTU, we can write M =[

M PTupT U pTu

], where M = PT U and M can be factored in

the form of

M =

[M PTu

pT U pTu

]=

[M 0aT ρ

] [I c0 1

],(3.19)

where aT = pT U, c = M−1PTu, and ρ = pTu− aT c = pT (u − UM−1PTu). Notethat |ρ| = ‖r‖∞, where r is defined in step 5 of Algorithm 1. Now, the inverse of Mis

M−1 =

[I −c0 1

] [M−1 0

−ρ−1aT M−1 ρ−1

](3.20)

=

[I −c0 1

] [I 0

−ρ−1aT ρ−1

] [M−1 00 1

](3.21)

=

{[I 00 0

]+ ρ−1

[c−1

] [aT ,−1]}[

M−1 00 1

].(3.22)

A bound for the 2-norm of M−1 is then given by

‖M−1‖2 ≤{∥∥∥∥

[I 00 0

]∥∥∥∥2

+ |ρ|−1

∥∥∥∥[

c−1

] [aT ,−1

]∥∥∥∥2

}∥∥∥∥[

M−1 00 1

]∥∥∥∥2

.(3.23)


Now observe that∥∥∥∥[

c−1

] [aT ,−1]∥∥∥∥

2

=

∥∥∥∥[U,u]

[c−1

] [aT ,−1]∥∥∥∥

2

(3.24)

≤ ∥∥Uc− u∥∥2

∥∥[aT ,−1]∥∥2

(3.25)

≤√1 + ‖a‖22

√n∥∥Uc− u

∥∥∞ ≤

√2n|ρ|.(3.26)

Substituting this into (3.23) gives

‖M−1‖2 ≤ [1 +√2n]‖M−1‖2 ≤ (1 +

√2n)m−1‖u1‖−1

∞ ,(3.27)

with the last inequality obtained by recursively applying this stepwise bound over them steps.

Since the DEIM procedure selects the index ℘� that maximizes |ρ|, it minimizesthe reciprocal 1

|ρ| , which controls the increment in the bound of ‖M−1‖2 at iteration

as shown in (3.23). Therefore, the selection process for the interpolation index in eachiteration of DEIM (line 6 of Algorithm 1) can be explained in terms of limiting growth

of the error bound of the approximation f . This error bound from Lemma 3.2 appliesto any nonlinear vector-valued function f(τ) approximated by DEIM. However, thebound in (3.10) is not useful as an a priori estimate, since it is very pessimistic andgrows far more rapidly than the actual observed values of ‖(PTU)−1‖2. In practice,we just compute this norm (the matrix is typically small) and use it to obtain an aposteriori estimate.

For a given dimension m of the DEIM approximation, the constant C does notdepend on f , and hence it applies to the approximation f (τ) of f(τ) from Definition 3.1for any τ ∈ D . However, the best approximation error

E∗ = E∗(f(τ))is dependent upon f(τ) and changes with each new value of τ . This would be quiteexpensive to compute, so an easily computable estimate is highly desirable. A rea-sonable estimate is available with the SVD of the nonlinear snapshot matrix

F = [f1, f2, . . . , fns ].

Let F = Range(F), and let F = UΣWT be its SVD, where U = [U, U] and U

represents the leading m columns of the orthogonal matrix U. Partition Σ =[Σ 00 ˜Σ

]to conform with the partitioning of U. The singular values are ordered as usual withσ1 ≥ σ2 ≥ · · · ≥ σm ≥ σm+1 ≥ · · · ≥ σn ≥ 0. The diagonal matrix Σ has the leading

m singular values on its diagonal. The orthogonal matrix W = [W,W] is partitionedaccordingly. Any vector f ∈ F may be written in the form

f = Fg = UΣg + UΣg,

where g = WT g and g = WT g. Thus

‖f − f∗‖2 = ‖(I−UUT )f‖2 = ‖UΣg‖2 ≤ σm+1‖g‖2.

For vectors f nearly in F , we have f = Fg+w, with wT Fg = 0, and thus

E∗ = E∗(f) ≈ σm+1(3.28)


is reasonable so long as ‖w‖2 is small (‖w‖2 = O(σm+1) ideally). The POD approach(and hence the resulting DEIM approach) is most successful when the trajectoriesare attracted to a low-dimensional subspace (or manifold). Hence, the vectors f(τ)should nearly lie in F , and this approximation will then serve for all of them.

To illustrate the error bound for DEIM approximation, we shall present numer-ical results for some examples of nonlinear parametrized functions defined on one-dimensional (1-D) and two-dimensional (2-D) discrete spatial points. These exper-iments show that the approximate error bound using σm+1 in place of E∗ is quitereasonable in practice.

3.3. Numerical examples of the DEIM error bound. This section demon-strates the accuracy and efficiency of the approximation from DEIM as well as itserror bound given in section 3.2. The examples here use the POD basis in the DEIMapproximation. The POD basis is constructed from a set of snapshots correspondingto a selected set of elements in D . In particular, we define

Ds = {μs1, . . . , μ

sns} ⊂ D ,(3.29)

used for constructing a set of snapshots given by

F = {f(μs1), . . . , f(μ

sns)},(3.30)

which is used for computing the POD basis {u�}m�=1 for the DEIM approximation.

