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Structure-preserving model order reduction of Hamiltonian systems

We discuss the recent developments of projection-based model order reduction (MOR) techniques targeting Hamiltonian problems. Hamilton's principle completely characterizes many high-dimensional models in mathematical physics, resulting in rich geometric structures, with examples in fluid dynamics, quantum mechanics, optical systems, and epidemiological models. MOR reduces the computational burden associated with the approximation of complex systems by introducing low-dimensional surrogate models, enabling efficient multi-query numerical simulations. However, standard reduction approaches do not guarantee the conservation of the delicate dynamics of Hamiltonian problems, resulting in reduced models plagued by instability or accuracy loss over time. By approaching the reduction process from the geometric perspective of symplectic manifolds, the resulting reduced models inherit stability and conservation properties of the high-dimensional formulations. We first introduce the general principles of symplectic geometry, including symplectic vector spaces, Darboux' theorem, and Hamiltonian vector fields. These notions are then used as a starting point to develop different structure-preserving reduced basis (RB) algorithms, including SVD-based approaches and greedy techniques. We conclude the review by addressing the reduction of problems that are not linearly reducible or in a non-canonical Hamiltonian form.

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07/26/2020

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1 Introduction

The discretization of partial differential equations (PDEs) by classical methods like finite element, spectral method, or finite volume leads to dynamical models with very large state-space dimensions, typically of the order of millions of degrees of freedom to obtain an accurate solution. MOR

[1] is an effective method for reducing the complexity of such models while capturing the essential features of the system state. Starting from the Truncated Balanced Realization, introduced by Moore [2] in 1981, several other reduction techniques have been developed and flourished during the last 40 years, including the Hankel-norm reduction [3], the proper orthogonal decomposition (POD) [4] and the Padé-via-Lanczos (PVL) algorithm [5]. More recently, there has been a focus on the physical interpretability of the reduced models. Failure to preserve structures, invariants, and intrinsic properties of the approximate model, besides raising questions about the validity of the reduced models, has been associated with instabilities and exponential error growth, independently of the theoretical accuracy of the reduced solution space. Stable reduced models have been recovered by enforcing constraints on the reduced dynamics obtained using standard reduction tools. Equality and inequality constraints have been considered to control the amplitude of the POD modes [6], the fluid temperature in a combustor [7], and the aerodynamic coefficients [8]. Other methods directly incorporate the quantity of interest into the reduced system, producing inf-sup stable [9], flux-preserving [10]

, and skew-symmetric

[11] conservative reduced dynamics. Even though great effort has been spent developing time integrators that preserve the symplectic flow underlying Hamiltonian systems, interest in geometric model order reduction initiated more recently, with efforts to preserve the Lagrangian structures [12].
The remainder of the paper is organized as follows. In Section 2, we present the structure characterizing the dynamics of Hamiltonian systems and the concept of symplectic transformations. In Section 3, we show that linear symplectic maps can be used to guarantee that the reduced models inherit the geometric formulation from the full dynamics. Different strategies to generate such maps are investigated in Section 4, with thoughts on optimality results and computational complexities. A novel approach deviating from the linearity of the projection map is briefly discussed in Section 5. Finally, we discuss applications of structure-preserving reduction techniques to two more general classes of problems in Section 6, and some concluding remarks are offered in Section 7.

2 Symplectic geometry and Hamiltonian systems

Let us first establish some definitions and properties concerning symplectic vector spaces.

Definition 2.1.

Let be a finite-dimensional real vector space and a bilinear map. is called anti-symmetric if

It is non-degenerate if

Definition 2.2.

Let be a finite-dimensional vector space with an anti-symmetric bilinear form on . The pair is a symplectic linear vector space if is non-degenerate. Moreover, has to be -dimensional.

Since we are interested in structure-preserving transformations, preserving the structure means to preserve the anti-symmetric bilinear form, as stated in the following definition.

Definition 2.3.

