Scientific Program

Approximating functions in high dimension is at the heart of many learning and scientific computing problems. To overcome the curse of dimension, it is necessary to exploit particular structures of functions, beyond classical regularity. This requires the introduction of dedicated approximation tools capable of capturing these structures, and algorithms for constructing these approximations.

In a first part of this course, we will first introduce benchmarks for function approximation, defining what we can expect from optimal algorithms. These benchmarks depend on how the quality of the approximation is measured, the nature of the available information (point evaluations of functions, gradient evaluations…) and the properties of the algorithms (linearity, stability…). We will present manifold approximation (or dimension reduction) methods which aim at constructing approximation tools dedicated to a certain task.

High-dimensional approximation tools are in general extremely nonlinear. The use of generic approaches from statistical learning and optimization generally lead to very high complexity, excessive data consumption, and accuracies that are not up to the expectations of scientific computing. A real challenge lies in generating or exploiting optimal data for a given task, thereby considerably reducing the complexity of algorithms. Data generation must be specific to the approximation tool used, and to the nature of the available information (function evaluations, residual of a partial differential equation in physics-informed learning…). In a second part of this course, we will give a short introduction to optimal sampling methods for approximation, and provide an overview of recent developments in the field.

Many approximation tools suffer from poor scalability as the dimensionality of a model inputs and outputs increases. A powerful approach to handling high-dimensional variables is to reduce their dimensionality, allowing the problem to be reformulated in a lower-dimensional space.

In this presentation, we will present a line of work that leverages gradient information to identify important low-dimensional subspaces within the input parameters of a given function. We will also demonstrate that these same gradients reveal critical low-dimensional subspaces in the output space, enabling data reduction in Bayesian inference and optimal sensor placement.

These subspaces emerge from error bounds derived using functional inequalities to approximate the input-output mapping. The error analysis not only provides a priori guarantees for the dimension reduction process but also offers criteria for selecting the dimensions of the reduced variables.

We will illustrate the advantages of this linear dimension reduction technique and further extend the methodology to perform nonlinear dimension reduction.

In this talk we will discuss about several aspects of the resolution of high-dimensional problems. In particular, with a particular focus on problems defined by a system of parametric Partial Differential Equations (PDEs).

First, a motivation on a real industrial case will be presented, aiming at illustrating the challenges that certain computations and tasks involve. Most of these practical challenges are inherently related to the curse of dimensionality, which we will examine through the lens of the Kolmogorov metric entropy and entropy numbers.

In the first part of the talk we will describe low-rank approximations and discuss two possible strategies to adapt them, and to start introducing more ‘flexible’ representations. The first one considers sums of tensor trains in which we do not fix a priori the order of the variables. The second one consists in a piece-wise tensor approximation in which we do not fix a priori the domain decomposition. In both, the adaptation is based on error and parsimony criteria. Solving the problems with non-linear approximations at hand poses undeniable challenges. We will describe several ways to precondition multilinear problems and will detail a method combining CUR factorisation and Krylov subspace iterations.

In the last part of the talk we will focus on structure preservation, i.e. the ability to construct numerical approximations which respect geometrical and physical properties of the system. We will restrict to PDEs which can be written as a sum of a Hamiltonian flow and a dissipation mechanism which is defined by a gradient flow. A mixed formulation will be introduced. This makes it possible to provably conserve several quantities of interest (Casimirs, invariants, equilibria) and have the correct dissipation rates if dissipation occurs. We will extend the formulation to dynamical discretisations (such as Dynamical Low Rank, neural Galerkin, shape morphing functions) and illustrate the method on some test-cases.

This talk discusses the relations between two approaches using positive definite kernels: probabilistic methods using Gaussian processes, and non-probabilistic methods using reproducing kernel Hilbert spaces (RKHS). They are widely studied and used in machine learning, statistics, and numerical analysis.

