arXiv subjects

Olga Mula

Publications and source records attributed to Olga Mula.

At least 19 recordsLinked to original sources

arXiv2026

Reduced Order Modeling and Applications to Inverse State Estimation

Solving parametric partial differential equations (PDEs) repeatedly for many parameter values,as needed in optimal control, uncertainty quantification, and inverse problems, is often prohibitive with classical discretizations. This chapter introduces Reduced Order Modeling (ROM), which builds compressed yet accurate representations of parametric solution sets for fast online evaluation. We formulate the approximation of parametric PDEs as a supervised learning task and review linear approximation, very effective for elliptic and parabolic problems, before turning to nonlinear methods needed when solutions exhibit discontinuities or steep gradients. We then show how to use reduced-order models to efficiently recover the state of a physical system from limited measurements, which is an inverse problem known as state estimation. Throughout, we favor results with theoretical guarantees while also discussing practical methods, and provide a companion Jupyter notebook implementing the main algorithms.

math.NA↗

arXiv2026

A Dynamical Approximation Scheme on the Stiefel manifold for Wasserstein Gradient Flows

We propose a meshless Lagrangian dynamical method for approximating Wasserstein gradient flows (WGFs). The evolving measure is represented as the pushforward of the initial measure $\mu_0$ through a transport map in the weighted Hilbert space $L^2_{\mu_0}$. We approximate this map in time-dependent linear subspaces of $L^2_{\mu_0}$, whose orthonormal frames are evolved by a Dirac--Frenkel dynamical principle on a Stiefel manifold constrained to a finite-dimensional background space, adaptively constructed via local approximations of the WGF velocity field. We prove that the resulting transport map induces an absolutely continuous curve of probability measures in Wasserstein space, whose velocity is obtained by projecting the exact WGF velocity onto the background space, and we show that the approximation preserves the energy dissipation structure up to the projection error of the velocity. Moreover, for geodesically convex energies, we derive an a posteriori estimate controlling such projection error through the adaptive construction of the background space, yielding as well a bound on the approximation error of the pushforward measure in the Wasserstein metric. Numerical experiments on linear and nonlinear Fokker--Planck equations, porous-medium diffusion, and interaction energies demonstrate the accuracy of the method, its energy-dissipation properties, and the advantages of the adaptive construction.

math.NA↗

arXiv2025

Stable Nonlinear Dynamical Approximation with Dynamical Sampling

We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and analyzing stability and accuracy of nonlinear dynamical approximations. The parameters of these functions are evolved in time by means of projections on finite dimensional subspaces of an ambient Hilbert space related to the PDE evolution. For practical computations of these projections, one usually needs to sample. We propose a dynamical sampling strategy which comes with stability guarantees, while keeping a low numerical complexity. We show the effectiveness of the method on several examples in moderate spatial dimension.

math.NA↗

arXiv2025

Leveraging Christoffel-Darboux Kernels to Strengthen Moment-SOS Relaxations

The classical Moment-Sum Of Squares hierarchy allows to approximate a global minimum of a polynomial optimization problem through semidefinite relaxations of increasing size. However, for many optimization instances, solving higher order relaxations becomes impractical or even impossible due to the substantial computational demands they impose. To address this, existing methods often exploit intrinsic problem properties, such as symmetries or sparsity. Here, we present a complementary approach, which enhances the accuracy of computationally more efficient low-order relaxations by leveraging Christoffel-Darboux kernels. Such strengthened relaxations often yield significantly improved bounds or even facilitate minimizer extraction. We illustrate the efficiency of our approach on several classes of important quadratically constrained quadratic Programs.

math.OC↗

arXiv2024

Towards a Real-Time Simulation of Elastoplastic Deformation Using Multi-Task Neural Networks

This study introduces a surrogate modeling framework merging proper orthogonal decomposition, long short-term memory networks, and multi-task learning, to accurately predict elastoplastic deformations in real-time. Superior to single-task neural networks, this approach achieves a mean absolute error below 0.40\% across various state variables, with the multi-task model showing enhanced generalization by mitigating overfitting through shared layers. Moreover, in our use cases, a pre-trained multi-task model can effectively train additional variables with as few as 20 samples, demonstrating its deep understanding of complex scenarios. This is notably efficient compared to single-task models, which typically require around 100 samples. Significantly faster than traditional finite element analysis, our model accelerates computations by approximately a million times, making it a substantial advancement for real-time predictive modeling in engineering applications. While it necessitates further testing on more intricate models, this framework shows substantial promise in elevating both efficiency and accuracy in engineering applications, particularly for real-time scenarios.