Let and be two symplectic vector spaces with . The differentiable map is called a symplectic transformation (symplectomorphism) if

where is the pull-back of with .

One of the essential properties of Euclidean spaces is that all the Euclidean spaces of equal dimensions are isomorphic. For the symplectic vector spaces, a similar result holds, since two -dimensional symplectic vector spaces are symplectomorphic to one another. They therefore are fully characterized by their dimensions (As a consequence of the following theorem).

Theorem 2.1 (Linear Darboux’ theorem [13]).

For any symplectic vector space , there exists a basis of such that

(1)

The basis is called Darboux’ chart or canonical basis.

The proof of Theorem 2.1

is based on a procedure similar to the Gram-Schmidt process to generate the symplectic basis, known as symplectic Gram-Schmidt

[14].
The canonical basis allows representing the symplectic form as

(2)

where are the expansion coefficients of with respect to the basis and

(3)

is known as the Poisson tensor, with

and denoting the zero and identity matrices, respectively. As a direct result, the matrix representation of the symplectic form in the canonical basis is . More generally, using a non-canonical basis, the form reduces to , with being an invertible constant skew-symmetric matrix.
While symplectic vector spaces are helpful for the analysis of dynamical problems in Euclidean spaces and to define geometric reduced-order models, the constraint to the Euclidean setting is not generally adequate. In particular, the abstraction of the phase spaces of classical mechanics over arbitrary manifolds requires the definition of more general symplectic manifolds. We refer the reader to [15] for a more comprehensive description of the topic. In this work, we limit ourselves to introducing a significant result regarding the evolution of the state of Hamiltonian systems.

Definition 2.4.

Let be a symplectic manifold and a -form. We refer to the unique vector field , which satisfies

as the Hamiltonian vector field related to , where denotes the contraction operator and is the exterior derivative. The function is called the Hamiltonian of the vector field .

Suppose is also compact, then is complete [16] and can be integrated, i.e., there exists an integral curve of , parametrized by the real variable , that is the solution of

(4)

Equation (4) is referred to as Hamilton’s equation of evolution or Hamiltonian system. Darboux’ theorem, as a generalization of Theorem 2.1, states that two symplectic manifolds are only locally symplectomorphic. Using this result, the Hamiltonian vector field admits the local representation

(5)

with is a local basis, leading to the following representation of (4), expressed directly in terms of .

Proposition 2.1.

Let be a -dimensional symplectic vector space and let be a canonical system of coordinates. Hamilton’s equation is defined by

(6)

for , which is a first order system in the -space, or generalized phase-space.

Thus, if the state vector is introduced, (6) takes the form

(7)

where is the naive gradient of . The flow of Hamilton’s equation has some interesting properties.

Proposition 2.2.

Let be the flow of a Hamiltonian vector field . Then is a symplectic transformation.

We rely on a geometric perspective of linear vector spaces to highlight the importance of Proposition 2.2. Given two coefficient vectors and in , the symplectic form (2) can be interpreted as the sum of the oriented areas of the orthogonal projection of the parallelogram defined by the two vectors on the planes. Definition 2.3, in case of -dimensional symplectic vector space with canonical coordinates, is equivalent to stating that a map is a symplectic transformation if and only if its Jacobian satisfies everywhere

(8)

Property (8) can be used to show that a symplectic transformation preserves the bilinear form in the sense that [15]

(9)

Hence, the symplectic map represents a volume-preserving transformation. However, being symplectic is a more restrictive condition than being volume-preserving, as shown in the Non-squeezing theorem [17].
We conclude this section by noting that if the Hamiltonian function does not depend explicitly on time, its value is conserved along the solution trajectory.

Proposition 2.3.

For Hamiltonian systems (7), the Hamiltonian function is a first integral.