Connections and equivalences between them are reviewed for fundamental topics such as regression, interpolation, numerical integration, distributional discrepancies, and statistical dependence, as well as for sample path properties of Gaussian processes. A unifying perspective for these equivalences is established, based on the equivalence between the Gaussian Hilbert space and the RKHS. This serves as a foundation for many other methods based on Gaussian processes and reproducing kernels, which are being developed in parallel by the two research communities.

Beyond their foundational roots in statistical and stochastic process theory, Gaussian processes (GPs) are now central to a versatile and widely used framework spanning applied mathematics and probabilistic machine learning. Despite well-known limitations, GP-based methods continue to show remarkable adaptability, often outlasting shifting methodological trends.

In this talk, I will highlight developments in modeling, prediction, and active learning with GPs, offering perspectives on their enduring role as a powerful building block for function approximation and uncertainty quantification.

Multi-objective optimization seeks solutions that balance competing objectives, leading not to a single optimum but to a set of Pareto-optimal trade-offs. This tutorial introduces the main ideas behind multi-objective Bayesian optimization (MOBO), a framework for solving expensive black-box optimization problems with multiple objectives using probabilistic surrogate models.

We will discuss different approaches to designing active learning schemes for selecting new observations in the multi-objective setting. Finally, we will see how MOBO can make a difference in industrial applications on the example of therapeutic antibody design.

In this presentation I will cover the latest development in reduced order modeling for parametrized systems described in terms of partial differential equations (PDEs), ranging from classical reduced basis methods to more recent deep learning based techniques.

After reviewing fundamental tools in reduced order modeling like, e.g., proper orthogonal decomposition and Galerkin reduced basis methods, I will introduce a series of strategies taking advantage of (i) neural networks to surrogate the reduced problem and to perform dimensionality reduction, (ii) sparse regression methods for discovery of latent dynamics models, (iii) shallow decoder architectures for the construction of ROMs capable of leveraging sensor data, and (iv) generative models to embed uncertainty quantification. Examples from applied sciences and engineering will be presented to showcase the accuracy, the efficiency, and the reliability of the proposed techniques.

A long-standing objective in applied science and engineering has been to reduce complex nonlinear differential equations and data sets to simple, low-dimensional models. The main topic of this lecture, model reduction to spectral submanifolds (SSMs), introduces a mathematically well-founded solution to this objective. SSMs are very low-dimensional attracting invariant manifolds tangent to eigenspaces (spectral subspaces) of linearizations of nonlinear systems at steady states. The internal dynamics of SSMs provide simple reduced-order models with which typical system trajectories synchronize exponentially fast.

We will discuss applications of SSM reduction include accelerated finite-element computations, reduced-order modeling of fluid-structure interactions from experimental data, derivation of equations of motion from videos, prediction of transitions to turbulence in pipe flows, model-predictive control of soft robots and other physical problems.

Computational Fluid Dynamics (CFD) is a cornerstone for the analysis of complex flow phenomena. However, the computational cost of high-fidelity simulations often restricts their use in many-query scenarios such as design optimization, uncertainty quantification, and parametric analysis. Reduced Order Models (ROMs) alleviate this limitation by constructing low-dimensional surrogates that retain the dominant flow dynamics at a significantly reduced computational cost [1]. This work presents a set of complementary data-driven strategies that combine reduced-order modeling and scientific machine learning to improve the accuracy, robustness, and generalization capabilities of ROMs for CFD applications. A first contribution addresses intrusive ROMs through parametric closure modeling, where a deep operator network is trained to learn reduced correction operators that compensate for unresolved dynamics, enabling reliable generalization across the parameter space [3].

A second contribution focuses on non-intrusive ROMs and introduces a space-dependent aggregation framework in which multiple surrogate models—obtained from different combinations of dimensionality reduction techniques, regression methods, or turbulence models—are locally combined through spatially varying convex weights learned from data. This approach enhances predictive accuracy in challenging benchmark problems, including transonic airfoil flows [2].

