Abstract
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.
Highlights
This paper is concerned with the existence of solutions of the so called DynamicalLow Rank Method (DLR) [6,7,16,17,20] to a semi-linear random parabolic evolutionary equation
The idea of the dynamical low rank (DLR) approximation is to approximate the solution of (1.1) at each time t > 0 as a linear combination of products of deterministic and stochastic basis functions, both of which evolve over time: the approximate solution is of the form uS(t) = U (t)Y (t), for some positive integer S ∈ N called the rank of the solution, where U(t) = (U1(t), . . . , US(t)) are linearly independent in H, and Y (t) = (Y1(t), . . . , YS(t)) are linearly independent in the space L2(Ω) of square-integrable random variables
Musharbash et al [16] pointed out that the Dynamically Orthogonal (DO) approximation can be related to the multi-configuration timedependent Hartree (MCTDH) method, by considering the so-called dynamically double orthogonal (DDO) formulation: yet another equivalent formulation of the DLR approach
Summary
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. 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. 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.
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