3 Symplectic Galerkin projection

The motivation of MOR is to reduce the computational complexity of dynamical systems in numerical simulations. In the context of structure-preserving projection-based reduction, two key ingredients are required to define a reduced model. First, we need a low-dimensional symplectic vector space that accurately represents the solution manifold of the original problem. Then, we have to define a projection operator to map the symplectic flow of the Hamiltonian system onto the reduced space, while preserving its delicate properties.
Let us assume there exists a canonical basis such that Hamilton’s equation can be written in canonical form

(10)

and the related symplectic vector space is denoted by . Symplectic projection-based model order reduction adheres to the key idea of more general projection-based techniques [18] to approximate in a low-dimensional symplectic subspace of dimension . In particular, we aim at to have a clear reduction, and therefore, significant gains in terms of computational efficiency. Let be a reduced basis for the approximate symplectic subspace and construct the linear map given by

(11)

where

belongs to the set of symplectic matrices of dimension , also known as the symplectic Stiefel manifold, defined by

Differential maps are often used to transfer structures from well-defined spaces to unknown manifolds. In this context, using the symplecticity of , it is possible to show [22] that Definition 2.3 holds, with the right inverse of represented by , and that there exists a symplectic form on given by

(12)

As a result, is a symplectic vector space. In the following, for the sake of notation, we use to indicate the reduced symplectic manifold paired with its bilinear form.
Given a symplectic matrix , its symplectic inverse is defined as

(13)

Even though different from the pseudo-inverse matrix , the symplectic inverse plays a similar role and, in the following proposition, we outline its main properties [21].

Proposition 3.1.

Suppose is a symplectic matrix and is its symplectic inverse. Then

  • .

  • .

  • .

  • If A is orthogonal then .

Using (12), the definition of and the symplectic Gram-Schmidt process, it is possible to construct a projection operator , that, differently from the POD orthogonal projection [19], can be used to approximate (10) with Hamiltonian system of reduced-dimension , characterized by the Hamiltonian function

(14)

In particular, in the framework of Galerkin projection, using (11) in (10) yields

(15)

with

being the residual term. Utilizing the chain rule and the second property of

in Proposition 3.1, the gradient of the Hamiltonian in (15) can be recast as

By assuming that the projection residual is orthogonal with respect to the symplectic bilinear form to the space spanned by , we recover

(16)

System (16) is known as a symplectic Galerkin projection of (10) onto . The pre-processing stage consisting of the collection of all the computations required to assemble the basis is known as the offline stage. The numerical solution of the low-dimensional problem (16) represents the online stage, and provides a fast approximation to the solution of the high-fidelity model (10) by means of (11). Even though the offline stage is possibly computationally expensive, this splitting is beneficial in a multi-query context, when multiple instances of (16) have to be solved, e.g. for parametric PDEs.
Traditional projection-based reduction techniques do not guarantee stability, even if the high-dimensional problem admits a stable solution [20], often resulting in a blowup of system energy. On the contrary, by preserving the geometric structure of the problem, several stability results hold for the reduced Hamiltonian equation (16). In [22, Proposition 15, page A2625], the authors show that the error in the Hamiltonian is constant for all . We detail two relevant results in the following, suggesting that structure and energy preservation are key for stability.

Theorem 3.1 (Boundedness result [21]).

Consider the Hamiltonian system (10), with Hamiltonian and initial condition such that , with symplectic basis. Let (16) be the reduced Hamiltonian system obtained as the symplectic Galerkin projection induced by of (10). If there exists a bounded neighborhood in such that , or , for all on the boundary of , then both the original system and the reduced system constructed by the symplectic projection are uniformly bounded for all .

Theorem 3.2 (Lyapunov stability [21, 22]).

Consider the Hamiltonian system (10) with Hamiltonian and the reduced Hamiltonian system (16). Suppose that is a strict local minimum of . Let be an open ball around such that and , for all and some , and for some , where is the boundary of . If there exists an open neighborhood of such that , then the reduced system (16) has a stable equilibrium point in .