To evaluate the accuracy, we apply the DEIM approximation f in (3.5) to theelements in the set

D = {μ1, . . . , μn} ⊂ D ,(3.31)

which is different from and larger than the set Ds used for the snapshots. Then weconsider the average error for DEIM approximation f over the elements in D given by

E(f) = 1

n

n∑i=1

‖f(μi)− f(μi)‖2(3.32)

and the average POD error in (3.9) for POD approximation f∗ from (3.11) over theelements in D given by

E∗(f) = 1

n

n∑i=1

‖f(μi)− f∗(μi)‖2 =1

n

n∑i=1

E∗(f(μi)).(3.33)

From Lemma 3.2, the average error bound is then given by

E(f) ≤ CE∗(f),(3.34)

with the corresponding approximation using (3.28):

E(f) � Cσm+1.(3.35)

This estimate is purely heuristic. Although it does seem to provide a reasonablequalitative estimate of the expected error, this quantity is clearly not a rigorousbound, and we are not optimistic that any such rigorous bound can be obtainedmathematically.


3.3.1. A nonlinear parametrized function with spatial points in onedimension. Consider a nonlinear parametrized function s : Ω×D �→ R defined by

s(x;μ) = (1− x) cos(3πμ(x + 1))e−(1+x)μ,(3.36)

where x ∈ Ω = [−1, 1] and μ ∈ D = [1, π]. This nonlinear function is from anexample in [27]. Let x = [x1, . . . , xn]

T ∈ Rn, with xi equidistantly spaced points in

Ω for i = 1, . . . , n, n = 100. Define f : D �→ Rn by

f(μ) = [s(x1;μ), . . . , s(xn;μ)]T ∈ R

n(3.37)

for μ ∈ D . This example uses 51 snapshots f(μsj) to construct POD basis {u�}m�=1,

with μs1, . . . , μ

s51 selected as equally spaced points in [1, π]. Figure 3.2 shows the

singular values of these snapshots and the corresponding first six POD basis vectorswith the first six spatial points selected from the DEIM algorithm using this PODbasis as an input. Figure 3.3 compares the approximate functions from DEIM ofdimension 10 with the original function of dimension 100 at different values of μ ∈ D .This demonstrates that DEIM gives a good approximation at arbitrary values μ ∈ D .Figure 3.4 illustrates the average errors defined in (3.32) and (3.33), with the averageerror bound and its approximation computed from the right-hand sides of (3.34) and(3.35), respectively, with μ1, . . . , μn ∈ D selected uniformly over D and n = 101.

0 10 20 30 40 50 6010

−15

10−10

10−5

100

105

Singular values of 51 Snapshots

−1 −0.5 0 0.5 1−5

−4

−3

−2

−1

0

1

2

3EIM points and POD bases (1−6)

PODbasis 1PODbasis 2PODbasis 3PODbasis 4PODbasis 5PODbasis 6EIM pts

Fig. 3.2. Singular values and the corresponding first six POD bases with DEIM points ofsnapshots from (3.37).

3.3.2. A nonlinear parametrized function with spatial points in twodimensions. Consider a nonlinear parametrized function s : Ω×D �→ R defined by

s(x, y;μ) =1√

(x− μ1)2 + (y − μ2)2 + 0.12,(3.38)

where (x, y) ∈ Ω = [0.1, 0.9]2 ⊂ R2 and μ = (μ1, μ2) ∈ D = [−1,−0.01]2 ⊂ R

2. Thisexample is modified from one given in [19]. Let (xi, yj) be uniform grid points in Ωfor i = 1, . . . , nx and j = 1, . . . , ny. Define s : D �→ R

nx×ny by

s(μ) = [s(xi, yj ;μ)] ∈ Rnx×ny(3.39)

for μ ∈ D and i = 1, . . . , nx, and j = 1, . . . , ny. In this example, the full dimensionis n = nxny = 400 (nx = ny = 20). Note that we can define a corresponding


−1 −0.5 0 0.5 1−2

−1

0

1

2μ = 1.17

exactDEIM approx

−1 −0.5 0 0.5 1−2

−1

0

1

2μ = 1.5

exactDEIM approx

−1 −0.5 0 0.5 1−2

−1

0

1

2μ = 2.3

exactDEIM approx

−1 −0.5 0 0.5 1−2

−1

0

1

2μ = 3.1

exactDEIM approx

Fig. 3.3. The approximate functions from DEIM of dimension 10 compared with the originalfunctions (3.37) of dimension n = 100 at μ = 1.17, 1.5, 2.3, 3.1.

0 10 20 30 40 5010

−15

10−10

10−5

100

105

m (Reduced dim)

Avg

Err

or

Avg Error and Avg Error Bound (1D)

Error PODError DEIMError BoundApprox Error Bound

Fig. 3.4. Comparison of average errors of POD and DEIM approximations for (3.37) with theaverage error bounds and their approximations given in (3.34) and (3.35), respectively.

vector-valued function f : D �→ Rn for this problem by reshaping the matrix s(μ)

to a vector of length n = nxny. The 225 snapshots constructed from uniformlyselected parameters μs = (μs

1, μs2) in parameter domain D are used for constructing

the POD basis. A different set of 625 pairs of parameters μ are used for testing (errorand CPU time). Figure 3.5 shows the singular values of these snapshots and thecorresponding first six POD basis vectors. Figure 3.6 illustrates the distribution ofthe first 20 spatial points selected from the DEIM algorithm using this POD basis asan input. Notice that most of the selected points cluster close to the origin, wherethe function s increases sharply. Figure 3.7 shows that the approximate functionsfrom DEIM of dimension 6 can reproduce the original function of dimension 400 verywell at arbitrarily selected value μ ∈ D . Figure 3.8 gives the average errors with thebounds from the last section and the corresponding average CPU times for differentdimensions of POD and DEIM approximations. The average errors of POD and DEIMapproximations are computed from (3.32) and (3.33), respectively. The average errorbounds and their approximations are computed from the right-hand sides of (3.34)


0 10 20 30 40 50 6010

−10

10−5

100

105

Singular Values of Snapshots

00.5

1

0

0.5

1−0.1

−0.05

0

x

POD basis #1

y 00.5

1

0

0.5

1−0.5

0

0.5

x

POD basis #2

y 00.5

1

0

0.5

1−0.2

0

0.2

x

POD basis #3

y

00.5

1

0

0.5

1−0.5

0

0.5

x

POD basis #4

y 00.5

1

0

0.5

1−0.5

0

0.5

x

POD basis #5

y 00.5

1

0

0.5

1−0.5

0

0.5

x

POD basis #6

y

Fig. 3.5. Singular values and the first six corresponding POD basis vectors of the snapshots ofthe nonlinear function (3.39).