Finally, a hybrid reduced-order modeling framework is developed for high- Reynolds-number fluid–structure interaction problems in an Arbitrary Lagrangian–Eulerian setting. The approach combines POD–Galerkin projection for the fluid equations with machine-learning-based models for the turbulent eddy viscosity and reduced interpolation techniques for mesh motion, enabling accurate and efficient simulation of flow-induced vibration phenomena [4]. Overall, the proposed methodologies advance hybrid reduced-order modeling by unifying surrogate aggregation, operator learning, and data-driven turbulence closure within a flexible and scalable framework, providing fast and reliable CFD predictions for complex, turbulent, compressible and parameter-dependent flow systems. Applications will be introduced as examples during the methodology overview.

REFERENCES

[1] Ivagnes, A., Khamlich, M., Siena, P., & Rozza, G. Reduced Order Modeling in Computational Fluid Dynamics: An Overview of Methods and Applications. In International Conference on Emerging Technologies in Computational Science for Industry, Sustainability and Innovation (pp. 1-20), LNCSE series, Math2Product volume, Springer Nature Switzerland. (2025).

[2] Ivagnes, A., Tonicello, N., Cinnella, P., & Rozza, G. Enhancing non-intrusive reduced-order models with space-dependent aggregation methods. Acta Mechanica, 1-30. (2024).

[3] Ivagnes, A., Stabile, G., & Rozza, G. Data-driven Closure Strategies for Parametrized Reduced Order Models via Deep Operator Networks. arXiv preprint arXiv:2505.17305 (2025), CMAME in press (2026)

[4] Ngan, V. N., Stabile, G., Mola, A., & Rozza, G. A hybrid reduced-order model for segregated fluid-structure interaction solvers in an ALE approach at high Reynolds number. Computers & Mathematics with Applications, 180, 299-321. (2025).

We will first discuss adaptive variance reduction techniques, in particular, for functions of Gaussian random vectors. We will next present the regression techniques that permit to approximate conditional expectations in view of American options pricing. The remaining of the lecture will be devoted to the discretization of stochastic differential equations : Euler-Maruyama and Milstein schemes, strong and weak errors, MultiLevel Monte-Carlo methods, schemes with high order of weak convergence like the Ninomiya-Victoir scheme. We will finally discuss the simulation of stochastic Volterra equations by combining Markovian approximations with time-discretization.

We will survey the strength and challenges of current machine learning approaches to financial modeling and computation on three examples: optimal investment, hedging, and trade execution. For these case studies, we will investigate the intertwined problems of (synthetic) data generation and optimisation of (over-)parametrised strategies. In the first case, we will theoretically examine the generalisation error achieved by optimal investment strategies learnt from finitely many samples and demonstrate good stability properties for entropy regularised mean-field neural networks. In the hedging context, we will propose an ambiguity-averse modification of a deep hedger which achieves more robustness to shifts in the market regime by minimising nested model and market risk measures constructed through feature clustering. Finally, we will exhibit synthetic data generators for limit order books using, alternatively, generative adversarial networks and nearest-neighbour resampling, and discuss implications on the construction of execution strategies.

Kernel Ridge Regression (KRR) is a powerful and well-established framework for non-parametric learning. However, its exact implementation suffers from severe computational bottlenecks on large-scale datasets, typically requiring cubic and quadratic time and memory complexity, respectively. A popular approach to alleviate this computational burden is the Nyström approximation, which is constructed upon selecting a set of representative points from a dataset. While random sampling is the standard practice for landmark selection, data-driven strategies—such as Greedy selection— are known to empirically provide better approximation of the regression function.In this work, we investigate the statistical properties of the Nyström approximated KRR estimator built upon a Greedy selection strategy. We present preliminary theoretical results analyzing the impact of this selection rule on the estimator’s performance. Our derivation relies on a novel analytical perspective: we demonstrate that the original regression problem inherently conceals a latent interpolation problem within a Reproducing Kernel Hilbert Space (RKHS) endowed with a modified metric.By framing the Greedy selection within this novel analytical perspective, we establish preliminary upper bounds on the training risk. Specifically, our theoretical analysis demonstrates a polynomial decay of the error with respect to the sample size, modulated by logarithmic factors and explicitly dependent on the regularization parameter and the intrinsic dimensionality of the problem.