For the time-discretization of (16), the use of a symplectic integrator [23] is crucial for preserving the symplectic structure at the discrete level. In particular, the discrete flow obtained using a symplectic integrator satisfies a discrete version of Proposition 2.2.
In the next section, we introduce different strategies to construct symplectic bases as results of optimization problems.

4 Proper symplectic decomposition

Let us consider the solution vectors (the so-called solution snapshots) obtained, for different time instances , , by time discretization of (10) using a symplectic integrator. Define the snapshot matrix

(17)

as the matrix collecting the solution snapshots as columns. In the following, we consider different algorithms stemming from the historical method of snapshots [4], as the base of the proper orthogonal decomposition (POD). To preserve the geometric structure of the original model, we focus on a similar optimization problem, the proper symplectic decomposition (PSD), which represents a data-driven basis generation procedure to extract a symplectic basis from . It is based on the minimization of the projection error of on and it results in the following optimization problem for the definition of the symplectic basis :

(18)

with and is the Frobenius norm. Problem (18) is similar to the POD minimization, but with the feasibility set of rectangular orthogonal matrices, also known as the Stiefel manifold

replaced by the symplectic Stiefel manifold. Recently there has been a great interest in optimization on symplectic manifolds, and a vast literature is available on the minimization of the least-squares distance from optimal symplectic Stiefel manifolds. This problem has relevant implications in different physical applications, such as the study of optical systems [24] and the optimal control of quantum symplectic gates [25]. Unfortunately, with respect to POD minimization, problem (18) is significantly more challenging for different reasons. The non-convexity of the feasibility set and the unboundedness of the solution norm precludes standard optimization techniques. Moreover, most of the attention is focused on the case , which is not compatible with the reduction goal of MOR.
Despite the interest in the topic, an efficient optimal solution algorithm has yet to be found for the PSD. Suboptimal solutions have been attained by focusing on the subset of the ortho-symplectic matrices, i.e.,

(19)

In [21], while enforcing the additional orthogonality constraint in (18), the optimization problem is further simplified by assuming a specific structure for . An efficient greedy method, not requiring any additional block structures to , but only its orthogonality and simplecticity, has been introduced in [22]. More recently, in [26], the orthogonality requirement has been removed, and different solution methods to the PSD problem are explored. In the following, we briefly review the abovementioned approaches.

4.1 SVD-based methods for orthonormal symplectic basis generation

In [21], several algorithms have been proposed to directly construct ortho-symplectic bases. Exploiting the SVD decomposition of rearranged snapshots matrices, the idea is to search for optimal matrices in subsets of . Consider the more restrictive feasibility set

Then holds if and only if , i.e. . Moreover, we have that . The cost function in (18) becomes

(20)

with , where and are the generalized phase-space components of . Thus, as a result of the Eckart–Young–Mirsky theorem, (20

) admits a solution in terms of the singular-value decomposition of the data matrix

. This algorithm, formally known as Cotangent Lift, owes its name to the interpretation of the solution to (20) in as the cotangent lift of linear mappings, represented by and , between vector spaces of dimensions and . Moreover, this approach constitutes the natural outlet in the field of Hamiltonian systems of the preliminary work of Lall et al. [12] on structure-preserving reduction of Lagrangian systems. However, there is no guarantee that the Cotangent Lift basis is close to the optimal of the original PSD functional.
A different strategy, known as Complex SVD decomposition, relies on the definition of the complex snapshot matrix , with being the imaginary unit. Let , with

, be the unitary matrix solution to the following accessory problem:

(21)

As for the Cotangent Lift algorithm, the solution to (21) is known to be the set of the left-singular vectors of

corresponding to its largest singular values. In terms of the real and imaginary parts of

, the orthogonality constraint implies

(22)

Consider the ortho-symplectic matrix, introduced in [21], and given by

(23)

Using (22), it can be shown that such an is the optimal solution of the PSD problem in

that minimizes the projection error of , also known as the rotated snapshot matrix, with given in (17). In [26], extending the result obtained in [27] for square matrices, it has been shown that (23) is a complete characterization of symplectic matrices with orthogonal columns, meaning that all the ortho-symplectic matrices admit a representation of the form (23), for a given , and hence . In the same work, Haasdonk et al. showed that an ortho-symplectic matrix that minimizes the projection error of is also a minimizer of the projection error of the original snapshot matrix , and vice versa. This is been achieved by using an equivalence argument based on the POD applied to the matrix . Thus, combining these two results, the Complex SVD algorithm provides a minimizer of the PSD problem for ortho-symplectic matrices.

