1 Introduction
The nonlinear Schrödinger equation (NLSE) is used frequently for modeling wave propagation phenomena in different areas of physics, chemistry, and engineering. It is one of the most important models of mathematical physics, with application to different fields such as plasma physics, nonlinear optics, water waves, bimolecular dynamics, and many other fields. AblowitzLadik equation (ALE) [1] represents an integrable discretization of NLSE [24, 36, 38]
. There are two classes of geometric integrators for Hamiltonian systems; symplectic and energy preserving integrators. Unfortunately, it is not possible, in general, to preserve both energy and the symplectic form at the same time. The ALE is a noncanonical Hamiltonian system with a statedependent Poisson structure, so that the symplectic integrators are not applicable. Therefore, the ALE is integrated in time with the energy preserving discrete gradient methods. The average vector field (AVF) method and the midpoint rule exactly preserve the energy (Hamiltonian) and the quadratic invariants such as the momentum and norm of the ALE
[16]. The linearly damped NLSE and the associated damped ALE are linearly perturbed noncanonical Hamiltonian systems, i.e., conservative systems with added linear damping. Discrete gradient methods have been also used in dissipative settings. In order to preserve the dissipation of energy, momentum, and norm with the correct rate, the damped ALE is integrated in time with exponential midpoint rule [30].Model order reduction (MOR) allows to construct lowdimensional reducedorder models (ROMs) for the high dimensional fullorder models (FOMs). The fullorder solutions are projected on low dimensional reduced spaces with the proper orthogonal decomposition (POD) [8, 37], which is a widely used ROM technique. Applying the POD with the Galerkin projection, the dominant POD modes are extracted from the snapshots of the fullorder solutions. We refer to the recent book [7] for an overview of the available MOR techniques. Conservation of nonlinear invariants like the energy is not, in general, guaranteed with conventional ROM. The violation of such invariants often results in a qualitatively wrong or unstable reduced system, even when the highfidelity system is stable. It is well known that conservation law plays an important role in conservative PDEs such as the NLSE with quasiperiodic solutions and solitons. The stability of the ROMs over longtime integration has been investigated in the context of Lagrangian systems [13], and for portHamiltonian systems [14]
. For canonical Hamiltonian systems, like the linear wave equation, SineGordon equation, NLSE, singular value decomposition (SVD) based symplectic model reduction techniques with POD Galerkin projection are constructed with orthogonal
[35], and nonorthogonal bases [10] capturing the symplectic structure of Hamiltonian systems in order to ensure long term stability of the reduced model. For parametric Hamiltonian systems, symplectic bases are generated using greedy approaches in [3, 12]. Parallel to these, energy (Hamiltonian) preserving ROMs have been developed for noncanonical Hamiltonian PDEs like the Kortewegde Vries (KdV) equation [20, 29, 39] and NLSE [25], and with statedependent Poisson structure such as rotating shallow water equations [26, 27]. These are all global ROM techniques, that maintain the globalized properties such as the symplectic structure or the Hamiltonian of the full order data in the reduced order representation do not possess a global lowrank structure. These approaches can provide robust and efficient reduced models, but they require a sufficiently large approximation space to achieve accurate solutions. This is due the fact that in Hamiltonian PDEs, nondissipative phenomena do not possess a global lowrank structure. Hence, local reduced spaces seem to be more effective for this systems. Recently, localized ROM have been developed for Hamiltonian PDEs, which are more effective than the global POD based ROMs. A reduced basis method is developed in [22] for noncanonical Hamiltonian systems that preserves general Poisson structure by ”freezing” the phase space manifold structure in each discrete temporal interval, then recasts the local problem in canonical form. In this way, a local reduced model is constructed in canonical Hamiltonian form. In [23], a rankadaptive structurepreserving dynamical reducedbased method is constructed relying on a residual error estimator. The FOM is approximated on local reduced spaces that are adapted in time using dynamical lowrank approximation techniques. In
