Existence of dynamical low rank approximations for random semi-linear evolutionary equations on the maximal interval

An existence result is presented for the dynamical low rank (DLR) approximation for random semi-linear evolutionary equations. The DLR solution approximates the true solution at each time instant by a linear combination of products of deterministic and stochastic basis functions, both of which evolve over time. A key to our proof is to find a suitable equivalent formulation of the original problem. The so-called Dual Dynamically Orthogonal formulation turns out to be convenient. Based on this formulation, the DLR approximation is recast to an abstract Cauchy problem in a suitable linear space, for which existence and uniqueness of the solution in the maximal interval are established.


Introduction
This paper is concerned with the existence of solutions of the so called Dynamical Low Rank Method (DLR) [6,7,16,17,20] to a semi-linear random parabolic evolutionary equation. For a separable R-Hilbert space (H, ·, · ) and a probability space . . , Y S (t)) are linearly independent in the space L 2 (Ω) of square-integrable random variables. We note that both bases depend on the temporal variable t. This dependence is intended to approximate well, with a fixed (possibly small) rank, the solution of stochastic dynamical systems such as (1.1), whose stochastic and spatial dependence may change significantly in time.
Numerical examples and error analysis suggests the method does indeed work well in a certain number of practical applications [17,20]. A fundamental open question regarding this approach is the unique existence of DLR solutions. The DLR approximation is given as a solution of a system of differential equations, and available approximation results are built upon the assumption that this solution exists, e.g. [6,16]. Nonetheless, to the best of our knowledge, the existence-let alone the uniqueness-of DLR solutions for an equation of the type (1.1) is not known. In this paper, we will establish a unique existence result.
A difficulty in proving the existence is the fact that the solution propagates in an infinite-dimensional manifold, and that we have an unbounded operator in the equation. Indeed, the DLR equations are derived so that the aforementioned approximation u S keeps the specified form in time, with the fixed rank S. By now it is well known that the collection of functions of this form admits an infinite-dimensional manifold structure [5,Sect. 3]. Besides the unbounded operator Λ, the resulting system of equations involves also a non-linear projection operator onto the tangent space to the manifold, which makes its analysis difficult and non-standard.
Our strategy is to work with a suitable set of parameters describing the manifold, that are elements of a suitable ambient Hilbert space, and invoke results for the evolutionary equations in linear spaces. In utilising such results, the right choice of parametrisation turns out to be crucial. Our choice of parameters leads us to the so-called Dual DO formulation introduced in [17].
A method similar to the DLR approximation is the multi-configuration timedependent Hartree (MCTDH) method, which has been considered in the context of computational quantum chemistry to approximate a deterministic Schrödinger equation. For the MCTDH method, several existence results have been established, e.g. [2,12,13]. The strategy used in these papers, first proposed by Koch and Lubich [13], is to consider a constraint called the gauge condition that is defined by the differential operator in the equation. With their choice of the gauge condition and their specific setting, the differential operator appears outside the projection operator, and this was a crucial step in [2,12,13] to apply the standard theory of abstract Cauchy problems. However, as we will see later in Sect. 2.4, the same approach does not work in our setting.
As mentioned above, our strategy is to work with the Dual DO formulation, by which we are able to show that the DLR approximation exists as long as a suitable full rank condition is satisfied. Further, we discuss the extendability of the approximation, beyond the point where we lose the full rankness.
The rest of this paper is organised as follows. In Sect. 2, we introduce the problem under study as well as the Dual DO formulation of the DLR equation. Section 3 introduces a parameter-equation that is equivalent to the Dual DO equations. Then, in Sect. 4 we prove our main result, namely the existence and uniqueness of a DLR solution on the maximal interval. The solution evolves in a manifold up to a maximal time. The solution cannot be continued in this manifold, but we will show that it can be extended in the ambient space, and the resulting continuation will take values in a different manifold with lower rank. Section 5 concludes the paper.