cs.CE↗

arXiv2024

Connected Components on Lie Groups and Applications to Multi-Orientation Image Analysis

We develop and analyze a new algorithm to find the connected components of a compact set $I$ from a Lie group $G$ endowed with a left-invariant Riemannian distance. For a given $\delta>0$, the algorithm finds the largest cover of $I$ such that all sets in the cover are separated by at least distance $\delta$. We call the sets in the cover the $\delta$-connected components of I (closely related to $\check{\text{C}}$ech complexes of radius $\delta/2$). The grouping relies on an iterative procedure involving morphological dilations with Hamilton-Jacobi-Bellman kernels on $G$ and notions of $\delta$-thickened sets. We prove that the algorithm converges in finitely many iteration steps. We find the optimal value for $\delta$ using persistence diagrams. We also propose specific affinity matrices that allow for grouping of $\delta$-connected components based on their local proximity and alignment. Among the many different applications of the algorithm, in this article, we focus on illustrating that the method can efficiently identify (possibly overlapping) branches in complex vascular trees on retinal images. This is done by applying an orientation score transform to the images that allows us to view them as functions from $\mathbb{L}_2(G)$ where $G=SE(2)$, the Lie group of roto-translations. By applying our algorithm in this Lie group, we illustrate that we obtain $\delta$-connected components that differentiate between crossing structures and that group well-aligned, nearby structures. This contrasts standard connected component algorithms in $\mathbb{R}^2$.

math.DG↗

arXiv2024

Predicting Cascading Failures with a Hyperparametric Diffusion Model

In this paper, we study cascading failures in power grids through the lens of information diffusion models. Similar to the spread of rumors or influence in an online social network, it has been observed that failures (outages) in a power grid can spread contagiously, driven by viral spread mechanisms. We employ a stochastic diffusion model that is Markovian (memoryless) and local (the activation of one node, i.e., transmission line, can only be caused by its neighbors). Our model integrates viral diffusion principles with physics-based concepts, by correlating the diffusion weights (contagion probabilities between transmission lines) with the hyperparametric Information Cascades (IC) model. We show that this diffusion model can be learned from traces of cascading failures, enabling accurate modeling and prediction of failure propagation. This approach facilitates actionable information through well-understood and efficient graph analysis methods and graph diffusion simulations. Furthermore, by leveraging the hyperparametric model, we can predict diffusion and mitigate the risks of cascading failures even in unseen grid configurations, whereas existing methods falter due to a lack of training data. Extensive experiments based on a benchmark power grid and simulations therein show that our approach effectively captures the failure diffusion phenomena and guides decisions to strengthen the grid, reducing the risk of large-scale cascading failures. Additionally, we characterize our model's sample complexity, improving upon the existing bound.

cs.SI↗

arXiv2024

Tree-Based Nonlinear Reduced Modeling

This paper is concerned with model order reduction of parametric Partial Differential Equations (PDEs) using tree-based library approximations. Classical approaches are formulated for PDEs on Hilbert spaces and involve one single linear space to approximate the set of PDE solutions. Here, we develop reduced models relying on a collection of linear or nonlinear approximation spaces called a library, and which can also be formulated on general metric spaces. To build the spaces of the library, we rely on greedy algorithms involving different splitting strategies which lead to a hierarchical tree-based representation. We illustrate through numerical examples that the proposed strategies have a much wider range of applicability in terms of the parametric PDEs that can successfully be addressed. While the classical approach is very efficient for elliptic problems with strong coercivity, we show that the tree-based library approaches can deal with diffusion problems with weak coercivity, convection-diffusion problems, and with transport-dominated PDEs posed on general metric spaces such as the $L^2$-Wasserstein space.