Most methods for generating synthetic populations rely on cross-sectional snapshots or pseudo- panels, which do not track individuals consistently over time. This paper proposes a general framework for constructing synthetic populations whose panel structure is specified by design. Individuals are represented through life-based trajectories defined independently of calendar time, and a deterministic mapping recovers their state at any time t, allowing the reconstruction of panel data and population distributions at arbitrary points in time. The framework is model-agnostic and enforces internal consistency through structural constraints embedded in the life representation. We further introduce a Bayesian updating mechanism that incorporates information from observed cross-sectional datasets. When data are available, the synthetic population is sampled from the posterior distribution, combining prior knowledge with the evidence contained in the observations. This allows cross-sectional information, such as census data, to inform the generation of coherent longitudinal populations.

Bayesian filtering and smoothing for high-dimensional nonlinear dynamical systems are fundamental yet challenging problems in many areas of science and engineering. In this work, we propose FLUID, a flow-based unified amortized inference framework for filtering and smoothing dynamics. The core idea is to encode each observation history into a fixed-dimensional summary statistic and use this shared representation to learn both a forward flow for the filtering distribution and a backward flow for the backward transition kernel. Specifically, a recurrent encoder maps each observation history to a fixed-dimensional summary statistic whose dimension does not depend on the length of the time series. Conditioned on this shared summary statistic, the forward flow approximates the filtering distribution, while the backward flow approximates the backward transition kernel. The smoothing distribution over an entire trajectory is then recovered by combining the terminal filtering distribution with the learned backward flow through the standard backward recursion. By learning the underlying temporal evolution structure, FLUID also supports extrapolation beyond the training horizon. Moreover, by coupling the two flows through shared summary statistics, FLUID induces an implicit regularization across latent state trajectories and improves trajectory-level smoothing. In addition, we develop a flow-based particle filtering variant that provides an alternative filtering procedure and enables ESS-based diagnostics when explicit model factors are available. Numerical experiments demonstrate that FLUID provides accurate approximations of both filtering distributions and smoothing paths.

Reactive granular assemblies with interstitial flows are frequently described using a combination of the discrete element method (DEM) and unresolved computational fluid dynamics (CFD). A particular characteristic of unresolved CFD is that small-scale flow structures are eliminated by application of a spatial filter. On part of the composition of the interstitial gas, this leads to closure challenges that are currently resolved by assuming the local composition to coincide with its filtered counterpart. However, in the presence of intricate small-scale flow structures, gas-particle mass and heat exchanges or chemical reactions, the composition may vary on scales that are smaller than the filter width, rendering the assumption of small-scale homogeneity invalid. In order to address this physical shortcoming, we adopt a statistical description of small-scale heterogeneity and present a probabilistic framework that is based on the one-point probability density function (PDF) associated with the spatial distribution of the gas phase composition inside the local filter volume. An evolution equation for the PDF is derived and reduced to the case of large-scale homogeneity. By recasting the reduced PDF equation in terms of a stochastic process and applying a time integration method, we obtain a Monte Carlo-type solution scheme. The modeling and solution framework is applied to the heating of a packed particle bed by a burning methane-air mixture and the influence of the mixing intensity on predictions of the mean temperature and conversion rates is quantified. Our contribution not only highlights the relevance of small-scale compositional heterogeneity, but also provides the foundation for coupled DEM-PDF models of reactive multiphase flows.