DLR formulation
In this section, we introduce the setting and recall some facts on the Dynamical Low Rank (DLR) approach that will be needed later.
We detail in Sect. 2.3 the precise assumptions on Λ, F and the initial conditions we will work with. For the moment, we just assume that a solution of (1.1) exists. We note, however, that the existence and uniqueness can be established by standard arguments. For instance, if Λ is self-adjoint and satisfies −Λx, x ≥ 0 for all x ∈ D H (Λ), by extending the definition of Λ to random functions u ∈ L 2 (Ω; H), where Λ : D(Λ) ⊂ L 2 (Ω; H) → L 2 (Ω; H) is applied pointwise in Ω, we have that Λ is densely defined, closed, and satisfies Together with a local Lipschitz continuity of F, existence of solutions can be established by invoking a standard theory of semi-linear evolution equations, see for example [18,21].
We define an element u S ∈ L 2 (Ω; H) to be an S-rank random field if u S can be expressed as a linear combination of S (and not less than S) linearly independent elements of H, and S (and not less than S) linearly independent elements of L 2 (Ω). Further, we letM S ⊂ L 2 (Ω; H) be the collection of all the S-rank random fields: .
It is known thatM S can be equipped with a differentiable manifold structure, see [5,17]. The idea behind the DLR approach is to approximate the curve t → u(t) ∈ L 2 (Ω; H) defined by the solution of the Eq. (1.1) by a curve t → u S (t) ∈M S given as a solution of the following problem: find u S ∈M S such that u S (0) = u 0S ∈M S , a suitable approximation of u 0 inM S , and for (almost) all t > 0 we have ∂u S ∂t (t) − (Λu S (t) + F(u S (t))) ∈ L 2 (Ω; H) and , and E[·] denotes expectation with respect to the underlying probability measure P.
In this paper, we search for the solution in the same set asM S but with a different parametrisation that is easier to work with. The set is the same subset of L 2 (Ω; H) asM S , and thus the above problem is equivalent when we seek solutions in M S instead ofM S . This leads us to the so-called Dual Dynamically Orthogonal (DO) formulation of the problem (2.1).
where, for an arbitrary H-orthonormal basis This operator P u S turns out to be the L 2 (Ω; H)-orthogonal projection to the tangent space T u S M S at u S = U Y , see [16,Proposition 3.3] together with [4]. Note that P u S is independent of the choice of the representation of u S . Using the above definitions, the problem we consider, equivalent to (2.1), can be formulated as follows: We consider two notions of solutions: the strong and classical solution.

Dual DO formulation
Our aim is to establish the unique existence of a DLR solution. Our strategy is to choose a suitable parametrisation of M S , and work in a linear space which the parameters belong to. For the parametrisation, we will choose the one proposed in [17], which results in a formulation of (2.4) called Dual DO, where we seek an approximate solution of the form u S is a solution to the following problem: , and satisfy the so-called gauge condition: for any t ∈ (0, T ), Noting that, since the operator Λ is deterministic and linear, we have (2.6) We define two notions of solutions to the initial value problem of (2.6) that correspond to those of the original problem as in Definitions 2.1-2.2.

Definition 2.3 (Strong dual DO solution)
is called a strong Dual DO solution if it satisfies the following conditions: Notice, in particular, that the condition 5 above implies that the matrix Z U is invertible for almost every t ∈ [0, T ]. Further, from (2.6) we necessarily have is called a classical Dual DO solution if it satisfies the following conditions:

Equivalence with the original formulation
In this section, we establish the equivalence of the original equation ( Moreover, such σ j > 0 is unique in the following sense: for any other representation with singular values σ j = σ j (K ) > 0, j = 1, . . . , S, see e.g. [9, Sects. III.4.3 and V.2.3]. Observe that, if we have another representation u S = S j=1 σ jṼ j W j , then upon relabelling if necessary we must have σ j = σ j .
To show (2.8), relabel {σ j (t)} S j=1 in the non-decreasing order and denote it by for any w ∈ L 2 (Ω), and thus the continuity of t → u S (t) implies that α j is continuous , which completes the proof.
In particular, u S (t) admits a representation u S = V W in M S with V =Ṽ Σ, with the specified smoothness.
To show Proposition 2.2, we will use an argument similar to what we will see in Sect. 4 below. Thus, we will defer the proof to Sect. 4.
Parametrisation of M S is determined by parameters up to a unique orthogonal matrix.

satisfying the linear independence and orthonormality conditions as in (2.2). Then, we have
From the L 2 (Ω)-orthonormality ofW and W , taking the expectation of both sides we conclude that Θ is an orthogonal matrix. The uniqueness is easy to see.
The above lemma implies the following corollary, which states that if both a DLR solution u S and a Dual DO solution (U, Y ) exist, and if further the DLR solution is unique, then (U, Y ) is determined by u S up to a unique orthogonal matrix. We stress that the next corollary does not guarantee the uniqueness of the Dual DO solution.