[32], nonlinear structurepreserving model reduction is proposed where the reduced phase space evolves in time. The reduced system is obtained by a symplectic projection of the Hamiltonian vector field onto the tangent space of the approximation manifold at each reduced state as in dynamical lowrank approximation. Also for dissipative Hamiltonian systems, structurepreserving ROMs have been developed in [2, 34]. We refer to [21] for an overview about the structurepreserving ROMs for Hamiltonian systems.In this paper, we develop ROMs for the conservative and damped ALEs while preserving its physical properties like conservation/dissipation of the Hamiltonian and the momentum which is a quadratic invariant. We construct energy preserving ROMs for conservative ALE with statedependent Poisson matrix following the approach in [20, 29] where the authors construct an energy preserving ROM for the KdV equation in noncanonical Hamiltonian form with constant skewsymmetric matrix and by the AVF or midpoint method. Following the same approach with an additional linear damping term, a ROM is constructed that preserves the dissipation of the energy and the momentum of the dissipative ALE. An important feature of the ROMs is the offlineonline decomposition. The computation of the FOM and the construction of the POD basis are performed in the offline stage, whereas the reduced system is solved by projecting the problem onto the lowdimensional reduced space in the online stage. The nonlinear term in the Poisson matrix is approximated with the hyperreduction technique, i.e., discrete empirical interpolation method (DEIM) [15, 17], so that offline and online stages are separated. For dynamical systems such as the ALEs with wavetype solutions, a relatively large number of POD modes are needed to represent the physical behavior of the system in reducedorder form. Therefore, the reducedorder system should be solved efficiently. Here, the quadratic reduced system is solved utilizing tensor techniques [5, 6, 26] by the use of MULTIPROD [28] in order to speed up online computations.
Construction of the efficient structurepreserving ROM for noncanonical Hamiltonian systems with a statedependent Poisson matrix, such as the conservative and dissipative ALEs, is challenging. The main contributions of the paper can be summarized below:

Considering the conservative ALE equation in skewgradient form and applying the energy preserving midpoint method, the reduced Hamiltonian and momentum are preserved, which guarantees the longterm stability of the solutions.

Dissipation of the reduced Hamiltonian and momentum of the linearly perturbed ALE is preserved with the correct rate by the ROMs by integrating in time with the exponential midpoint method.

Approximation of the nonlinear terms with DEIM enables offlineonline decomposition. The solutions of the resulting linearquadratic system are accelerated by applying tensor techniques.

The soliton solutions of the conservative and dissipative ALEs are captured accurately by the ROMs in long time integration.
The paper is organized as follows. In Section 2, the high fidelity discretization that preserves the conservation/dissipation property of the conservative/damped ALEs is presented. In Section 3, the structurepreserving ROMs with POD and DEIM are described. Two numerical tests in Section 4 demonstrate the structurepreserving properties of the ROMs. The paper ends with some conclusions in Section 5.
2 Discretization of the conservative and dissipative AblowitzLadik equations
The NLSE is a wellknown nonlinear partial differential equation (PDE) with a broad spectrum of applications, ranging from wave propagation in nonlinear media to nonlinear optics, molecular biology, quantum physics, quantum chemistry, and plasma physics. We consider the NLSE described by
(1) 
under periodic boundary conditions, , for with a final time , and with a prescribed initial condition, for , where denotes the complexvalued wave function. The NLSE (1) is completely integrable for , i.e., there exists infinitely many integrals such as energy, momentum and norm [40, 38].