4.2 SVD-based methods for non-orthonormal symplectic basis generation

In the previous section, we showed that the basis provided by the Complex SVD method is not only near-optimal in , but is optimal for the cost functionals in the space of ortho-symplectic matrices. The orthogonality of the resulting basis is beneficial [28], among others, for reducing the condition number associated with the fully discrete formulation of (16). A suboptimal solution to the PSD problem not requiring the orthogonality of the feasibility set is proposed in [21], as an improvement of the SVD-based generators of ortho-symplectic bases using the Gappy POD [29], under the name of nonlinear programming approach (NLP). Let be a basis of dimension generated using the Complex SVD method. The idea of the NLP is to construct a target basis , with , via the linear mapping

(24)

with . Using (24) in (18) results in a PSD optimization problem for the coefficient matrix , of significant smaller dimension ( parameters) as compared to the original PSD problem ( parameters) with unknown. However, no optimality results are available for the NLP method.
A different direction has been pursued in [26], based on the connection between traditional SVD and Schur forms and the matrix decompositions, related to symplectic matrices, as proposed in the following theorem.

Theorem 4.1 (SVD-like decomposition [30, Theorem 1, page 6]).

If , then there exists , and of the form

(25)

with , , such that

(26)

Moreover, rank and are known as symplectical singular values.

Let us apply the SVD-like decomposition to the snapshot matrix (17), where represents the number of snapshots, and define its weighted symplectic singular values as

with being the -th column of and the Euclidean norm. The physical interpretation of the classical POD approach characterizes the POD reduced basis as the set of a given cardinality that captures most of the energy of the system. The energy retained in the reduced approximation is quantified as the sum of the squared singular values corresponding to the left singular vectors of the snapshot matrix representing the columns of the basis. A similar guiding principle is used in [26], where the energy of the system, i.e., the Frobenius norm of the snapshot matrix, is connected to the weighted symplectic singular values as

(27)

Let be the set of indices corresponding to the largest energy contributors in (27),

(28)

Then, the PSD SVD-like decomposition defines a symplectic reduced basis by selecting the pairs of columns from the symplectic matrix corresponding to the indices set

(29)

Similarly to the POD, the reconstruction error of the snapshot matrix depends on the magnitude of the discarded weighted symplectic singular values as

(30)

Even though there are no proofs that the PSD SVD-like algorithm reaches the global optimum in the sense of (18), some analysis and numerical investigations suggest that it provides superior results as compared to orthonormal techniques [26].

4.3 Greedy approach to symplectic basis generation

The reduced basis methodology is motivated and applied within the context of real-time and multi-queries simulations of parametrized PDEs. In the framework of Hamiltonian systems, we consider the following parametric form of (10)

(31)

with being a -dimensional parameter space. Let be the set of solutions to (31) defined as