Corollary 2.4 Suppose that a strong DLR solution u S
, satisfying the linear independence and orthonormality conditions defined in (2.2). Furthermore, suppose that a Dual DO solution (U(t), Y (t)) exists in the strong sense. Then, we have Then, from the uniqueness of the DLR solution we have Thus, in view of Lemma 2.3 the statement follows.
In Corollary 2.4, we assumed the existence of both the DLR solution and the Dual DO solution, and deduced the existence of a unique orthogonal matrix. The following lemma shows that the existence of a Dual DO solution is implied by the existence of a DLR solution.

Lemma 2.5 Let a strong DLR solution
We show that such Θ(t), i.e. an orthogonal matrix Θ(t) for which the pair (Θ V , Θ W ) is a Dual DO solution, uniquely exists. Note that again from Corollary 2.4, it suffices to consider an arbitrarily fixed representation (V , W ). We will obtain Θ as a solution of an ordinary differential equation we will now derive. If and from (2.7) we must have where in the last line we used E[W (t)W (t) ] = I . Using the orthonormality of Θ yields the equatioṅ Now, from the assumptions we have where · F denotes the Frobenius norm, and thus −E[W (·)Ẇ (·) ] ∈ R S×S is integrable on (0, T ). Thus, from a standard fixed-point argument we obtain that a solution We claim that (U(t), Y (t)) is a Dual DO solution. First, we note that U is linearly independent, and that Y is orthonormal and satisfies the gauge condition. Indeed, we have det(  [17], also [16,20]) we conclude that We are ready to state the following equivalence of the original problem (2.4) and the Dual DO formulation (Definitions 2.3-2.4).

Assumptions
In view of Proposition 2.6, we establish the unique existence of the Dual DO solution. We work under the following assumptions. Assumptions 1 and 2 will be used for the existence in the strong sense, and in addition, Assumption 3 will be used for the classical sense. Further, the stability Assumptions 4 and 5 will be used to establish the extendability of the strong solution, and respectively the classical solution, to the maximal time interval.
In the above assumption, note that given the first condition, the second condition is implied by F(a) L 2 (Ω;H) < ∞ for a point a ∈ L 2 (Ω; H).
To establish the existence of the Dual DO solution in the classical sense, we use the following further regularity of F.

Assumption 3
In addition to Assumption 2, assume that for every r > 0 and every v 0 ∈ L 2 (Ω; H) with Λv 0 ∈ L 2 (Ω; H) such that Λv 0 L 2 (Ω;H) ≤ q, there exists a constant C q,r > 0 such that Since Λ is closed, D H (Λ) admits a Hilbert space structure with respect to the graph inner product ·, · + Λ·, Λ· , which we denote V. Then, Assumptions 2-3 imply that for a constantC q,r > 0 we have The following uniform stability condition will be used to establish the existence of a strong Dual DO solution in the maximal interval . Here, uniform means that the constant C Λ,F below is independent of bounds of v.

Assumption 4
The pair (Λ, F) satisfies the following: for every v ∈ L 2 (Ω; H) such that Λv ∈ L 2 (Ω; H) we have For example, this condition holds when Λ satisfies Λx, x ≤ 0 for x ∈ D H (Λ) and F satisfies the uniform linear growth condition To establish the existence of the classical Dual DO solution in the maximal interval, we use the following stronger uniform stability condition, where we again note that the constant is independent of bounds of v.
The following examples satisfy the above assumptions.