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1. 2.

3.

4.

5.

6.7.

8.

9. 10.

11.

12.

13.

14.15.

16.

17.

18.

19.

20.

DEIM points

x

y

Fig. 3.6. First 20 points selected by DEIM for the nonlinear function (3.39).

and (3.35), respectively. This example uses μ1, . . . , μn ∈ D selected uniformly over Dand n = 625. The CPU times are averaged over the same set D .

3.4. Application of DEIM to nonlinear FD discretized systems. TheDEIM approximation (3.4) developed in the previous section may now be used toapproximate the nonlinear term in (2.9) and the Jacobian in (2.10) with nonlinearapproximations having computational complexity proportional to the number of re-duced variables obtained with POD.

In the case of nonlinear time dependent PDEs, from (3.4), set τ = t and f(t) =F(Vky(t)); then the nonlinear function in (2.5) approximated by DEIM can be writtenas

F(Vky(t)) ≈ U(PTU)−1PTF(Vky(t))(3.40)

= U(PTU)−1F(PTVky(t)).(3.41)


0.20.4

0.60.8

0.20.4

0.60.8

1

2

3

4

x

Full dim= 400,[μ1,μ

2] = [−0.05,−0.05]

y

s(x,

y;μ)

0.20.4

0.60.8

0.20.4

0.60.8

1

2

3

4

x

POD: dim = 6, L2 error: 8.2e−3

y

s(x,

y;μ)

0.20.4

0.60.8

0.20.4

0.60.8

1

2

3

4

x

DEIM: dim = 6, L2 error: 1.8e−2

y

s(x,

y;μ)

Fig. 3.7. Compare the original nonlinear function (3.39) of dimension 400 with the POD andDEIM approximations of dimension 6 at parameter μ = (−0.05,−0.05).

0 5 10 15 2010

−5

100

105

m (Reduced dim)

Avg

Err

or

Avg Error and Avg Error Bound (2D)

Error PODError DEIMError BoundApprox Error Bound

0 5 10 15 2010

−2

10−1

100

101

Reduced dim

Tim

e (s

ec)

Avg CPU time

PODDEIM

Fig. 3.8. Left: Average errors of POD and DEIM approximations for (3.39) with the averageerror bounds given in (3.34) and their approximations given in (3.35). Right: Average CPU timefor evaluating the POD and DEIM approximations.

The last equality in (3.41) follows from the fact that the function F evaluates compo-nentwise at its input vector. The nonlinear term in (2.9) can thus be approximatedby

N(y) ≈ VTk U(PTU)−1︸︷︷︸

precomputed:k×m

F(PTVky(t))︸︷︷︸m×1

.(3.42)

Note that the term VTk U(PTU)−1 in (3.42) does not depend on t, and therefore it can

be precomputed before solving the system of ODEs. Note also thatPTVky(t) ∈ Rm in

(3.42) can be obtained by extracting the rows ℘1, . . . , ℘m of Vk and then multiplyingagainst y, which requires 2mk operations. Therefore, if α(m) denotes the cost ofevaluating m components of F, the complexity for computing this approximationof the nonlinear term becomes roughly O(α(m) + 4km), which is independent ofdimension n of the full-order system (2.1).

Similarly, in the case of steady parametrized nonlinear PDEs, from (3.4), setτ = μ and f(μ) = F(Vky(μ)). Then the nonlinear function in (2.6) approximated byDEIM can be written as

F(Vky(μ)) ≈ U(PTU)−1F(PTVky(μ)),(3.43)


and the approximation for the Jacobian of the nonlinear term (2.10) is of the form

JF(y(μ)) ≈ VTk U(PTU)−1︸︷︷︸

precomputed:k×m

JF(PTVky(μ))︸︷︷︸m×m

PTVk︸︷︷︸m×k

,(3.44)

where

JF(PTVky(μ)) = JF(y

r(μ)) = diag{F ′(yr1(μ)), . . . , F

′(yrm(μ))},

and yr(μ) = PTVky(μ), which can be computed with complexity independent of nas noted earlier. Therefore, the computational complexity for the approximation in(3.44) is roughly O(α(m) + 2mk + 2γmk + 2mk2), where γ is the average number ofnonzero entries per row of the Jacobian.

The approximations from DEIM are now in the form of (3.42) and (3.44) thatrecover the computational efficiency of (2.9) and (2.10), respectively.

Note that the nonlinear approximations from DEIM in (3.41) and (3.43) are ob-tained by exploiting the special structure of the nonlinear function F being evaluatedcomponentwise at y. The next section provides a completely general scheme.

3.5. Interpolation of general nonlinear functions. The very simple caseof F(y) = [F (y1), . . . , F (yn)]

T has been discussed for purposes of illustration and isindeed important in its own right. However, DEIM extends easily to general nonlinearfunctions. MATLAB notation is used here to explain this generalization:

[F(y)]i = Fi(y) = Fi(yji1,yji2

,yji3, . . . ,yjini

) = Fi(y(ji)),(3.45)

where Fi : Yi → R, Yi ⊂ Rni , and the integer vector ji = [ji1, j

i2, j

i3, . . . , j

ini]T denotes

the indices of the subset of components of y required to evaluate the ith componentof F(y) for i = 1, . . . , n.