ALE [1] represents an integrable Hamiltonian semidiscretization of the NLSE (1) [24, 36, 38]
(2) 
with . The solutions of the equation (2) converge to the solutions of the NLSE (1) when the stepsize . Under unitary time dependent transformation , the ALE (2) can be written as a noncanonical Hamiltonian system [24, 38]
(3) 
Separating the real and complex parts as , the equation (3) yields the coupled system
(4)  
Setting and , the ALE (4) is given in matrixvector form as the following noncanonical Hamiltonian system
(5) 
with the quadratic Hamiltonian , and the matrix with nonlinear terms are given by
(6) 
Under the above setting, the gradients of the Hamiltonian reduce to
(7) 
Besides the Hamiltonian, there exist quadratic invariants, i.e., Casimirs, such as the momentum whose discrete form is given by
(8) 
For the solution vector , the ALE (5) is a dimensional skewgradient ODE of the form
(9) 
where the skewsymmetric matrix is given by
Being a discrete gradient integrator, the AVF method given below, preserves the Hamiltonian (6) and the quadratic invariants such as the momentum (8), of the ALE (5) [16]
(10) 
The AVF method is equivalent to the midpoint rule for quadratic Hamiltonians such as the ALE (5)
(11) 
The damped NLSE
(12) 
with the damping factor , describes resonant phenomena in nonlinear media, the nonlinear Faraday resonance in a vertically oscillating water and the effect of phasesensitive amplifiers on solitons in optical fibers [19]. The AblowitzLadik discretization of the damped NLSE (12) yields
(13)  
which represens a linearly perturbed noncanonical Hamiltonian system [30]. In matrixvector form, the system (13) is a skewgradient system with a damping term
(14) 
The energy or the Hamiltonian and the momentum dissipate like
Dissipation of the energy is expressed in form of the energy balance equation [30] given by
(15) 
Similarly, the dissipation of the momentum (8) is given as [30]
(16) 
The AVF method and the midpoint rule do not guarantee the correct rate of energy dissipation, i.e., the energy may be over or under damped [30]. The exponential midpoint rule preserves the correct dissipation rate of the energy and dissipative dynamics of the quadratic Casimirs such as the momentum and norm [30]. Approximation of the gradient of the Hamiltonian with
(17) 
yields the exponential midpoint rule [9, 30]
(18) 
where . The exponential midpoint rule (18) preserves the dissipation rate of the invariants of the damped ALE (13)
3 Reducedorder modelling
The ROM for the damped ALE differs only by adding the linear damping term, therefore we describe the construction of the ROMs for the conservative ALE (5) whose compact form is given by (9).
3.1 POD reduced skewgradient system
The lowdimensional ROMs are constructed by projecting the fullorder system onto a low dimensional reduced space spanned by POD basis. The computation of the POD basis modes rely on a set of discrete solutions of the FOMs. To this end, let and denote the matrix of solution snapshots given by
where , , are the fully discrete solution vectors of the fullorder ALE (5) through the midpoint rule (11
). Then, the POD basis modes are taken as the left singular vectors related to the most dominant singular values from the singular value decomposition (SVD) of the snapshot matrices
andwhere for , and are orthonormal matrices whose column vectors are the left and right singular vectors, respectively, and is the diagonal matrix containing the singular values . For some positive integer , let denotes the truncated matrix of POD modes consisting of the first left singular vectors from . For easy notation, we take the same number of POD modes for each component , but it can be taken different number of POD modes for either component and . The POD matrix is the minimizer of the least squares error
where is the Frobenius norm. Once the POD modes are obtained, an approximation to the fullorder solutions of (5) from the reduced space spanned by the POD modes, can be written as
(19) 
where are the coefficient vectors. The coefficient vectors are the solution of the dimensional reduced system
(20) 
which is constructed by the Galerkin projection of the FOM (5) (or (9)) onto the reduced space. In the ROM (20), is the vector of coefficients, is the vector of reduced approximations, and the block diagonal matrix contains the matrix of POD modes for each state variable
On the other hand, the reduced system (20) is not a skewgradient system like the FOM (9). To recover a reduced skewgradient system from the reduced system (20), we formally insert between and [20, 29]. This results in the POD reduced skewgradient system
(21) 
where is the reduced skewsymmetric matrix and is the reduced discrete gradient of the Hamiltonian. Then, the skewgradient structure of the reduced system (21) yields
which means that the Hamiltonian is preserved by the POD reduced skewgradient system (21).
The approach that preserves the skewsymmetry of the reduced system (21) also preserves the energy dissipation property for dissipative systems such as the damped ALE [20]. Both the reduced systems are solved by the same time integrators as for the related FOMs, i.e., the POD reduced ALE (21) is solved in time with the implicit midpoint rule (11), whereas the POD reduced system of the dissipative ALE (14) is solved with the exponential midpoint rule (18).