Example 2.1
For a bounded domain D ⊂ R d , let H = L 2 (D). Further, letΛ be a second order uniformly elliptic differential operator with zero Dirichlet boundary condition. For the non-linear term, let a, b ∈ L ∞ (Ω; L ∞ (D)), c ∈ L 2 (Ω; L 2 (D)), and let f : R → R be a differentiable function such that sup s∈R | f (s)| < ∞. Consider the following multiplicative and additive noise: where · denotes the point-wise multiplication. Then, the pair (Λ,F) satisfies Assumptions 1, 2, and 4.

On the choice of the dual DO formulation
To establish uniqueness and existence of the DLR approximation we work with the Dual DO formulation (2.6). We have chosen this formulation with care. This section provides a discussion on choosing a good formulation.
The DLR approach to the stochastic dynamical system such as (1.1) was first introduced by Sapsis and Lermusiaux [20]. The formulation they introduced is called the Dynamically Orthogonal (DO) formulation: they imposed the orthogonality of the spatial basis. Musharbash et al. [16] pointed out that the DO approximation can be related to the MCTDH method, by considering the so-called dynamically double orthogonal (DDO) formulation: yet another equivalent formulation of the DLR approach. Through this relation of the DDO approximation to the MCTDH method, Musharbash et al. further developed an error estimate of the DO method. The error analysis obtained by Musharbash et al. was partially built upon results regarding the MCTDH method.
A reasonable strategy to establish the existence of the DLR approximation would thus be to establish the existence of the DDO approximation. Namely, following the argument of Koch and Lubich [13], it is tempting to apply the gauge condition defined by the differential operator Λ to the DDO formulation. It turns out that this approach does not work, since the aforementioned gauge condition turns out to be vacuous unless Λ is skew-symmetric, as we illustrate hereafter.
In the DDO formulation, we seek an approximant of the form whereŨ(t) = (U 1 (t), . . . , U S (t)) , and Y (t) = (Y 1 (t), . . . , Y S (t)) are orthonormal in H, and in L 2 (Ω) respectively; and A(t) ∈ R S×S is a full-rank matrix. The triplet (Ũ, A, Y ) is given as a solution of the set of equations: We note that in the Eq. (2.14) forŨ we have the composition of the unbounded operator Λ and the projection operator PŨ , where we note that the mapŨ → PŨ is non-linear. Koch and Lubich [13] had a similar situation in the MCTDH setting. As outlined above, they got away with this problem by considering a different gauge condition. We will explain below an analogous strategy and why it does not work in our setting.
First, from the orthonormality condition onŨ it is necessary to have d dt Ũ ,Ũ =  [13] noted this, and to establish an existence result they considered a suitable gauge condition, which enabled them to take the differential operator out of the projection. The gauge condition that is formally analogous to [13] may be given as ∂ ∂tŨ ,Ũ = ΛŨ,Ũ , for Λ not necessarily skew-symmetric. One can check that this condition formally allows us to take the operator Λ out of the projection PŨ , but for example when Λ is self-adjoint, the solutioñ U will not stay orthonormal. This is not acceptable, since we use the orthonormality to derive the Eq. (2.14), and thus we necessarily have to consider a different gauge condition or a different formulation.

exists}. Then, the Dual DO solution, if it exists, satisfies the following Cauchy problem for a semi-linear abstract evolution equation in
where the initial condition (U 0 , Y 0 ) ∈ X satisfies suitable assumptions detailed below. Conversely, later in Sect. 4 we will see that the strong solution of this Cauchy problem is a Dual DO solution, and that it gives a DLR solution. We first establish the unique existence of the mild solution of (3.2): We will use the following result, which is a variation of a standard local existence and uniqueness theorem for mild solutions, e.g. see [18,Theorem 6.1.4] Further, suppose that for some C α,β > 0 we have G(Û,Ŷ ) X ≤ C α,β . Then, the problem (3.2) starting at t 0 ≥ 0 with the initial condition (Û,Ŷ ) ∈ X : has a unique mild solution on an interval of length δ ∈ (0, 1], where δ depends on α, β, sup s∈[t 0 ,t 0 +1] e s A , and r = r (Û,Ŷ ).
To invoke this proposition, we start with checking that the operator A defined above generates a C 0 semigroup.