Active Inference is an emerging framework providing a quantitative account of behavioral processes in neuroscience and a principled approach to decision-making under uncertainty. Its application to agency problems is natural, offering an autopoietic interpretation of action while addressing classical challenges such as the exploration-exploitation trade-off. Recently, Active Inference has been applied to digital twin scenarios for adaptive and predictive modeling of complex systems. In this work, we extend Active Inference to multi-agent digital twins in which agents interact within a shared environment while maintaining decentralized generative models. Our multi-agent framework features two innovations: (i) contextual inference to improve adaptability in dynamic environments, and (ii) the integration of streaming machine learning within agents’ generative structures, enabling tunable goal-oriented behavior while preserving efficiency and scalability. The framework is illustrated through a Cournot competition example, providing a digital twin representation of a socio-economic system and highlighting its potential for coordinated decision-making in multi-agent contexts.

Bayesian inference for models with intractable likelihoods requires balancing accuracy and computational cost. We propose an amortized MCMC-based approach that matches the accuracy of the exchange algorithm while significantly reducing computation. A gradient-informed grid selection combined with Hermite interpolation yields an accurate and efficient surrogate model, as demonstrated by application a Potts model.

Inverse problems for nonlinear diffusion equations aim to infer a system’s properties from limited observations of its evolution, rendering them an important tool in various fields such as geophysics and medical imaging. As these problems are inherently ill posed due to measurement noise, a major challenge is to develop robust and efficient methods for uncertainty quantification. We focus on parameter inference for the porous medium equation (PME) which is particularly challenging, being a nonlinear degenerate parabolic equation. To recover these parameters, observations may be available at discrete spatial locations over time which reflects realistic measurement constraints. We employ Bayesian inversion which allows us to incorporate measurement noise, as well as prior information into our model. In this setting, we examine the theoretical properties of the posterior, which represents the inverse problems solution, including the problem’s well-posedness. A main contribution is the reformulation of the inverse problem in a way that exploits the structure of the PME, leading to an auxiliary elliptic inverse problem that may be solved instead. This allows us to formulate an efficient numerical method for solving the inverse problem which has mainly been discussed in theory so far. We present numerical experiments, demonstrating the computational performance of the proposed method in practice.

We consider the problem of estimating rare event probabilities for stochastic reaction networks, a problem that is challenging for Monte Carlo (MC) methods due to high relative variance. We use the importance sampling (IS) technique to reduce the variance of the MC estimator. Finding the optimal IS parameters leads to a stochastic optimal control problem, which is prone to the curse of dimensionality and is infeasible to solve for multi-dimensional systems. In this work, we explore a low-rank tensor train approximation to estimate the optimal IS parameters.

We present a novel kernel-based method for learning multivariate stochastic differential equations (SDEs). The method follows a two-step procedure: we first estimate the drift term function, then the (matrix-valued) diffusion function given the drift. Occupation kernels are integral functionals on a reproducing kernel Hilbert space (RKHS) that aggregate information over a trajectory. Our approach leverages vector-valued occupation kernels for estimating the drift component of the stochastic process. For diffusion estimation, we extend this framework by introducing operator-valued occupation kernels, enabling the estimation of an auxiliary matrix valued function as a positive semi-definite operator, from which we readily derive the diffusion estimate.

We present a systematic analysis of estimation errors for a class of optimal transport based algorithms for filtering and data assimilation. Along the way, we extend previous error analyses of Brenier maps to the case of conditional Brenier maps that arise in the context of simulation based inference. We then apply these results in a filtering scenario to analyze the optimal transport filtering (OTF) algorithm. An extension of that algorithm along with numerical benchmarks on various non-Gaussian and high-dimensional examples are provided to demonstrate its effectiveness and practical potential.

