arXiv subjects

Gianluca Ceruti

Publications and source records attributed to Gianluca Ceruti.

At least 19 recordsLinked to original sources

arXiv2025

A low-rank, high-order implicit-explicit integrator for three-dimensional convection-diffusion equations

This paper presents a rank-adaptive implicit-explicit integrator for the tensor approximation of three-dimensional convection-diffusion equations. In particular, the recently developed Reduced Augmentation Implicit Low-rank (RAIL) integrator is extended from the two-dimensional matrix case to the three-dimensional tensor case. The solutions are approximated using a Tucker tensor decomposition. The RAIL integrator first discretizes the partial differential equation fully in space and time using traditional methods. Here, spectral methods are considered for spatial discretizations, and implicit-explicit Runge-Kutta (IMEX RK) methods are used for time discretization. At each RK stage: the bases computed at the previous stages are augmented and reduced to construct projection subspaces. After updating the bases in a dimension-by-dimension manner, a Galerkin projection is performed to update the coefficients stored in the core tensor. As such, the algorithm balances high-order accuracy from spanning as many bases as possible from previous stages, with efficiency from leveraging low-rank structures in the solution. A post-processing step follows to maintain a low-rank solution while conserving mass, momentum, and energy. We validate the proposed method on a number of convection-diffusion problems, including a Fokker-Planck model, and a 3d viscous Burgers' equation.

math.NA↗

arXiv2025

A Galerkin Alternating Projection Method for Kinetic Equations in the Diffusive Limit

The numerical approximation of high-dimensional evolution equations poses significant computational challenges, particularly in kinetic theory and radiative transfer. In this work, we introduce the Galerkin Alternating Projection (GAP) scheme, a novel integrator derived within the Dynamical Low-Rank Approximation (DLRA) framework. We perform a rigorous error analysis, establishing local and global accuracy using standard ODE techniques. Furthermore, we prove that GAP possesses the Asymptotic-Preserving (AP) property when applied to the Radiative Transfer Equation (RTE), ensuring consistent behavior across both kinetic and diffusive regimes. In the diffusive regime, the K-step of the GAP integrator directly becomes the limit equation. In particular, this means that we can easily obtain schemes that even in the diffusive regime are free of a CFL condition, do not require well prepared initial data, and can have arbitrary order in the diffusive limit (in contrast to the semi-implicit and implicit schemes available in the literature). Numerical experiments support the theoretical findings and demonstrate the robustness and efficiency of the proposed method.

math.NA↗

arXiv2024

A parallel Basis Update and Galerkin Integrator for Tree Tensor Networks

Computing the numerical solution to high-dimensional tensor differential equations can lead to prohibitive computational costs and memory requirements. To reduce the memory and computational footprint, dynamical low-rank approximation (DLRA) has proven to be a promising approach. DLRA represents the solution as a low-rank tensor factorization and evolves the resulting low-rank factors in time. A central challenge in DLRA is to find time integration schemes that are robust to the arising small singular values. A robust parallel basis update & Galerkin integrator, which simultaneously evolves all low-rank factors, has recently been derived for matrix differential equations. This work extends the parallel low-rank matrix integrator to Tucker tensors and general tree tensor networks, yielding an algorithm in which all bases and connecting tensors are evolved in parallel over a time step. We formulate the algorithm, provide a robust error bound, and demonstrate the efficiency of the new integrators for problems in quantum many-body physics, uncertainty quantification, and radiative transfer.

math.NA↗

arXiv2024

GeoLoRA: Geometric integration for parameter efficient fine-tuning

Low-Rank Adaptation (LoRA) has become a widely used method for parameter-efficient fine-tuning of large-scale, pre-trained neural networks. However, LoRA and its extensions face several challenges, including the need for rank adaptivity, robustness, and computational efficiency during the fine-tuning process. We introduce GeoLoRA, a novel approach that addresses these limitations by leveraging dynamical low-rank approximation theory. GeoLoRA requires only a single backpropagation pass over the small-rank adapters, significantly reducing computational cost as compared to similar dynamical low-rank training methods and making it faster than popular baselines such as AdaLoRA. This allows GeoLoRA to efficiently adapt the allocated parameter budget across the model, achieving smaller low-rank adapters compared to heuristic methods like AdaLoRA and LoRA, while maintaining critical convergence, descent, and error-bound theoretical guarantees. The resulting method is not only more efficient but also more robust to varying hyperparameter settings. We demonstrate the effectiveness of GeoLoRA on several state-of-the-art benchmarks, showing that it outperforms existing methods in both accuracy and computational efficiency.