The nonlinear function of the reduced-order system obtained from the POD-Galerkin method by projecting on the space spanned by columns of Vk ∈ R

n×k isin the form of F(Vky), where the components of y ∈ R

k are the reduced variables.Recall that the DEIM approximation of order m for F(Vky) is given by

F(Vky) ≈ U(PTU)−1︸︷︷︸k×m

PTF(Vky)︸︷︷︸m×1

,(3.46)

where U ∈ Rn×m is the projection matrix for the nonlinear function F, P = [e℘1 , . . . ,

e℘m ] ∈ Rn×m, and ℘1, . . . , ℘m are interpolation indices from the DEIM point selection

algorithm. In the simple case when F is evaluated componentwise at y, we havePTF(Vky) = F(PTVky), where PTVk can be obtained by extracting rows of Vk

corresponding to ℘1, . . . , ℘m, and hence its computational complexity is independentof n. However, this is clearly not applicable to the general nonlinear vector-valuedfunction.

An efficient method for computing PTF(Vky) in the DEIM approximation (3.46)of a general nonlinear function is possible using a certain sparse matrix data structure.Notice that, since yj ≈ Vk(j, :)y, an approximation to F(y) is provided by

F(Vky) = [F1(Vk(j1, :)y), . . . , Fn(Vk(jn, :)y)]T ∈ R

n,(3.47)

and thus

PTF(Vky) = [F℘1(Vk(j℘1 , :)y), . . . , F℘m(Vk(j℘m , :)y)]T ∈ Rm.(3.48)


The complexity for evaluating each component ℘i, i = 1, . . . ,m, of (3.48)

F℘i (y) := F℘i(Vk(j℘i , :)y)(3.49)

is n℘i × k flops plus the complexity of evaluating the nonlinear scalar-valued functionF℘i of the n℘i variables indexed by j℘i .

The sparse evaluation procedure may be implemented using a compressed sparserow data structure as used in sparse matrix factorizations. Two linear integer arraysare needed: irstart is a vector of length m + 1 containing pointers to locations inthe vector jrow, which is of length n�℘ =

∑mi=1 n℘i . The successive ni entries of

jrow(irstart(i)) indicate the dependence of the i component of F(y) on the selectedvariables from y. In particular,

• irstart(i) contains the location of the start of the ith row with irstart(m +

1) = n�℘ + 1. I.e., irstart(1) = 1, and irstart(i) = 1 +∑i−1

j=1 n℘j for i =2, . . . ,m+ 1.• jrow contains the indices of components in y required to compute the ℘ithfunction F℘i in locations irstart(i) to irstart(i+1)− 1 for i = 1, . . . ,m. I.e.,

irstart(1) irstart(2) irstart(m)↓ ↓ ↓

jrow =[j℘1

1 , . . . , j℘1n℘1︸︷︷︸

j℘1

, j℘2

1 , . . . , j℘2n℘2︸︷︷︸

j℘2

, . . . , j℘m

1 , . . . , j℘mn℘m︸︷︷︸

j℘m

]T∈ Z

n�℘

+ .

Given Vk and y, the following demonstrates how to compute the approximationF℘i(y) in (3.49), for i = 1, . . . ,m, from the vectors irstart and jrow:

for i = 1 : mj℘i = jrow(irstart(i) : irstart(i + 1)− 1)

F℘i(y) = F℘i(Vk(j℘i , :)y)end

Typically, the Jacobians of large-scale problems are sparse, and this scheme willbe very efficient. However, if the Jacobian is dense (or nearly so), the complexitywould be on the order of mn, where m is the number of interpolation points.

The next section will discuss the computational complexity used for constructingand solving the reduced-order systems. It will also illustrate in terms of complexityas well as computation time that solving the POD reduced system could be moreexpensive than solving the original full-order system.

3.6. Computational complexity. Recall that the POD-DEIM reduced systemfor the unsteady nonlinear problem (2.1) is

d

dty(t) = Ay(t) +B F(V�℘y(t)),(3.50)

and the approximation for the steady state problem (2.2) is given by

Ay(t) +B F(V�℘y(t)) = 0,(3.51)

where A = VTk AVk ∈ R

k×k, and B = VTk UU−1

�℘ ∈ Rk×m, with U�℘ = PTU and

V�℘ = PTVk. This section summarizes the computational complexity for constructingand solving the POD-DEIM reduced system compared to both the original full-order


Table 3.1

Computational complexity for constructing a POD-DEIM reduced-order system.

Procedure Complexity

Snapshots Problem dependent

SVD: POD basis O(nn2s)

DEIM algorithm: m interpolation indices O(m4 +mn)

Precompute: A = VTk AVk

{ O(n2k + nk2) for dense A,O(nk + nk2) for sparse A

Precompute: B = VTk UU−1

�℘O(nkm+m2n+m3)

system and the POD reduced system. Table 3.1 gives the computational complexityfor constructing a POD-DEIM reduced system.

Note that, for large snapshot sets, it is far more efficient to compute the dominantsingular values and vectors iteratively via ARPACK (or svds in MATLAB) [24]. Thecomputational work shown in Table 3.1 has to be done only once before solving thePOD-DEIM reduced systems. The constant coefficient matrices A and B are pre-computed, stored, and reused while solving the reduced systems.