3.2 PODDEIM reduced skewgradient system
The explicit form of the POD reduced skewgradient system (21) reads as
(22) 
where is given as in (6), i.e., with , . Using the diagonal structure of the matrix , setting the nonlinear vector , and inserting the discrete gradients of the Hamiltonian in (7), i.e., and , the system (22) can be rewritten in the following explicit form
(23)  
with denoting the elementwise product of vectors. The POD reduced system (23) still depends on the dimension of the FOM, due to the nonlinear vector function . This can be circumvent applying the DEIM [17, 15]. The idea is to interpolate the nonlinear vector using only entries. For this goal, one needs to compute the interpolation basis and an operator which selects the interpolation points and calculate the empirical basis. Let denotes the dimensional interpolation basis. The interpolation basis is constructed by applying the POD to the snapshot matrix of the nonlinear vector, given by
(24) 
where denotes the nonlinear vector computed by the fully discrete solutions of the FOM (5) at . In other words, the DEIM basis modes are determined as the left singular vectors related to the first largest singular values from the SVD of the snapshot matrix . Once the DEIM basis matrix is obtained, the nonlinearity can be approximated as
(25) 
where is the vector of timedependent coefficients to be determined. In order to uniquely solve for the coefficient vector , the overdetermined system (25) needs to be projected by multiplication from left by a matrix, say , so that the product is invertible. The matrix is indeed a selection matrix which is determined by a greedy algorithm based on the system residual [15]. Alternatively, the QDEIM [17] uses a different selection criteria for the sampling points than the original DEIM algorithm [15]. The QDEIM leads to better accuracy and stability properties of the computed selection matrix using the pivoted QRfactorization of . In the sequel, we use QDEIM for the calculation of the selection matrix , see Algorithm 1.
After computation of the selection matrix , the coefficient vector is uniquely determined by solving the projected linear system , by which the approximation (25) becomes
(26) 
where the matrix is a constant matrix which can be precomputed in the offline stage. The reduced nonlinear vector , is computed in the online stage with the reduced dimension . In fact, the reduced nonlinear vector is nothing but selected entries among the entries of the nonlinear vector .
Inserting the DEIM approximation into the reduced system (23), we obtain the PODDEIM reduced skewgradient system
(27)  
3.3 The reduced quadratic system
The system (27) contains the precomputable constant matrices to be computed in the offline stage, and the reduced terms to be computed in the online stage, but they are still not separated. The precomputable constant matrices and the reduced terms can be separated by the use of tensor techniques, so that the solution of the PODDEIM reduced skewgradient system (27) is accelerated. We first rewrite the reduced system (27) in which Kronecker product takes place of componentwise product . This can be handled by using a selection matrix satisfying the identity for any vectors . The matrix is generally called as a matricized tensor. Then, in terms of Kronecker product, the reduced system (27) yields
(28)  
Using the properties of Kronecker product, the system (28) further reduces to
(29)  
where it needs only the computation of the nonlinear vectors and in the online stage, and the terms and are constant matrices to be computed in the offline stage. By DEIM approximation with tensor setting, online computations scale with , whereas it scales with if DEIM approximation is not used.
Apart from the computational efficiency in the online stage, we also follow a computationally efficient approach in the offline stage for the calculation of the constant matrices and . Let us consider the calculation of the constant matrix . Since it scales with the dimension of FOM, the explicit calculation of the matrix is inefficient. Using the structure of , the matrix is given in MATLAB notation without constructing the matricized tensor explicitly by
(30) 
However, the computation in (30) needs for loops in which a matrix product is done. This drawback can be overcome by the use of MULTIPROD [28] which handles multiple multiplications of the multidimensional arrays via virtual array expansion. More clearly, by the properties of Kronecker product, the th row of the matrix in (30) is given equivalently by
(31) 
For each , all the operations in (31) can be done at once by MULTIPROD as follows: we reshape the twodimensional array (matrix) as a threedimensional array , and then we compute MULTIPROD of and in nd and th dimensions, which results in the threedimensional array
(32) 
where the tensor collects all the matrix product of two matrices of sizes and within iterations. Finally, the required matrix is obtained by reshaping the tensor in (32) into a twodimensional array of dimension . Hence, utilizing the MULTIPROD, all the matrix products are done simultaneously in a single loop, which decreases the computational cost in the offline stage.
4 Numerical results
In this section, we illustrate the energy and momentum preserving properties for the conservative ALE and preservation of the dissipative structure of the damped ALE. The POD and DEIM basis are truncated according to the following relative cumulative energy criterion
(33) 
where is a userspecified tolerance, and or .