cs.LG↗

arXiv2024

Geometry-aware training of factorized layers in tensor Tucker format

Reducing parameter redundancies in neural network architectures is crucial for achieving feasible computational and memory requirements during training and inference phases. Given its easy implementation and flexibility, one promising approach is layer factorization, which reshapes weight tensors into a matrix format and parameterizes them as the product of two small rank matrices. However, this approach typically requires an initial full-model warm-up phase, prior knowledge of a feasible rank, and it is sensitive to parameter initialization. In this work, we introduce a novel approach to train the factors of a Tucker decomposition of the weight tensors. Our training proposal proves to be optimal in locally approximating the original unfactorized dynamics independently of the initialization. Furthermore, the rank of each mode is dynamically updated during training. We provide a theoretical analysis of the algorithm, showing convergence, approximation and local descent guarantees. The method's performance is further illustrated through a variety of experiments, showing remarkable training compression rates and comparable or even better performance than the full baseline and alternative layer factorization strategies.

cs.LG↗

arXiv2024

Randomized low-rank Runge-Kutta methods

This work proposes and analyzes a new class of numerical integrators for computing low-rank approximations to solutions of matrix differential equation. We combine an explicit Runge-Kutta method with repeated randomized low-rank approximation to keep the rank of the stages limited. The so-called generalized Nyström method is particularly well suited for this purpose; it builds low-rank approximations from random sketches of the discretized dynamics. In contrast, all existing dynamical low-rank approximation methods are deterministic and usually perform tangent space projections to limit rank growth. Using such tangential projections can result in larger error compared to approximating the dynamics directly. Moreover, sketching allows for increased flexibility and efficiency by choosing structured random matrices adapted to the structure of the matrix differential equation. Under suitable assumptions, we establish moment and tail bounds on the error of our randomized low-rank Runge-Kutta methods. When combining the classical Runge-Kutta method with generalized Nyström, we obtain a method called Rand RK4, which exhibits fourth-order convergence numerically -- up to the low-rank approximation error. For a modified variant of Rand RK4, we also establish fourth-order convergence theoretically. Numerical experiments for a range of examples from the literature demonstrate that randomized low-rank Runge-Kutta methods compare favorably with two popular dynamical low-rank approximation methods, in terms of robustness and speed of convergence.

math.NA↗

arXiv2024

Low-rank Tree Tensor Network Operators for Long-Range Pairwise Interactions

Compactly representing and efficently applying linear operators are fundamental ingredients in tensor network methods for simulating quantum many-body problems and solving high-dimensional problems in scientific computing. In this work, we study such representations for tree tensor networks, the so called tree tensor network operators (TTNOs), paying particular attention to Hamiltonian operators that involve long-range pairwise interactions between particles. Generalizing the work by Lin, Tong, and others on matrix product operators, we establish a direct connection between the hierarchical low-rank structure of the interaction matrix and the TTNO property. This connection allows us to arrive at very compact TTNO representations by compressing the interaction matrix into a hierarchically semi-separable matrix. Numerical experiments for different quantum spin systems validate our results and highlight the potential advantages of TTNOs over matrix product operators.

math.NA↗

arXiv2024

From low-rank retractions to dynamical low-rank approximation and back

In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Projected Ralston--Hermite (PRH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. Numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new methods.

math.NA↗

arXiv2024

A robust second-order low-rank BUG integrator based on the midpoint rule

Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments.