Proposition 3.2 Let Assumption 1 hold. Then, A
We will invoke the Hille-Yosida theorem, see for example [18,Theorem 1.5.2]. From Assumption 1, every μ > 0 is in the resolvent set of Λ. Thus, but Assumption 1 implies (μI − Λ) −n [H] S ≤ K Λ /μ n , and thus we obtain In view of the Hille-Yosida theorem the statement now follows.
Furthermore, we establish a Lipschitz continuity of the non-linear term G. We start with the Lipschitz continuity of the projection operator. Then, we have and thus the assumption on κ implies κ 2 + 2κβ < σŶ 2 . On the other hand, we have Next, we note that the identity holds for any g ∈ L 2 (Ω): indeed, we have ( g. This type of identity was shown by Wedin in the finite dimensional setting, see [22, (4.2)]. In view of this identity, the first inequality in (3.4) can be shown as where we used the assumption onŴ ,Ŵ and (3.5). Finally, we apply the inequality (3.6). Then, noting that the assumption on κ implies κ(σŶ , β)) and C κ,β,σŶ are as in Lemma 3.3.

Proof In view of Lemma 3.3, it suffices to show
We will invoke a perturbation result on pairs of projections, [10,Lemma 221], see also [9,Theorem I.6.34]. In this regard, first we will show the following identity of finite dimensional vector subspaces It suffices to show that Im(PŴ | Im(PŴ ) ) cannot be a proper subspace of Im(PŴ ). We will verify that the dimension of Im(PŴ | Im(PŴ ) ) and Im(PŴ ) are the same. In view of (3.5) in the proof of Lemma 3.3, we have dim(Im(PŴ )) = S = dim(Im(PŴ )).

Lemma 3.5 Suppose thatÛ,Û ∈ [H] S are linearly independent and that for somẽ
Proof For componentsÛ j ,Û k ofÛ; andÛ j ,Û k ofÛ , we have and thus there exists a constant C α,S depending on S such that ZÛ − ZÛ 2 ≤ Noting that the matrix ZÛ is non-singular whenÛ is linear independent, we recall that the Fréchet derivative of the mapping R S×S B → B −1 =: Inv(B) ∈ R S×S at B acting on W ∈ R S×S is given by D Inv(B)[W ] = −B −1 W B −1 (see, e.g. [1,Appendix A.5]). Then, with the notation D Inv(ZÛ ) R S×S →R S×S := max W ∈R S×S :

, in view of [3, Corollary 3.2] we have
Now, for r ∈ [0, 1] given, since Z −1 U and Z −1 U are symmetric positive definite, from [15] we have c r ZÛ Now the statement follows.
As a consequence, we obtain the following.
Thus, withα := α + 1 and C α,S := 8Cα ,S in Lemma 3.5 the statement follows. Lemmata 3.4 and 3.6 established above give the following local Lipschitz continuity of the non-linear term G we need.