math.NA↗

arXiv2024

Optimal Transport on the Lie Group of Roto-translations

The roto-translation group SE2 has been of active interest in image analysis due to methods that lift the image data to multi-orientation representations defined on this Lie group. This has led to impactful applications of crossing-preserving flows for image de-noising, geodesic tracking, and roto-translation equivariant deep learning. In this paper, we develop a computational framework for optimal transportation over Lie groups, with a special focus on SE2. We make several theoretical contributions (generalizable to matrix Lie groups) such as the non-optimality of group actions as transport maps, invariance and equivariance of optimal transport, and the quality of the entropic-regularized optimal transport plan using geodesic distance approximations. We develop a Sinkhorn like algorithm that can be efficiently implemented using fast and accurate distance approximations of the Lie group and GPU-friendly group convolutions. We report valuable advancements in the experiments on 1) image barycentric interpolation, 2) interpolation of planar orientation fields, and 3) Wasserstein gradient flows on SE2. We observe that our framework of lifting images to SE2 and optimal transport with left-invariant anisotropic metrics leads to equivariant transport along dominant contours and salient line structures in the image. This yields sharper and more meaningful interpolations compared to their counterparts on R^2

cs.CV↗

arXiv2024

State estimation of urban air pollution with statistical, physical, and super-learning graph models

We consider the problem of real-time reconstruction of urban air pollution maps. The task is challenging due to the heterogeneous sources of available data, the scarcity of direct measurements, the presence of noise, and the large surfaces that need to be considered. In this work, we introduce different reconstruction methods based on posing the problem on city graphs. Our strategies can be classified as fully data-driven, physics-driven, or hybrid, and we combine them with super-learning models. The performance of the methods is tested in the case of the inner city of Paris, France.

cs.LG↗

arXiv2024

High order recovery of geometric interfaces from cell-average data

We consider the problem of recovering characteristic functions $u:=\chi_\Omega$ from cell-average data on a coarse grid, and where $\Omega$ is a compact set of $\mathbb{R}^d$. This task arises in very different contexts such as image processing, inverse problems, and the accurate treatment of interfaces in finite volume schemes. While linear recovery methods are known to perform poorly, nonlinear strategies based on local reconstructions of the jump interface $\Gamma:=\partial\Omega$ by geometrically simpler interfaces may offer significant improvements. We study two main families of local reconstruction schemes, the first one based on nonlinear least-squares fitting, the second one based on the explicit computation of a polynomial-shaped curve fitting the data, which yields simpler numerical computations and high order geometric fitting. For each of them, we derive a general theoretical framework which allows us to control the recovery error by the error of best approximation up to a fixed multiplicative constant. Numerical tests in 2d illustrate the expected approximation order of these strategies. Several extensions are discussed, in particular the treatment of piecewise smooth interfaces with corners.

math.NA↗

arXiv2024

A Dynamical Neural Galerkin Scheme for Filtering Problems

This paper considers the filtering problem which consists in reconstructing the state of a dynamical system with partial observations coming from sensor measurements, and the knowledge that the dynamics are governed by a physical PDE model with unknown parameters. We present a filtering algorithm where the reconstruction of the dynamics is done with neural network approximations whose weights are dynamically updated using observational data. In addition to the estimate of the state, we also obtain time-dependent parameter estimations of the PDE parameters governing the observed evolution. We illustrate the behavior of the method in a one-dimensional KdV equation involving the transport of solutions with local support. Our numerical investigation reveals the importance of the location and number of the observations. In particular, it suggests to consider dynamical sensor placement.

math.NA↗

arXiv2023

Dynamical approximation and sensor placement for filtering problems

We consider the inverse problem of reconstructing an unknown function $u$ from a finite set of measurements, under the assumption that $u$ is the trajectory of a transport-dominated problem with unknown input parameters. We propose an algorithm based on the Parameterized Background Data-Weak method (PBDW) where dynamical sensor placement is combined with approximation spaces that evolve in time. We prove that the method ensures an accurate reconstruction at all times and allows to incorporate relevant physical properties in the reconstructed solutions by suitably evolving the dynamical approximation space. As an application of this strategy we consider Hamiltonian systems modeling wave-type phenomena, where preservation of the geometric structure of the flow plays a crucial role in the accuracy and stability of the reconstructed trajectory.

math.NA↗

arXiv2023

Accuracy Controlled Schemes for the Eigenvalue Problem of the Radiative Transfer Equation