The accuracy of the ROMs is measured by the time averaged relative norm errors between FOM and ROM solutions
(34) 
Conservation of the discrete invariants such as the energy (6) and momentum (8) of the conservative ALE (2) are measured using the timeaveraged relative errors for the FOM and ROM
(35) 
Similarly the dissipation rate of the discrete invariants such as the energy (15) and momentum (16) of the damped ALE (13) are measured using the timeaveraged relative errors for the FOM and ROM
(36) 
4.1 Conservative ALE
We consider the ALE (2) with soliton solution for . The initial data is taken as [36]
where , with , , , and . The time step is set as for . The snapshot matrices are computed by saving the full discrete solutions and at every five time steps, and they are of size . Figure 1 shows the normalized singular values, where normalized means that the first normalized singular value is one. The singular values of the snapshots corresponding to the state variables and to the nonlinear term decay slowly. The slow decay of the singular values is the characteristic for the problems with complex wave and transport phenomena [4, 31]. The rate of the decay of singular values is related to the Kolmogorov width which is a classical concept of nonlinear approximation theory as it describes the error arising from a projection onto the bestpossible space of a given dimension . It determines the linear reducibility of the underlying systems, which can be connected to the POD spectrum [33], therefore the selection of optimal number of POD/DEIM modes is important. By the use of much larger number of POD and DEIM modes, the conserved quantities in the reduced form are preserved, resulting accurate and stable solutions in longterm integration In our simulations, we set the tolerances and in (33), giving POD and DEIM modes. The relative FOMROM error (34) is 9.55e03, whereas, the relative energy and momentum errors are 1.90e04 and 2.29e04, respectively.
In Figure 2 the Hamiltonian and momentum are preserved over the time without a drift, which ensures the stability of the solitons. In Table 1, the relative FOMROM errors (34) and timeaveraged errors 35 of the invariants do not decrease much with increasing number of POD/DEIM modes. Therefore, the ROM solutions in Figure 3 accurately capture the fullorder solutions.
POD/DEIM Modes  

20  1.06e02  2.95e04  2.95e04 
30  1.56e03  4.82e06  3.50e06 
40  1.73e03  3.06e07  3.29e06 
50  1.79e03  2.21e07  5.70e06 
4.2 Damped ALE
We consider the damped NLSE (12) with one soliton solution with and [19]. The solutions are computed with a time step in the time interval , and on the spatial domain with the mesh size . The resulting ALE is of size . The Initial condition is taken as
with the initial phase . The full order solutions are saved at every five time steps, for which the snapshot matrices are of size .
In Figure 4, singular values of the snapshots for the damped ALE show a similar decay behavior as for the conservative ALE. The number of the POD and DEIM modes are calculated as and , corresponding to the tolerances and , respectively. In order to fulfill the energy and momentum balance in Figures 56, the tolerances are taken as one order smaller than for the conservative ALE. The relative FOMROM error (34) is 7.22e03.
Setting the tolerances and in (33), results in and POD and DEIM modes, respectively. The relative FOMROM error (34) is 7.09e03, whereas the relative energy and momentum balance errors (36) are 1.3804 and 5.24e04, respectively. The solution error and residuals of the energy/momentum balances are saturated around 50 POD/DEIM modes in Table 2, similar to the conservative ALE in Figure 1. The FOM/ROM solutions in Figure 7 are almost identical, and the energy and momentum dissipate with correct rates in Figures 56.
POD/DEIM Modes  

20  2.24e01  9.64e02  1.95e01 
30  3.89e02  3.19e03  8.55e03 
40  7.07e03  1.38e04  5.25e04 
50  1.70e03  5.39e05  1.65e04 
60  7.23e04  5.04e05  1.51e04 
5 Conclusions
In this paper, structurepreserving ROMs are constructed for the conservative and dissipative ALEs. The reduced system can be identified by accurately preserved reduced energy and momentum for the conservative and dissipative ALEs in long time integration, that mimics those of the highfidelity system and ensures the stability and robustness of the reducedorder soliton solutions. Relatively large number of POD and DEIM modes show the limitation of the linear MOR techniques such as POD and DEIM for problems associated with transport and wave type phenomena as in this paper. The nonlinear model order reduction on manifolds [11] or kernel methods [18] can produce accurate solutions in low dimensional reduced spaces, which will be subject of future research.
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