For the sake of simplicity, in the previous sections we have only considered the non-parametric case. The extension of SVD-based methods for basis generations to (31) is straightforward on paper, but it is often computationally problematic in practice as the number of snapshots increases. Similar to other SVD-based algorithms, the methods described in the previous sections require the computation of the solution to (31) corresponding to a properly chosen discrete set of parameters and time instances , defined a priori, and constituting the sampling set . Random or structured strategies exist to define the set , such as the Monte Carlo sampling, Latin hypercube sampling, and sparse grids [31], while is a subset of the time-discretization, usually dictated by the integrator of choice. The set of snapshots corresponding to the sampling set must provide a “good” approximation of the solution manifold and should not miss relevant parts of the time-parameter domain. Once the sampling set has been fixed, the matrix , , or , depending on the method of choice, is assembled, and its singular value decomposition is computed. Even though a certain amount of computational complexity is tolerated in the offline stage to obtain a significant speed-up in the online stage, the evaluation of the high-fidelity solution for a large sampling set and the SVD of the corresponding snapshot matrix are often impractical or not even feasible. Hence, an efficient approach is an incremental procedure. The reduced basis, in which the column space represents the approximating manifold, is improved iteratively by adding basis vectors as columns. The candidate basis vector is chosen as the maximizer of a much cheaper optimization problem. This summarizes the philosophy of the greedy strategy applied to RB methods [33, 34], which requires two main ingredients: the definition of an error indicator and a process to add a candidate column vector to the basis.
Let be an orthonormal reduced basis produced after steps of the algorithm. In its idealized form, introduced in [32], the greedy algorithm uses the projection error

(32)

to identify the snapshot that is worst approximated by the column space of over the entire sampling set . Let be the vector obtained by orthonormalizing with respect to . Then the basis is updated as . To avoid the accumulation of rounding errors, it is preferable to utilize backward stable orthogonalization processes, such as the modified Gram-Schmidt orthogonalization. The algorithm terminates when the basis reaches the desired dimension, or the error (32) is below a certain tolerance. In this sense, the basis is hierarchical because its column space contains the column space of its previous iterations. This process is referred to as strong greedy

method. Even though introduced as a heuristic procedure, interesting results regarding algebraic and exponential convergence have been formulated in

[33, 34], requiring the orthogonality of the basis in the corresponding proofs. However, in this form, the scheme cannot be efficiently implemented: the error indicator (32) is expensive to calculate because it requires all the snapshots of the training set to be accessible, relieving the computation only of the cost required for the SVD.
An adjustment of the strong greedy algorithm, known as weak greedy algorithm, assembles the snapshot matrix corresponding to iteratively while expanding the approximating basis. The idea is to replace (32) with a surrogate indicator that does not demand the computation of the high-fidelity solution for the entire time-parameter domain.
In the case of elliptic PDEs, an a-posteriori residual-based error indicator requiring a polynomial computational cost in the approximation space dimension has been introduced in [35]. The substantial computational savings allow the choice of a more refined, and therefore representative, sampling set . One might also use a goal-oriented indicator as the driving selection in the greedy process to obtain similar computational benefits. In this direction, in the framework of structure-preserving model order reduction, [22] suggests the Hamiltonian as a proxy error indicator. Suppose , with , is a given ortho-symplectic basis and consider

(33)

By [22, Proposition 15], the error in the Hamiltonian depends only on the initial condition and the symplectic reduced basis. Hence, the indicator (33) does not require integrating in time the full system (31) over the entire set , but only over a small fraction of the parameter set, making the procedure fast. Hence, the parameter space can be explored first,

(34)

to identify the value of the parameter that maximizes the error in the Hamiltonian as a function of the initial condition. This step may fail if , . Then (31) is temporally integrated to collect the snapshot matrix . Finally, the candidate basis vector is selected as the snapshot that maximizes the projection error

(35)

Standard orthogonalization techniques, such as QR methods, fail to preserve the symplectic structure [36]. In [22], the SR method [37], based on the symplectic Gram-Schimidt, is employed to compute the additional basis vector that conforms to the geometric structure of the problem. To conclude the -th iteration of the algorithm, the basis is expanded in

We stress that, with this method, known as symplectic greedy RB, two vectors, and , are added to the symplectic basis at each iteration, because of the structure of ortho-symplectic matrices. A different strategy, known as PSD-Greedy algorithm and partially based on the PSD SVD-like decomposition, has been introduced in [38], with the feature of not using orthogonal techniques to compress the matrix . In [22], following the results given in [33], the exponential convergence of the symplectic strong greedy method has been proved.