The computational complexity for solving the standard POD reduced system caneven exceed the complexity of solving the original full-order system due to the orthog-onal projection of the nonlinear term at each iteration, especially when A ∈ R

n×n

represents the discretization of a linear differential operator and its sparsity is em-ployed in the computation. To illustrate the inefficiency of the POD reduced system,we consider the nonlinear 2-D steady state problem introduced later in section 4. Ta-ble 3.2 summarizes the computational complexity (flops) for computing one Newtoniteration of the full-order system as well as the POD and POD-DEIM reduced-ordersystems. We see that, in the case of the sparse full-order system, the complexityO(k3 + nk2) used in solving the POD reduced system could become higher than thecomplexity O(n2) used in solving the original system once O(k2) becomes propor-tional to O(n). In practice, the CPU time may not be directly proportional to thesepredicted flops, since there are many other factors that might affect the CPU times.However, this analysis does reflect the relative computational requirements and maybe useful for predicting expected relative computational times.

This inefficiency of the POD reduced system indeed occurs in this computation.From Figure 3.9, the average CPU time for solving the POD reduced system in eachtime step exceeds the CPU time for solving the original system as soon as its dimensionreaches around 80. Also, Figure 4.8 in section 4 shows that, while the POD reducedsystem of dimension 15 gives an O(10) reduction in computation time as comparedto the full-order system, the POD-DEIM reduced system with both POD and DEIMhaving dimension 15 gives an O(100) reduction in computation time with the same

Table 3.2

Comparison of the computational work for each Newton iteration of the steady state problem.

System Complexity

Full Dense A: O(n3), Sparse A: O(n2)

POD O(k3 + nk2)

POD-DEIM O(k3 +mk2)


0 20 40 60 80 10010

−2

10−1

100

101

102

103

k (POD dim)

time

(sec

)

Average CPU time (scaled) for each Newton Iteration

DEIM1DEIM10DEIM20DEIM30DEIM40DEIM50DEIM60DEIM70DEIM80DEIM90DEIM100PODFull:n=2500(dense)Full:n=2500(sparse)

Fig. 3.9. Average CPU time (scaled with the CPU time for the full-sparse system) in eachNewton iteration for solving the steady state 2-D problem.

order of accuracy. These demonstrate the inefficiency of the POD reduced systemthat has been remedied by the introduction of DEIM.

4. Numerical results. The efficiency and accuracy of the approximation fromDEIM will be demonstrated through two problems. The first is a 1-D nonlinear PDEarising in neuron modeling. The second is a nonlinear 2-D steady state problem whosesolution is obtained by solving its FD discretized system by using Newton’s method.In both experiments, computation time was reduced roughly by a factor of 100.

4.1. The FitzHugh–Nagumo (F–N) system. The F–N system is used inneuron modeling. It is a simplified version of the Hodgkin–Huxley model, whichmodels in a detailed manner activation and deactivation dynamics of a spiking neuron.This system is given by (4.1)–(4.4): Let x ∈ [0, L], t ≥ 0,

εvt(x, t) = ε2vxx(x, t) + f(v(x, t)) − w(x, t) + c,(4.1)

wt(x, t) = bv(x, t)− γw(x, t) + c,(4.2)

with nonlinear function f(v) = v(v − 0.1)(1− v) and initial conditions and boundaryconditions

v(x, 0) = 0, w(x, 0) = 0, x ∈ [0, L],(4.3)

vx(0, t) =− i0(t), vx(L, t) = 0, t ≥ 0,(4.4)

where the parameters are given by L = 1, ε = 0.015, b = 0.5, γ = 2, and c = 0.05. Thestimulus i0(t) = 50000t3 exp(−15t). The variables v and w are voltage and recoveryof voltage, respectively. The dimension of the full-order system (finite difference) is1024. The POD basis vectors are constructed from 100 snapshot solutions obtainedfrom the solutions of the full-order FD system at equally spaced time steps in theinterval [0, 8]. This is not a scalar equation and requires a slight generalization of theproblem setting discussed earlier. However, the FD discretization does indeed yield asystem of ODEs of the same form as (2.1).

Figure 4.1 shows the fast decay around the first 40 singular values of the snapshotsolutions for v and w and the nonlinear snapshots f(v). The solution of this systemhas a limit cycle for each spatial variable x. We therefore illustrate the solutions v andw through the plots of a phase-space diagram as shown in Figure 4.2 for the solutionsof the full-order system and the POD-DEIM reduced system using both POD and


0 20 40 60 80 10010

−20

10−10

100

1010 Singular values of the snapshots

Singular Val of vSingular Val of wSingular Val of f(v)

Fig. 4.1. The singular values of the 100 snapshot solutions for v, w, and f(v) from the full-orderFD system of the F–N system.

0

0.5

1

−0.50

0.51

1.50

0.05

0.1

0.15

0.2

x

Phase−Space diagram of reduced system(POD=5/DEIM=5)

v(x,t)

w(x

,t)

Full1024 POD5/EIM5

−0.5 0 0.5 1 1.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

v(x,t)

w(x

,t)

Phase−Space diagram of reduced system(POD=5/DEIM=5)

Full1024 POD5/EIM5

Fig. 4.2. Left: Phase-space diagram of v and w at different spatial points x from the FD system(dim 1024) and the POD-DEIM reduced systems (dim 5). Right: Corresponding projection of thesolutions onto the v-w plane.

DEIM of dimension 5. We see that this reduced-order system captures the limit cycleof the original full-order system very well. The average relative errors of the solutionsof the reduced systems and the average CPU time (scaled with the CPU time fromthe sparse full-order system) for each time step from different dimensions of POD andDEIM are presented in Figure 4.3.