The Shifted Proper Orthogonal Decomposition (Shifted POD) is a modal decomposition for the dimensionality reduction of solver data in transport-dominated problems. It takes the form of a sum of basis modes, each of which is transported by a certain amount. Most previous work in the literature assumes that the correct amount of transport is known a-priori. We discuss a fully non-intrusive approach for determining the Shifted POD representation of solver data, and proceed to interpolate it with Radial Basis Functions. Our contributions are threefold. First, we propose to determine the shifts alongside the modes by optimizing with respect to an objective that includes the Wasserstein distance, which, we argue, is more suited to align features than the L_2 norm. Second, we use Radial Basis Functions to interpolate the coefficients. Third, we use physics-based constraints to improve the convergence behavior. The resulting interpolator can be used as a surrogate model by itself or to regularize the Galerkin projection/residual minimization problem. Synthetic numerical examples and the reconstruction of data from an aerodynamics solver demonstrate the effectiveness of the proposed approach, with or without constraints.

For real-time monitoring, such as digital twins, we want to predict a system behavior on the fly. We therefore need a low complexity model as accurate as possible. Assuming availability of spatially sparse measurements, we can enhance these observations with a priori knowledge on the system. It leads to a two step approach involving a stochastic Reduced-Order Model (sROM) [4] and data assimilation [3]. The sROM learns a priori knowledge from high-fidelity simulations and physics equations through a low dimensional model. One can see it either as a generative model or as a predictor including uncertainty quantification. The data assimilation step combines the sROM prediction with the accessible measurements.

In this work, we focus on the hyperreduction part of the sROM. The core of the sROM is the dimensionality reduction that we perform with POD-Galerkin, where POD stands for Proper Orthogonal Decomposition. It mainly consists of projecting linearly the physics partial differential equation onto a linear reduced basis built through PCA (POD). However, there is no complexity reduction when applying this method directly to nonlinear terms of the original physical equation. Hyperreduction comes into play to preserve dimensionality reduction when projecting nonlinear terms.

There are two main types of hyperreduction method, referred to as the Discrete Empirical Interpolation Method (DEIM) [1] and Empirical Cubature [2]. Both methods are approximations, so neither gives an exact projection. The modeling of their errors to enhance the uncertainty quantification in the sROM is the main point of the presentation. We focus on DEIM, which interpolates the nonlinear terms from few local evaluations before projecting onto the PCA modes.

We show that the DEIM method is similar to Gaussian process regression (kriging) with a covariance built from an additional PCA learned on a training set of the nonlinear terms. This PCA suits well reduced models, also built from a training data set of high fidelity simulations. Similitude between DEIM and kriging is particularly helpful to quantify the hyperreduction uncertainty. It includes contributions from the additional PCA truncation and Bayesian quadrature. This second contribution results from the projection of the kriging approximation onto the PCA modes of the sROM.

We apply this methodology in the context of 3D unsteady turbulent fluid dynamics. The fields are statistically non-stationary in both space and time. Results focus on time extrapolation of the nonlinear terms inside the sROM. Quality of the deterministic hyperreduction prediction and its associated confidence interval are assessed.

REFERENCES

[1] Saifon Chaturantabut and Danny C. Sorensen. Nonlinear Model Reduction via Discrete Empirical Interpolation. SIAM Journal on Scientific Computing, 32(5):2737–2764, January 2010. Publisher: Society for Industrial and Applied Mathematics.

[2] J. A. Hern´andez, M. A. Caicedo, and A. Ferrer. Dimensional hyper-reduction of nonlinear finite element models via empirical cubature. Computer Methods in Applied Mechanics and Engineering, 313:687–722, January 2017.

[3] Valentin Resseguier, Matheus Ladvig, and Dominique Heitz. Real-time estimation and prediction of unsteady flows using reduced-order models coupled with few measurements. Journal of Computational Physics, 471:111631, 2022.

[4] Valentin Resseguier, Agustin M. Picard, Etienne Memin, and Bertrand Chapron. Quantifying TruncationRelated Uncertainties in Unsteady Fluid Dynamics Reduced Order Models. SIAM/ASA Journal on Uncertainty Quantification, 9(3):1152–1183, January 2021.

