Dynamical Low-Rank Approximation (DLRA)

Updated 22 July 2025

DLRA is a numerical methodology that represents high-dimensional, time-dependent solutions on a fixed or adaptive low-rank manifold.
It employs integrators like projector-splitting and BUG techniques to evolve basis functions while minimizing approximation errors.
DLRA is applied in fields such as radiative transfer, kinetic equations, and quantum dynamics to achieve memory-efficient and accurate simulations.

Dynamical Low-Rank Approximation (DLRA) is a numerical methodology for representing and evolving the solution of high-dimensional time-dependent problems on a manifold of fixed or adaptively varying low rank. DLRA enables reduction of both computational memory and time required to simulate complex systems, particularly in applications such as kinetic equations, radiative transfer, quantum dynamics, stochastic differential equations, and large-scale control problems. The approach constructs a time-dependent decomposition where the factors (typically interpreted as orthonormal bases in chosen dimensions) and a small coupling matrix are evolved such that the overall approximation remains as close as possible to the full solution, but with a complexity set by the chosen rank.

1. Mathematical Foundation and General Framework

DLRA is predicated on the tensor or matrix low-rank decomposition of the solution variable. For a time-dependent function or tensor $\psi(\mathbf{x}, t)$ (for instance, $\psi(z, \mu, t)$ in radiative transfer), the approximation is written as

$\psi(z, \mu, t) \approx \sum_{i=1}^r \sum_{j=1}^r X_i(z, t)\, S_{ij}(t)\, W_j(\mu, t)$

where $X_i(z, t)$ and $W_j(\mu, t)$ form time-dependent orthonormal bases in the respective variables and $S(t)$ is a small $r\times r$ matrix. Analogous tensor decompositions are used in higher dimensions or with more variables, e.g., space, angle, and energy in kinetic equations.

The evolution is governed by requiring that the time derivative of the approximated solution $Y(t)$ remains within the tangent space $\mathcal{T}\mathcal{M}_r$ to the manifold of rank- $r$ representations. Formally, this is a projection of the full evolution equation onto $\mathcal{T}\mathcal{M}_r$ at each time,

$\dot{Y}(t) = P_{Y(t)}\big(F(Y(t))\big)$

where $F$ is the right-hand side of the governing PDE and $P_{Y(t)}$ is the tangent space projector with respect to the chosen low-rank manifold (Peng et al., 2019, Peng et al., 2019).

This “Dirac–Frenkel time-dependent variational principle” ensures that the approximation error, measured in a Frobenius or $L^2$ norm, is minimized in a local instantaneous sense.

2. Algorithms and Integrator Strategies

DLRA algorithms proceed by evolving the factors ( $X$ , $W$ , $S$ ) instead of the full solution. The evolution equations for these factors are derived through projection and are typically nonlinear and coupled. Several integrator strategies have been developed:

Projector-splitting Integrator: The tangent space projection is split into three substeps, each updating only one factor at a time while holding others fixed. For matrices, this results in K-step (left basis), S-step (core matrix), and L-step (right basis), with updates such as

$\text{K-step:} \quad K'(t) = F(K(t)V_0^T)V_0$

and analogous steps for $S$ and $L$ (Peng et al., 2019, Peng et al., 2019, Kusch et al., 2021).

Unconventional (BUG) Integrator: This method performs the K- and L-steps in parallel before a Galerkin projection (S-step), avoiding “backward in time” substeps and improving stability. This approach is especially important for stiff, high-curvature manifolds or when small singular values appear (Einkemmer et al., 2022).
Rank-adaptive Schemes: These dynamically augment (and truncate via SVD) the bases, allowing the rank to increase or decrease during evolution as needed to achieve the required accuracy (Baumann et al., 2023, Hauck et al., 2022).
Operator Splitting: When the PDE contains terms corresponding to different physical processes (e.g., advection, scattering, absorption), operator splitting techniques are used to decouple and efficiently integrate each component, always within the tangent space projection paradigm (Peng et al., 2019, Peng et al., 2019).

3. Discretization and Implementation Techniques

DLRA relies critically on compatible discretizations in space, angle, and possible stochastic/uncertainty dimensions:

Finite Volume in Space: Piecewise constant spatial basis functions with a constant mesh are typical. The finite volume approach respects conservation properties and is amenable to operator splitting (Peng et al., 2019).
Spectral Basis in Angular/Velocity Variables: Legendre polynomials (1D angle) or spherical harmonics (higher-dimensional angles) serve as the angular basis, capitalizing on their orthogonality and facilitating integration of the transport operator (Peng et al., 2019).
Weighted and Adaptive Bases: To ensure conservation of invariants and facilitate efficient adaptivity, weighted orthonormality (with respect to natural problem-dependent norms or measures) is often imposed on the basis functions (Yin et al., 2023, Einkemmer et al., 2022).
Randomized SVD for Adaptivity: Low-rank residuals or corrections are computed using randomized algorithms, which allow efficient truncation and adaptation without explicit construction of large intermediate matrices (Hauck et al., 2022).
Time-stepping Integration: Both explicit and implicit time-stepping schemes are combined with the operator splitting and factor evolution. Special care is taken to maintain energy stability and suitable CFL conditions in stiff or strongly coupled regimes (Einkemmer et al., 2022, Baumann et al., 2023).