Existence and regularity
We will now show the existence of the Dual DO solution on the maximal interval. We start with local existence of the mild solution Proof In view of Proposition 3.7, the statement follows from Proposition 3.1.
A regularity of the initial condition gives us the existence of the strong solution.  To see the orthonormality, first note that, from the absolute continuity of Y (t), the function E[Y j Y k ] is absolutely continuous on [0, T ]. But following the same argument as (2.7), we . Therefore, from the orthonormality of the initial condition, for every t ∈ [0, With a further regularity of F, we obtain the classical Dual DO solution. Proof We first observe that G : With these in mind, we see that a result analogous to Proposition 3.1 holds in [V] S ⊕[L 2 (Ω)] S . Then, in view of the discussion in [18, pages 190-191], the statement follows from the similar argument as in the proof of Corollary 4.3.
We now extend the solution to the maximal time interval.
We start with the following bound.
Proof Following the same argument as (2.7), we have E Y j   2 ds. Therefore, the Gronwall's inequality implies that the second statement holds for almost every t. Noting that the mapping t → [ is continuous, this is true for every t ∈ [0, t * ].
We are ready to establish the existence of a Dual DO solution until U becomes linearly dependent.
Theorem 4.6 (Dual DO-strong, maximal) Suppose that Assumptions 1, 2, and 4 are satisfied, and that the initial condition U 0 ∈ [H] S is linearly independent in H, and Proof Under the condition (U 0 , Y 0 ) ∈ D(A), it suffices to show the maximality of the mild solution. We show that t max < ∞ implies lim t↑t max Z −1 U(t) 2 = ∞. In this regard, we first show lim sup t↑t max Z −1 U(t) 2 = ∞. We argue by contradiction and assume t max < ∞ and lim sup t↑t max Z −1 and thus in view of Proposition 3.7 we have [G 1 (Y (s))](U(s)) [H] S ≤ C α max ,S for any s ∈ [0, t max ). If 0 < t < t < t max then letting K Λ = sup r ∈[0,t max ] e r Λ we have , the dominated convergence theorem implies that the right hand side of tends to zero as t, t tend to t max . Hence, Therefore, U admits a continuous extension lim t↑t max U(t) = U(t max ). This allows us to extend Z −1 and thus we have lim t↑t max Z −1 U(t) = Z * ∈ R S×S with Z * 2 ≤ K , but we must have we see that lim t↑t max Y (t) = Y (t max ) exists, and from Corollary 4.3 we have E[Y (t max )Y (t max ) ] = I . But in view of Proposition 3.7 these consequences imply that we can extend the solution beyond t max , which contradicts the maximality of [0, t max ). Hence, lim sup t↑t max Z −1 U(t) 2 = ∞. To conclude the proof we will show lim t↑t max If this is false, then there exist a sequence t n ↑ t max and γ > 0 such that Z −1 U(t n ) 2 ≤ γ for all n ≥ 0. But since lim sup t↑t max Z −1 U(t) 2 = ∞ there is a sequence s k ↑ t max such that Z −1 U(s k ) 2 ≥ γ + 1 for all k ≥ 0. We take a subsequence (s k n ) n so that t n < s k n for all n. From the continuity of t → Z −1 U(t) 2 on [t n , s k n ], there exists h n ∈ [0, s k n − t n ] such that Z −1 U(t n +h n ) 2 = γ + 1. Now, from Lemma 3.5 we have for any n ≥ 0 which is absurd since |h n | ≤ |s k n − t max | + |t max − t n | → 0 as n → ∞ and U is continuous on [0, t max ). Hence, the proof is complete.
Under a stronger assumption on the non-linear term F, we obtain the following bound for ΛU(t) [H] S . This bound will be used to establish the existence in the classical sense on the maximal interval.
where the constant C F > 0 is from Assumption 5.

Proof
We have and thus, as Assumption 1 implies e sΛ Then, applying the Gronwall's inequality completes the proof.  We are now interested in continuing the DLR approximation u S beyond the maximal time t max . A difficulty arising is the full rank condition imposed on M S : at t max the spatial basis becomes linearly dependent, and thus the solution will not stay in M S . But from a practical point of view this should be favourable-roughly speaking, at the maximal time a smaller basis is sufficient to capture the same information as U does. This observation motivates us to leave M S : to extend the approximation beyond t max we consider the extension to t max in the ambient space L 2 (Ω; H). To do so, we go back to the original formulation (2.4). Then, upon extending the solution to t max , one can re-start from t max with a suitable decomposition as the initial condition. and F(u S (r )) L 2 (Ω;H) ≤ C t max ,K Λ ,F for some constant C t max ,K Λ ,F > 0. Hence, we obtain and thus u S admits a continuous extension u S (t) → u * =: u S (t max ) as t ↑ t max .

Proof of Proposition 2.2
Finally, we will show the existence of a smooth parametrisation given a smooth curve

Conclusions
We established the existence of the dynamical low rank (DLR) approximation for random semi-linear evolutionary equations on the maximal interval. A key was to consider an equivalent formulation, the Dual DO formulation. After showing that the Dual DO formulation is indeed equivalent, we showed the unique existence of the solution in the strong and classical sense, by invoking results for the abstract Cauchy problem in the vector spaces. Further, we considered a continuation of the DLR approximation beyond the maximal time interval.