We study lower bounds for estimating the lengthscale of a Gaussian kernel in Gaussian process interpolation using maximum likelihood and cross-validation. For several classes of smooth functions, we show that the estimated lengthscale grows with the number of observations. These results describe the asymptotic behavior of the estimators and are supported by numerical experiments.

We present Regenerative Rejection Sampling (RRS), a novel approximate sampling method inspired by classical Rejection Sampling and Markov Chain Monte Carlo (MCMC) methods. The method constructs a continuous time regenerative process whose stationary distribution coincides with a target density which is known only up to a multiplicative constant. Unlike standard Rejection Sampling, RRS does not require the existence of a finite constant that upper-bounds the likelihood ratio. In this work, we introduce the RRS algorithm and derive its convergence rate. We compare its performance with state-of-the-art MCMC methods. A numerical experiment of a Bayesian logistic regression performed on a medical dataset demonstrates that RRS can exhibit lower autocorrelations and faster effective mixing. Moreover, we show that the bias of the time average estimator constructed from the RRS method is of order $O(1/t^2)$, not of the usual MCMC order of $O(1/t)$, and provide an easy-to-estimate non-asymptotic error bound for this bias.

When dealing with parameter-dependent partial differential equations (PDEs), classical numerical methods lead to costly numerical simulations. One usually relies on model order reduction to approximate the original problem by a reduced one which we can solve more efficiently. It becomes possible to fastly compute a solution for a given input (parameters, initial or boundary conditions) in real time, or to compute a bunch of solutions in a many-query scenario or uncertainty quantification.

Different reduction methods have been developed for time-dependent problems such as the Proper Orthogonal Decomposition (POD) [1, 3]. The goal is to construct a reduced basis of an approximation space that captures the essential dynamic of the problem. However, POD needs to precompute the global solution trajectory in order to build such a reduced space. This means having to solve sequentially in time (at least for some part of the global time interval) before projecting the dynamic onto a reduced space.

An alternative has been developed in [4] where it has been proposed to construct the reduced basis in a parallel manner in time. It consists in solving the full dimensional problem on small chunks of the global time interval.

The objective is to construct approximation spaces capturing the action of the transfer operator mapping initial condition to final state on a time interval. For linear PDE operators, learning the range of the affine transfer operators can be split into two tasks, the approximation of solutions on subintervals with zero initial conditions, or with zero source term. The latter task is related to the approximation of the range of a linear operator, for which we rely on randomized numerical linear algebra methods (RNLA) (range finder or random probes [2]), that translates into the resolution of the equation with random initial conditions. Using random initial conditions on each sub-interval, the local trajectories are completely independent from each other, hence allowing parallelization.

The main interest of such an approach is to capture local in time features of the problem and construct relevant reduced spaces that can be used afterwards for an efficient resolution over the whole time interval.

In this work, we extend the approach from [4] to the resolution of evolution problems in tensor spaces (e.g. resulting from the discretization of high-dimensional PDEs or stochastic PDEs) and rely on RNLA for tensors and dynamical low-rank methods for the resolution of dynamical systems in tensor spaces.

We will present numerical examples for parabolic PDEs.

REFERENCES

[1] G. Berkooz, P. Holmes, and J. L. Lumley. The proper orthogonal decomposition in the analysis of turbulent flows. Annual Review of Fluid Mechanics, 25:539–575, 1993.

[2] N. Halko, P-G. Martinsson, and J. A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, 2010.

[3] K. Kunisch and S. Volkwein. Galerkin proper orthogonal decomposition methods for parabolic problems. Numerische Mathematik, 90(1):117–148, 2001.

[4] J. Schleuß, K. Smetana, and L. ter Maat. Randomized quasi-optimal local approximation spaces in time. SIAM J. Sci. Comput., 45(3):1066–1096, 2023.