Theorem 4.2 ([33, Theorem 20, page A2632]).

Let be a compact subset of . Assume that the Kolmogorov -width of defined as

decays exponentially fast, namely with . Then there exists such that the symplectic basis generated by the symplectic strong greedy algorithm provides exponential approximation properties,

(36)

for all and some .

Theorem 4.2 holds only when the projection error is used as the error indicator instead of the error in the Hamiltonian. However, it has been observed for different symplectic parametric problems [22] that the symplectic method using the loss in the Hamiltonian converges with the same rate of (36). The orthogonality of the basis is used to prove the convergence of the greedy procedure. In the case of a non-orthonormal symplectic basis, supplementary assumptions are required to ensure the convergence of the algorithm.

5 Dynamical low-rank reduced basis methods for Hamiltonian systems

The Kolmogorov m-width of a compact set describes how well this can be approximated by a linear subspace of a fixed dimension . A problem (31) is informally defined reducible if decays sharply with , implying the existence of a low-dimensional representation of . A slow decay limits the accuracy of any efficient projection-based reduction on linear subspaces, including all the methods discussed so far. For Hamiltonian problems, often characterized by the absence of physical dissipation due to the conservation of the Hamiltonian, we may have in case of discontinuous initial condition [39] for wave-like problems. Several techniques, either based on nonlinear transformations of the solution manifold to a reducible framework [40] or presented as online adaptive methods to target solution manifolds at fixed time [41], have been introduced to overcome the limitations of the linear approximating spaces. In different ways, they all abandon the framework of symplectic vector spaces. Therefore, none of them guarantees conservation of the symplectic structure in the reduction process. Musharbash et al. [42] proposed a dynamically orthogonal (DO) discretization of stochastic wave PDEs with a symplectic structure. In the following, we outline the structure-preserving dynamic RB method for parametric Hamiltonian systems, proposed by Pagliantini [44] in the spirit of the geometric reduction introduced in [43]. In contrast with traditional methods that provide a global basis, which is fixed in time, the gist of a dynamic approach is to evolve a local-in-time basis to provide an accurate approximation of the solution to the parametric problem (31). The idea is to exploit the local low-rank nature of Hamiltonian dynamics in the parameter space. From a geometric perspective, the approximate solution evolves according to naturally constrained dynamics, rather than weakly enforcing the required properties, such as orthogonality or symplecticity of the RB representation, via Lagrange multipliers. This result is achieved by viewing the flow of the reduced model as prescribed by a vector field that is everywhere tangent to the desired manifold.
Suppose we are interested in solving (31) for a set of vector-valued parameters , sampled from . Then the Hamiltonian system, evaluated at , can be recast as a set of ODEs in the matrix unknown ,

(37)

where is a vector-valued Hamiltonian function, the -th column of is such that , and . We consider an approximation of the solution to (37) of the form

(38)

where , and is such that its -th column collects coefficients, with respect to the basis , of the approximation of . Despite being cast in the same framework of an RB approach, a stark difference between (38) and (11) lies in the time-dependency of the basis in (38).
Consider the manifold of matrices having at most rank , and defined as

(39)

with the technical requirement

(40)

This represents a full-rank condition on to ensure uniqueness of the representation (38) for a fixed basis. The tangent vector at is given by , where and correspond to the tangent directions for the time-dependent matrices and , respectively. Applying the orthogonality and symplecticity condition on , for all times , results in

(41)

respectively. Using (41) and an additional gauge constraint to uniquely parametrize the tangent vectors by the displacements and , the tangent space of at can be characterized as

The reduced flow describing the evolution of the approximation is derived in [44] by projecting the full velocity field in (37) onto the tanget space of at , i.e.,

(42)

To preserve the geometric structure of the problem, the projection operator is a symplectomorphism (see Definition 2.3) for each realization of the parameter , in the sense given in the following proposition.