4. Accuracy, Stability, and Conservation Properties

DLRA achieves significant reduction in computational requirements, but only with rigorous attention to accuracy and the preservation of physical invariants:

Error and Accuracy: For fixed memory or degrees of freedom, DLRA solutions have been demonstrated to attain lower error than comparable full-rank solutions in practical radiative transport problems (Peng et al., 2019, Peng et al., 2019). Error is controlled via the rank and truncation tolerances, which can be dynamically adjusted.
Stability Analysis: Certain integrators may produce instability if tangent space projection and discretization are not carefully ordered. Analysis shows that applying DLRA to the continuous equation prior to discretization and adopting stabilized splitting strategies can recover classical CFL-type stability conditions (Kusch et al., 2021).
Asymptotic-Preserving and Energy-Stable Schemes: Approaches such as the unconventional (BUG) integrator are specifically constructed to both capture the correct diffusion limit of kinetic equations (as scattering becomes dominant) and to ensure non-increasing energy, even under coarse discretization (Einkemmer et al., 2022, Baumann et al., 2023).
Conservation Laws: Extensions of the standard DLRA, such as mass-/momentum-conserving modifications (e.g., incorporation of fixed basis functions for conserved moments, Petrov–Galerkin projections, or basis augmentation strategies), have been devised to guarantee that discrete mass and momentum are rigorously conserved (Baumann et al., 2023, Uschmajew et al., 13 Mar 2025).

5. Applications and Performance in High-Dimensional and Stiff Problems

DLRA is particularly effective for problems characterized by high-dimensional phase spaces or where the curse of dimensionality renders traditional full-rank solvers infeasible:

Radiative Transport: In simulations for radiation transport (including therapeutic dose calculation in medical applications), DLRA methods using operator splitting and low-rank factorization in space and angle can reduce memory and computational costs by orders of magnitude, while maintaining accuracy (Peng et al., 2019, Kusch et al., 2021).
Kinetic and Boltzmann-type Equations: For the Vlasov-Poisson and Boltzmann-BGK equations, DLRA provides a means to efficiently capture the evolution of distribution functions and their moments, even in the presence of challenging boundary conditions or stiff collision operators. Multiplicative splitting and conservative integrators are key in these settings (Baumann et al., 11 Nov 2024, Yin et al., 2023, Uschmajew et al., 2023).
Uncertainty Quantification: DLRA, combined with generalized polynomial chaos expansions, provides a scalable strategy to represent the propagation of uncertainty in hyperbolic PDEs, dramatically reducing the memory footprint compared with full stochastic Galerkin methods (Kusch et al., 2021).
Nonlinear Control and Inverse Problems: Recent work demonstrates DLRA-based strategies for optimal feedback control and inverse kinetic parameter identification, where the low-rank constraint is leveraged in conjunction with adjoint-based optimization or state-dependent Riccati computations to solve high-dimensional PDE-constrained problems efficiently (Saluzzi et al., 13 Jan 2025, Baumann et al., 26 Jun 2025).
Large-Scale Matrix/Tensor Differential Equations: Tucker tensor cross strategies, such as DEIM-FS, enable DLRA to be applied to very high-dimensional nonlinear PDEs by compressing the right-hand side on-the-fly, bypassing the need to access the full tensor (Ghahremani et al., 8 Jan 2024).

6. Adaptivity, Parallelization, and Recent Advances

DLRA research is rapidly evolving, with ongoing advances that address adaptivity, parallel efficiency, and higher-order integration:

Rank Adaptivity: Algorithms that dynamically add or remove basis functions (using singular value decomposition and residual estimates) allow DLRA to efficiently track transients and varying solution complexity, while minimizing resource usage (Hauck et al., 2022, Appelö et al., 8 Feb 2024, Baumann et al., 2023).
Parallel Integrators: Recent developments include fully parallel, second-order robust time integrators for DLRA that achieve higher order of accuracy without increasing the rank beyond 2r (where r is the current rank) and maintain norm preservation independently of the curvature of the low-rank manifold (Kusch, 5 Mar 2024).
Multiplicative Splitting and Conservative Schemes: Multiplicative splitting, where the solution is written as $f = M g$ (with $M$ representing an equilibrium or reference state), often results in lower rank for $g$ and enhances stability, particularly in kinetic equations (Baumann et al., 11 Nov 2024, Baumann et al., 5 Feb 2025).
Mesh and Rank Adaptivity in Coupled Discretizations: Combining DLRA with discontinuous Galerkin spatial methods and mesh adaptivity enables scalable and conservative simulation of transport problems even in two or more dimensions (Uschmajew et al., 13 Mar 2025).

7. Broader Implications and Outlook

DLRA has proven effective in reducing simulation costs and memory requirements across a spectrum of challenging high-dimensional and stiff dynamical systems. Its adaptability, the capacity to rigorously enforce stability and conservation requirements, and the demonstrated potential in scientific and engineering applications—ranging from radiation therapy to neutron transport, plasma physics, and optimal control—align it with the broader movement toward data-driven, structure-preserving, and efficient model reduction techniques. Future research continues to address algorithmic robustness for highly nonlinear, time-varying, and multi-physics problems, deeper integration with advanced discretization schemes, and parallelization and scaling to extreme computational environments.