4.2. A nonlinear 2-D steady state problem (from [19]). In this subsection,we apply the POD-DEIM method to nonlinear PDEs in a 2-D spatial domain. Theequations are

−∇2u(x, y) + s(u(x, y);μ) = 100 sin(2πx) sin(2πy),(4.5)

s(u;μ) =μ1

μ2(eμ2u − 1),(4.6)

where the spatial variables (x, y) ∈ Ω = (0, 1)2 and the parameters are μ = (μ1, μ2) ∈D = [0.01, 10]2 ⊂ R

2, with a homogeneous Dirichlet boundary condition. We nu-merically solve this system by applying Newton’s method to the nonlinear equationsresulting from an FD discretization. The spatial grid points (xi, yj) are equally spaced


0 20 40 60 80 10010

−12

10−10

10−8

10−6

10−4

10−2

100

POD dim

Ave

rag

e re

lati

ve E

rro

r

Error DEIM (periodic FN):(1/nt)sum

t||yFD(t) −yDEIM(t)||/||yFD(t)||

DEIM 1DEIM 3DEIM 5DEIM 10DEIM 20DEIM 30DEIM 40DEIM 50DEIM 60DEIM 70DEIM 80DEIM 90DEIM 100 POD

0 20 40 60 80 10010

−2

10−1

100

101

k (POD dim)

tim

e (s

ec)

CPU time (scaled) for each semi−backward Euler iteration

DEIM1DEIM10DEIM20DEIM30DEIM40DEIM50DEIM60DEIM70DEIM80DEIM90DEIM100PODFull:n=1024(dense)Full:n=1024(sparse)

Fig. 4.3. Left: Average relative errors from the POD-DEIM reduced system (solid lines) andfrom POD reduced systems (dashed line) for the F–N system. Once the dimension of DEIM reaches40, the approximations from DEIM and POD (optimal) are indistinguishable. Right: Average CPUtime (scaled with the CPU time for the full-sparse system) in each time step of the semi-implicit(backward) Euler method.

in Ω for i, j = 1, . . . , 50. The full dimension is then n = 2500. Figures 4.4 and 4.5show the singular values and the first six corresponding POD bases of the uniformlyselected 144 sampled snapshot solutions for (4.5) and of the uniformly selected 144nonlinear snapshots for (4.6). Figure 4.6 shows the distribution of the first 30 pointsin Ω selected from the DEIM algorithm. Figure 4.7 shows that the POD-DEIM re-duced system (with POD and DEIM having dimension 6) can accurately reproducethe solution of the full-order system of dimension 2500 with error O(10−3). The av-erage errors and the average CPU time (scaled with the CPU time from the sparsefull-order system) for each Newton iteration of the reduced systems with differentdimensions of POD and DEIM are presented in Figure 4.8. The average CPU timesfor higher dimensions are shown earlier in section 3.6. These errors are averaged overa set of 225 parameters μ that were not used to obtain the sample snapshots. Thissuggests that the DEIM-POD reduced-order system can give a good approximationto the original system with any value of parameter μ ∈ D .

0 50 100 15010

−15

10−10

10−5

100

105

Singular Values

Snapshot solsSnapshot nonlin

Fig. 4.4. Singular values of the snapshot solutions u from (4.5) and the nonlinear snapshotss(u;μ) from (4.6).


Fig. 4.5. The first six dominant POD basis vectors of the snapshot solutions u from (4.5) andof the nonlinear snapshots s(u;μ) from (4.6).

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

DEIM points

x

y

Fig. 4.6. First 30 points selected by DEIM.

00.5

1

0

0.5

1

−1

−0.5

0

0.5

x

Full dim=2500,[μ1,μ

2]=[0.3,9]

y

u(x

,y;μ

)

00.5

1

0

0.5

1

−1

−0.5

0

0.5

x

POD6/DEIM6,[μ1,μ

2] = [0.3,9],err: 0.0011115

y

u(x

,y;μ

)

0 0.5 10

0.2

0.4

0.6

0.8

1 Error POD6/DEIM6, [μ

1,μ

2] = [0.3,9]

x

y

0

0.2

0.4

0.6

0.8

1

x 10−3

Fig. 4.7. Numerical solution from the full-order system (dim = 2500) with the solution fromthe POD-DEIM reduced system (POD dim = 6, DEIM dim = 6) for μ = (μ1, μ2) = (0.3, 9). Thelast plot shows the corresponding errors at the grid points.


0 5 10 15 2010

−6

10−4

10−2

100

102

k (POD dim)

Avg

Err

or

Error POD/EIM:(1/nμ)sum

μ||u(μ) −uEIM(μ)||

2

DEIM1DEIM3DEIM5DEIM11DEIM13DEIM15DEIM19POD

0 5 10 15 2010

−2

10−1

100

101

102

103

k(POD dim)

tim

e (s

ec)

Normalized Average CPU time for each Newton Iteration

DEIM1DEIM3DEIM5DEIM11DEIM13DEIM15DEIM19PODFull2500(sparse)Full2500(dense)

Fig. 4.8. Average error from the POD-DEIM reduced systems and average CPU time (scaled)in each Newton iteration for solving the steady state 2-D problem.

5. Conclusions. We have demonstrated through several basic examples thatDEIM is a promising approach to overcoming the deficiencies of POD with respect togeneral nonlinearities. In two recent papers, we have demonstrated the effectiveness ofDEIM in two very different applications. One is in neural modeling, where we reducenumerous examples of full Hodgkin–Huxley models of realistic spiking neurons from5000 down to 100 variables [21] with corresponding reduction in computation time.The other is in two-phase miscible flow in porous media with varying Peclet number,both with and without chemistry at the interface of the different fluids [12]. Thismiscible flow experiment verifies the potential of the DEIM approach. It was able toreduce a 15000 variable system to 40 variables with little loss in accuracy and a factorof 1000 reduction in computation time. By comparison, standard POD resulted inonly a factor of 10 reduction in computation time.