Proposition 5.1 ([44, Proposition 4.3, page 420]).

Let . Then, the map

(43)

is a symplectic projection, in the sense that

where is the symplectic form associated with the parameter .

The optimality of the reduced dynamics, in the Frobenius norm, follows from (42), where the flow of is prescribed by the best low-rank approximation of the Hamiltonian velocity field vector into the tangent space of the reduced manifold . Using (43) and (42), it is straightforward to derive the evolution equations for and :

(44)

with .
The coefficients evolve according to a system of independent Hamiltonian equations, each in unknowns, corresponding to the symplectic Galerkin projection onto for each parameter instance in , similarly to the global symplectic RB method (16). In (44), however, the basis evolves in time according to a matrix equation in unknowns, affecting the projection. A crucial property of the structure of is given in the following proposition.

Proposition 5.2 ([44, Proposition 4.5, page 423]).

If then solution of (44) satisfies for all .

Standard numerical integrators, applied to (44), do not preserve, at the time-discrete level, the property in Proposition 5.2 and the ortho-symplectic structure is compromised after a single time step. In [44], two different intrinsic integrators have been investigated to preserve the ortho-symplecticity of the basis, based on Lie groups and tangent techniques. Both methods require the introduction of a local chart defined on the tangent space of the manifold at , with

and being the vector space of skew-symmetric and Hamiltonian real square matrices. In terms of differential manifolds, represents, together with the Lie bracket defined as the matrix commutator , with , the Lie algebra corresponding to the Lie group . The idea is to recast the basis equation in (44) in an evolution equation in the corresponding Lie algebra. The linearity of Lie algebras allows to compute, via explicit Runge-Kutta methods, numerical solutions that remain on the Lie algebra. Finally, the Cayley transform is exploited to generate local coordinate charts and retraction /inverse retraction maps, used to recover the solution in the manifold of rectangular ortho-symplectic matrices. In [45]

, the structure-preserving dynamical RB-method has been paired with a rank-adaptive procedure, based on a residual error estimator, to dynamically update also the dimension of the basis.

6 Extensions to more general Hamiltonian problems

6.1 Dissipative Hamiltonian systems

Many areas of engineering require a more general framework than the one offered by classical Hamiltonian systems, requiring the inclusion of energy-dissipating elements. While the principle of energy conservation is still used to describe the state dynamics, dissipative perturbations must be modelled and introduced in the Hamiltonian formulation (10). Dissipative Hamiltonian systems, with so-called Rayleigh type dissipation, are considered a special case of forced Hamiltonian systems, with the state , with , following the time evolution given by

(45)

where is a velocity field, introducing dissipation, of the form

(46)

We require to satisfy , , to represent a dissipative term and therefore

(47)

In terms of Rayleigh dissipation theory, there exists a symmetric positive semidefinite matrix such that and (47) reads

Several strategies have been proposed to generate stable reduced approximations of (45), based on Krylov subspaces or POD [46, 48]. In [47], without requiring the symplecticity of the reduced basis, the gradient of the Hamiltonian vector field is approximated using a projection matrix , i.e., , which results in a non-canonical symplectic reduced form. The stability of the reduced model is then achieved by preserving the passivity of the original formulation. A drawback of such an approach is that, while viable for nondissipative formulations, it does not guarantee the same energy distribution of (45) between dissipative and null energy contributors. In the following, we show that the techniques based on symplectic geometry introduced in the previous sections can still be used in the dissipative framework described in (45) with limited modifications to obtain consistent and structured reduced models. Let us consider an ortho-symplectic basis and the reduced basis representation , with being the reduced coefficients of the representation and being the generalized phase coordinates of the reduced model. The basis can be represented as

(48)

with being the blocks, the indices of which are chosen to represent the interactions between the generalized phase coordinates of the two models, such that and . Following [49], the symplectic Galerkin projection of (45) reads

(49)

with