The criticality problem in nuclear engineering asks for the principal eigenpair of a Boltzmann operator describing neutron transport in a reactor core. Being able to reliably design, and control such reactors requires assessing these quantities within quantifiable accuracy tolerances. In this paper we propose a paradigm that deviates from the common practice of approximately solving the corresponding spectral problem with a fixed, presumably sufficiently fine discretization. Instead, the present approach is based on first contriving iterative schemes, formulated in function space, that are shown to converge at a quantitative rate without assuming any a priori excess regularity properties, and that exploit only properties of the optical parameters in the underlying radiative transfer model. We develop the analytical and numerical tools for approximately realizing each iteration step within judiciously chosen accuracy tolerances, verified by a posteriori estimates, so as to still warrant quantifiable convergence to the exact eigenpair. This is carried out in full first for a Newton scheme. Since this is only locally convergent we analyze in addition the convergence of a power iteration in function space to produce sufficiently accurate initial guesses. Here we have to deal with intrinsic difficulties posed by compact but unsymmetric operators preventing standard arguments used in the finite dimensional case. Our main point is that we can avoid any condition on an initial guess to be already in a small neighborhood of the exact solution. We close with a discussion of remaining intrinsic obstructions to a certifiable numerical implementation, mainly related to not knowing the gap between the principal eigenvalue and the next smaller one in modulus.

math.NA↗

arXiv2023

Approximation and Structured Prediction with Sparse Wasserstein Barycenters

We develop a general theoretical and algorithmic framework for sparse approximation and structured prediction in $\mathcal{P}_2(Ω)$ with Wasserstein barycenters. The barycenters are sparse in the sense that they are computed from an available dictionary of measures but the approximations only involve a reduced number of atoms. We show that the best reconstruction from the class of sparse barycenters is characterized by a notion of best $n$-term barycenter which we introduce, and which can be understood as a natural extension of the classical concept of best $n$-term approximation in Banach spaces. We show that the best $n$-term barycenter is the minimizer of a highly non-convex, bi-level optimization problem, and we develop algorithmic strategies for practical numerical computation. We next leverage this approximation tool to build interpolation strategies that involve a reduced computational cost, and that can be used for structured prediction, and metamodelling of parametrized families of measures. We illustrate the potential of the method through the specific problem of Model Order Reduction (MOR) of parametrized PDEs. Since our approach is sparse, adaptive and preserves mass by construction, it has potential to overcome known bottlenecks of classical linear methods in hyperbolic conservation laws transporting discontinuities. It also paves the way towards MOR for measure-valued PDE problems such as gradient flows.

math.NA↗

arXiv2022

Moment-SoS Methods for Optimal Transport Problems

Most common Optimal Transport (OT) solvers are currently based on an approximation of underlying measures by discrete measures. However, it is sometimes relevant to work only with moments of measures instead of the measure itself, and many common OT problems can be formulated as moment problems (the most relevant examples being $L^p$-Wasserstein distances, barycenters, and Gromov-Wasserstein discrepancies on Euclidean spaces). We leverage this fact to develop a generalized moment formulation that covers these classes of OT problems. The transport plan is represented through its moments on a given basis, and the marginal constraints are expressed in terms of moment constraints. A practical computation then consists in considering a truncation of the involved moment sequences up to a certain order, and using the polynomial sums-of-squares hierarchy for measures supported on semi-algebraic sets. We prove that the strategy converges to the solution of the OT problem as the order increases. We also show how to approximate linear quantities of interest, and how to estimate the support of the optimal transport map from the computed moments using Christoffel-Darboux kernels. Numerical experiments illustrate the good behavior of the approach.

math.NA↗

arXiv2022

Nonlinear approximation spaces for inverse problems

This paper is concerned with the ubiquitous inverse problem of recovering an unknown function u from finitely many measurements possibly affected by noise. In recent years, inversion methods based on linear approximation spaces were introduced in [MPPY15, BCDDPW17] with certified recovery bounds. It is however known that linear spaces become ineffective for approximating simple and relevant families of functions, such as piecewise smooth functions that typically occur in hyperbolic PDEs (shocks) or images (edges). For such families, nonlinear spaces [Devore98] are known to significantly improve the approximation performance. The first contribution of this paper is to provide with certified recovery bounds for inversion procedures based on nonlinear approximation spaces. The second contribution is the application of this framework to the recovery of general bidimensional shapes from cell-average data. We also discuss how the application of our results to n-term approximation relates to classical results in compressed sensing.

math.NA↗

arXiv2022

Waserstein model reduction approach for parametrized flow problems in porous media

The aim of this work is to build a reduced-order model for parametrized porous media equations. The main challenge of this type of problems is that the Kolmogorov width of the solution manifold typically decays quite slowly and thus makes usual linear model-order reduction methods inappropriate. In this work, we investigate an adaptation of the methodology proposed in a previous work, based on the use of Wasserstein barycenters, to the case of non-conservative problems. Numerical examples in one-dimensional test cases illustrate the advantages and limitations of this approach and suggest further research directions that we intend to explore in the future.

math.NA↗