math.NA↗

arXiv2023

Robust low-rank training via approximate orthonormal constraints

With the growth of model and data sizes, a broad effort has been made to design pruning techniques that reduce the resource demand of deep learning pipelines, while retaining model performance. In order to reduce both inference and training costs, a prominent line of work uses low-rank matrix factorizations to represent the network weights. Although able to retain accuracy, we observe that low-rank methods tend to compromise model robustness against adversarial perturbations. By modeling robustness in terms of the condition number of the neural network, we argue that this loss of robustness is due to the exploding singular values of the low-rank weight matrices. Thus, we introduce a robust low-rank training algorithm that maintains the network's weights on the low-rank matrix manifold while simultaneously enforcing approximate orthonormal constraints. The resulting model reduces both training and inference costs while ensuring well-conditioning and thus better adversarial robustness, without compromising model accuracy. This is shown by extensive numerical evidence and by our main approximation theorem that shows the computed robust low-rank network well-approximates the ideal full model, provided a highly performing low-rank sub-network exists.

cs.LG↗

arXiv2023

A parallel rank-adaptive integrator for dynamical low-rank approximation

This work introduces a parallel and rank-adaptive matrix integrator for dynamical low-rank approximation. The method is related to the previously proposed rank-adaptive basis update & Galerkin (BUG) integrator but differs significantly in that all arising differential equations, both for the basis and the Galerkin coefficients, are solved in parallel. Moreover, this approach eliminates the need for a potentially costly coefficient update with augmented basis matrices. The integrator also incorporates a new step rejection strategy that enhances the robustness of both the parallel integrator and the BUG integrator. By construction, the parallel integrator inherits the robust error bound of the BUG and projector-splitting integrators. Comparisons of the parallel and BUG integrators are presented by a series of numerical experiments which demonstrate the efficiency of the proposed method, for problems from radiative transfer and radiation therapy.

math.NA↗

arXiv2023

Numerical simulations of long-range open quantum many-body dynamics with tree tensor networks

Open quantum systems provide a conceptually simple setting for the exploration of collective behavior stemming from the competition between quantum effects, many-body interactions, and dissipative processes. They may display dynamics distinct from that of closed quantum systems or undergo nonequilibrium phase transitions which are not possible in classical settings. However, studying open quantum many-body dynamics is challenging, in particular in the presence of critical long-range correlations or long-range interactions. Here, we make progress in this direction and introduce a numerical method for open quantum systems, based on tree tensor networks. Such a structure is expected to improve the encoding of many-body correlations and we adopt an integration scheme suited for long-range interactions and applications to dissipative dynamics. We test the method using a dissipative Ising model with power-law decaying interactions and observe signatures of a first-order phase transition for power-law exponents smaller than one.

cond-mat.stat-mech↗

arXiv2022

Low-rank lottery tickets: finding efficient low-rank neural networks via matrix differential equations

Neural networks have achieved tremendous success in a large variety of applications. However, their memory footprint and computational demand can render them impractical in application settings with limited hardware or energy resources. In this work, we propose a novel algorithm to find efficient low-rank subnetworks. Remarkably, these subnetworks are determined and adapted already during the training phase and the overall time and memory resources required by both training and evaluating them are significantly reduced. The main idea is to restrict the weight matrices to a low-rank manifold and to update the low-rank factors rather than the full matrix during training. To derive training updates that are restricted to the prescribed manifold, we employ techniques from dynamic model order reduction for matrix differential equations. This allows us to provide approximation, stability, and descent guarantees. Moreover, our method automatically and dynamically adapts the ranks during training to achieve the desired approximation accuracy. The efficiency of the proposed method is demonstrated through a variety of numerical experiments on fully-connected and convolutional networks.

cs.LG↗

arXiv2022

Rank-adaptive time integration of tree tensor networks

A rank-adaptive integrator for the approximate solution of high-order tensor differential equations by tree tensor networks is proposed and analyzed. In a recursion from the leaves to the root, the integrator updates bases and then evolves connection tensors by a Galerkin method in the augmented subspace spanned by the new and old bases. This is followed by rank truncation within a specified error tolerance. The memory requirements are linear in the order of the tensor and linear in the maximal mode dimension. The integrator is robust to small singular values of matricizations of the connection tensors. Up to the rank truncation error, which is controlled by the given error tolerance, the integrator preserves norm and energy for Schrodinger equations, and it dissipates the energy in gradient systems. Numerical experiments with a basic quantum spin system illustrate the behavior of the proposed algorithm.