In this paper, we developed an error bound showing the obtained approximationto be nearly optimal. The average errors for the POD-DEIM approach in Figures 4.3and 4.8 show that the accuracy of the approximation depends on the dimensions ofboth POD and DEIM. The numerical results demonstrate that the POD-DEIM ap-proach not only gives an accurate reduced system that is substantially smaller thanthe original system with a general nonlinearity, but it also preserves the steady statebehavior (e.g., the limit cycle) of the original system. The POD-Galerkin approachcombined with DEIM approximation is therefore a promising dimension reductiontechnique for FD discretized systems of time dependent and/or parametrized nonlin-ear PDEs. However, the method clearly applies to fairly arbitrary systems of ODEsregardless of their origin.

We have avoided discussing systems arising from FE discretization of the spatialoperators in a time dependent system of PDEs. This semidiscretization approachleads to large systems of ODEs in the coefficients of the FE basis representation of theapproximate solution. Formally, there is no reason DEIM could not be applied to theODEs describing the coefficients. However, there are approximation issues concerninghow inaccuracies in the coefficients might affect the FE approximation to the solution.The original EIM approach and recent variants are derived in conjunction with theFE approximation, and as a consequence these interpolant approximation questionsare automatically dealt with. We intend to investigate the effectiveness of DEIM onthe FEM coefficients in future research.


Acknowledgments. The authors wish to thank Professor Mark Embree forseveral enlightening discussions and for pointing out [39]. DCS would also like toacknowledge Professor Jacob White for a lively discussion about DEIM for generalnonlinearities at the SIAM CSE 09 meeting. This discussion led directly to the ma-terial in section 3.5.

REFERENCES

[1] P. Astrid, Fast reduced order modeling technique for large scale LTV systems, in Proceedingsof the 2004 American Control Conference, Vol. 1, 2004, pp. 762–767.

[2] P. Astrid, Reduction of Process Simulation Models: A Proper Orthogonal DecompositionApproach, Ph.D. thesis, Department of Electrical Engineering, Eindhoven University ofTechnology, Eindhoven, The Netherlands, 2004.

[3] P. Astrid and S. Weiland, On the construction of POD models from partial observations,in Proceedings of the 44th IEEE Conference on Decision and Control and 2005 EuropeanControl Conference (CDC-ECC 2005), 2005, pp. 2272–2277.

[4] P. Astrid, S. Weiland, K. Willcox, and T. Backx, Missing point estimation in modelsdescribed by proper orthogonal decomposition, in Proceedings of the 43rd IEEE Conferenceon Decision and Control (CDC 2004), Vol. 2, 2004, pp. 1767–1772.

[5] P. Astrid, S. Weiland, K. Willcox, and T. Backx, Missing point estimation in modelsdescribed by proper orthogonal decomposition, IEEE Trans. Automat. Control, 53 (2008),pp. 2237–2251.

[6] Z. Bai, Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems,Appl. Numer. Math., 43 (2002), pp. 9–44.

[7] M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera, An “empirical interpolation”method: Application to efficient reduced-basis discretization of partial differential equa-tions, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 667–672.

[8] T. Bechtold, M. Striebel, K. Mohaghegh, and E. J. W. ter Maten, Nonlinear modelorder reduction in nanoelectronics: Combination of POD and TPWL, Proc. Appl. Math.Mech., 8 (2008), pp. 10057–10060.

[9] T. Bui-Thanh, M. Damodaran, and K. Willcox, Aerodynamic data reconstruction and in-verse design using proper orthogonal decomposition, AIAA J., 42 (2004), pp. 1505–1516.

[10] M. A. Cardoso and L. J. Durlofsky, Linearized reduced-order models for subsurface flowsimulation, J. Comput. Phys., 229 (2010), pp. 681–700.

[11] M. A. Cardoso, L. J. Durlofsky, and P. Sarma, Development and application of reduced-order modeling procedures for subsurface flow simulation, Internat. J. Numer. MethodsEngrg., 77 (2009), pp. 1322–1350.

[12] S. Chaturantabut and D. C. Sorensen, Application of POD and DEIM to dimension re-duction of nonlinear miscible viscous fingering in porous media, Math. Comput. Model.Dyn. Syst., to appear.

[13] S. Chaturantabut and D. C. Sorensen, Discrete empirical interpolation for nonlinear modelreduction, in Proceedings of the 48th IEEE Conference on Decision and Control and the28th Chinese Control Conference (CDC/CCC 2009), 2009, pp. 4316–4321.

[14] Y. Chen, Model Order Reduction for Nonlinear Systems, Master’s thesis, Massachusetts Insti-tute of Technology, Cambridge, MA, 1999.

[15] Y. Chen and J. White, A quadratic method for nonlinear model order reduction, in Tech-nical Proceedings of the 2000 International Conference on Modeling and Simulation ofMicrosystems, 2000, pp. 477–480.

[16] N. Dong and J. Roychowdhury, Piecewise polynomial nonlinear model reduction, in Pro-ceedings of the Design Automation Conference, IEEE Computer Society, Los Alamitos,CA, 2003, pp. 484–489.

[17] D. Galbally, K. Fidkowski, K. Willcox, and O. Ghattas, Non-linear model reduction foruncertainty quantification in large-scale inverse problems, Internat. J. Numer. MethodsEngrg., 81 (2010), pp. 1581–1608.

[18] W. R. Graham, J. Peraire, and K. Y. Tang, Optimal control of vortex shedding using low-order models. Part I—Open-loop model development, Internat. J. Numer. Methods Engrg.,44 (1999), pp. 945–972.

[19] M. A. Grepl, Y. Maday, N. C. Nguyen, and A. T. Patera, Efficient reduced-basis treatmentof nonaffine and nonlinear partial differential equations, M2AN Math. Model. Numer.Anal., 41 (2007), pp. 575–605.


[20] M. Hinze and S. Volkwein, Proper orthogonal decomposition surrogate models for nonlineardynamical systems: Error estimates and suboptimal control, in Dimension Reduction ofLarge-Scale Systems, Lect. Notes Comput. Sci. Eng. 45, Springer, Berlin, 2005, pp. 261–306.

[21] A. R. Kellems, S. Chaturantabut, D. C. Sorensen, and S. J. Cox, Morphologicallyaccurate reduced order modeling of spiking neurons, J. Comput. Neurosci., 28 (2010), pp.477–494.

[22] K. Kunisch and S. Volkwein, Control of the Burgers equation by a reduced-order approachusing proper orthogonal decomposition, J. Optim. Theory Appl., 102 (1999), pp. 345–371.

[23] K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for ageneral equation in fluid dynamics, SIAM J. Numer. Anal., 40 (2002), pp. 492–515.

[24] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution ofLarge-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM,Philadelphia, 1998.

[25] L. Machiels, Y. Maday, I. B. Oliveira, A. T. Patera, and D. V. Rovas, Output boundsfor reduced-basis approximations of symmetric positive definite eigenvalue problems, C.R. Acad. Sci. Paris Ser. I Math., 331 (2000), pp. 153–158.

[26] Y. Maday, A. T. Patera, and G. Turinici, A priori convergence theory for reduced-basisapproximations of single-parameter elliptic partial differential equations, J. Sci. Comput.,17 (2002), pp. 437–446.

[27] N. C. Nguyen, A. T. Patera, and J. Peraire, A “best points” interpolation method forefficient approximation of parametrized functions, Internat. J. Numer. Methods Engrg.,73 (2007), pp. 521–543.

[28] N. C. Nguyen and J. Peraire, An efficient reduced-order modeling approach for non-linearparametrized partial differential equations, Internat. J. Numer. Methods Engrg., 76 (2008),pp. 27–55.

[29] N. C. Nguyen, G. Rozza, and A. T. Patera, Reduced basis approximation and a posteriorierror estimation for the time-dependent viscous Burgers’ equation, Calcolo, 46 (2009),pp. 157–185.

[30] A. T. Patera and G. J. Rozza, Reduced Basis Methods and A Posteriori Error Estimationfor Parametrized Partial Differential Equations, c©MIT, Cambridge, MA, 2006-08, MITPappalardo Graduate Monographs in Mechanical Engineering, to appear; available onlinefrom http://mathicse.epfl.ch/∼rozza/publications.html.

[31] J. R. Phillips, Projection frameworks for model reduction of weakly nonlinear systems, inDAC ’00: Proceedings of the 37th Annual Design Automation Conference, ACM, NewYork, 2000, pp. 184–189.

[32] C. Prud’homme, D. V. Rovas, K. Veroy, L. Machiels, Y. Maday, A. T. Patera, and

G. Turinici, Reliable real-time solution of parametrized partial differential equations:Reduced-basis output bound methods, J. Fluids Eng., 124 (2002), pp. 70–80.

[33] S. S. Ravindran, A reduced-order approach for optimal control of fluids using properorthogonal decomposition, Internat. J. Numer. Methods Fluids, 34 (2000), pp. 425–448.

[34] M. J. Rewienski, A Trajectory Piecewise-Linear Approach to Model Order Reduction ofNonlinear Dynamical Systems, Ph.D. thesis, Massachusetts Institute of Technology,Cambridge, MA, 2003.

[35] M. Rewienski and J. White, A trajectory piecewise-linear approach to model order reductionand fast simulation of nonlinear circuits and micromachined devices, in Proceedings ofthe International Conference on Computer-Aided Design, 2001, pp. 252–257.

[36] M. Rewienski and J. White, A trajectory piecewise-linear approach to model order reductionand fast simulation of nonlinear circuits and micromachined devices, IEEE Trans.Comput. Aided Des. Integr. Circuits Syst., 22 (2003), pp. 155–170.

[37] M. Rewienski and J. White, Model order reduction for nonlinear dynamical systems based ontrajectory piecewise-linear approximations, Linear Algebra Appl., 415 (2006), pp. 426–454.

[38] C. W. Rowley, T. Colonius, and R. M. Murray, Model reduction for compressible flowsusing POD and Galerkin projection, Phys. D, 189 (2004), pp. 115–129.

[39] D. B. Szyld, The many proofs of an identity on the norm of oblique projections, Numer.Algorithms, 42 (2006), pp. 309–323.

[40] A. Verhoeven, Redundancy Reduction of IC Models by Multirate Time-Integration andModel Order Reduction, Ph.D. thesis, Department of Mathematics and Computer Science,Eindhoven University of Technology, Eindhoven, The Netherlands, 2008.

[41] K. Veroy, D. V. Rovas, and A. T. Patera, A posteriori error estimation for reduced-basisapproximation of parametrized elliptic coercive partial differential equations: “Convexinverse” bound conditioners, ESAIM Control Optim. Calc. Var., 8 (2002), pp. 1007–1028.

[42] K. Willcox, Unsteady flow sensing and estimation via the gappy proper orthogonaldecomposition, Comput. Fluids, 35 (2006), pp. 208–226.

Date post:	19-Aug-2020
Category:	Documents
Upload:	others
View:	2 times
Download:	1 times

NONLINEAR MODELREDUCTION VIA DISCRETE EMPIRICAL … · SIAM J. SCI. COMPUT. c 2010 Society for...

Documents