math.NA↗

arXiv2021

On the stability of robust dynamical low-rank approximations for hyperbolic problems

The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform a $L^2$ stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector splitting integrator when first discretizing the equations and then applying the DLRA. Based on this we propose a projector splitting integrator, based on applying DLRA to the continuous system before performing the discretization, that recovers the classic CFL condition. We also show that the unconventional integrator has more favorable stability properties and explain why the projector splitting integrator performs better when approximating higher moments, while the unconventional integrator is generally superior for first order moments. Furthermore, an efficient and stable dynamical low-rank update for the scattering term in kinetic transport is proposed. Numerical experiments for kinetic transport and uncertainty quantification, which confirm the results of the stability analysis, are presented.

math.NA↗

arXiv2021

Dynamical low-rank approximation for Burgers' equation with uncertainty

Quantifying uncertainties in hyperbolic equations is a source of several challenges. First, the solution forms shocks leading to oscillatory behaviour in the numerical approximation of the solution. Second, the number of unknowns required for an effective discretization of the solution grows exponentially with the dimension of the uncertainties, yielding high computational costs and large memory requirements. An efficient representation of the solution via adequate basis functions permits to tackle these difficulties. The generalized polynomial chaos (gPC) polynomials allow such an efficient representation when the distribution of the uncertainties is known. These distributions are usually only available for input uncertainties such as initial conditions, therefore the efficiency of this ansatz can get lost during runtime. In this paper, we make use of the dynamical low-rank approximation (DLRA) to obtain a memory-wise efficient solution approximation on a lower dimensional manifold. We investigate the use of the matrix projector-splitting integrator and the unconventional integrator for dynamical low-rank approximation, deriving separate time evolution equations for the spatial and uncertain basis functions, respectively. This guarantees an efficient approximation of the solution even if the underlying probability distributions change over time. Furthermore, filters to mitigate the appearance of spurious oscillations are implemented, and a strategy to enforce boundary conditions is introduced. The proposed methodology is analyzed for Burgers' equation equipped with uncertain initial values represented by a two-dimensional random vector. The numerical results show a reduction of the memory requirements, and that the important characteristics of the original system are well captured.

math.NA↗

arXiv2021

A rank-adaptive robust integrator for dynamical low-rank approximation

A rank-adaptive integrator for the dynamical low-rank approximation of matrix and tensor differential equations is presented. The fixed-rank integrator recently proposed by two of the authors is extended to allow for an adaptive choice of the rank, using subspaces that are generated by the integrator itself. The integrator first updates the evolving bases and then does a Galerkin step in the subspace generated by both the new and old bases, which is followed by rank truncation to a given tolerance. It is shown that the adaptive low-rank integrator retains the exactness, robustness and symmetry-preserving properties of the previously proposed fixed-rank integrator. Beyond that, up to the truncation tolerance, the rank-adaptive integrator preserves the norm when the differential equation does, it preserves the energy for Schrödinger equations and Hamiltonian systems, and it preserves the monotonic decrease of the functional in gradient flows. Numerical experiments illustrate the behaviour of the rank-adaptive integrator.

math.NA↗

arXiv2020

An unconventional robust integrator for dynamical low-rank approximation

We propose and analyse a numerical integrator that computes a low-rank approximation to large time-dependent matrices that are either given explicitly via their increments or are the unknown solution to a matrix differential equation. Furthermore, the integrator is extended to the approximation of time-dependent tensors by Tucker tensors of fixed multilinear rank. The proposed low-rank integrator is different from the known projector-splitting integrator for dynamical low-rank approximation, but it retains the important robustness to small singular values that has so far been known only for the projector-splitting integrator. The new integrator also offers some potential advantages over the projector-splitting integrator: It avoids the backward time integration substep of the projector-splitting integrator, which is a potentially unstable substep for dissipative problems. It offers more parallelism, and it preserves symmetry or anti-symmetry of the matrix or tensor when the differential equation does. Numerical experiments illustrate the behaviour of the proposed integrator.

math.NA↗