Uncertainty Quantification and parameter estimation in time dependent systems of parametric PDEs are challenging as the computational burden to solve them is often prohibitive. In this talk we will present some contributions on low-rank tensor methods to provide a numerical approximation of parametric PDEs solutions. In the first part of the talk, we will discuss the set up of multilinear GMRES methods and several ways to precondition them. In the second part of the talk, we will detail how we could use low-rank tensor approximations to parsimoniously discretise a sequential Bayesian parameter estimation. On top of a low-rank multi-linear system solver, an interpolation step is needed. We will discuss how the low-rank tensor approximation can be used to define a projection based interpolator. The interplay between the resolution and interpolation steps contributes to alleviate the memory burden and provides a substantial acceleration of the Bayesian filter computation. This enables the estimation of the a posteriori density of the parameters and the Uncertainty Quantification of the quantities of interest. Several examples will be presented, ranging from the estimation of the boundary conditions in a parabolic problem, to a more challenging 3D fluid-structure interaction problem. We will conclude by presenting some perspectives on possible ways to go beyond tensor decompositions.

The Stochastic Landau-Lifshitz-Gilbert equation (SLLG) is a nonlinear stochastic PDE model for a magnetic body in a heat bath. The strong nonlinearities and the “roughness” of the stochastic component make its numerical approximation particularly challenging. We present a novel methodology to reduce this stochastic PDE to a parametric coefficient PDE. The high dimensionality of the resulting parameter-to-solution map precludes the use of any scheme affected by the curse of dimensionality. To tackle this issue, we prove the existence of a holomorphic extension of the parameter-to-solution map and quantify the size of its domain. Based on this information, some schemes may circumvent the Curse of Dimensionality. We consider: Sparse grid interpolation: A high-dimensional interpolation method that allows fine-tuning the number of collocation nodes assigned to each parameter. When tuned well, it achieves the approximation error of tensor-product interpolation with dramatically fewer nodes. POD (Proper Orthogonal Decomposition) reduced basis: We compute a Singular Value Decomposition (SVD) of appropriate samples of the SLLG dynamics. The resulting basis of the solution manifold provides the same approximation quality of a nodal finite elements' basis with fewer functions. This in turn allows faster Galerkin approximation. The presentation is based on the publication [An, Dick, Feischl, S, Tran. 2025] and ongoing joint work with Fernando Henriquez and Michael Feischl.

Many PDE applications exhibit hidden low-dimensional structure in some form, allowing for storage and computations in terms of relatively few degrees of freedom. Low-rank tensor decompositions are low-parametric representations relying on well-established techniques of numerical linear algebra and optimization in adaptively capturing the multivariate or multilevel structure in data and solutions. In particular, the multilevel version of the tensor network proposed under the names of «matrix product states» (MPS) in computational quantum physics and «tensor train» (TT) in computational mathematics allows for generic but extravagantly large discretizations and leads to data-driven computations based on effective discretizations adapted to the problem class and to the data instead of problem-dependent discretizations designed analytically. This approach has been shown, both theoretically and experimentally, to lead to the efficient approximation of algebraic singularities, boundary layers and high-frequency oscillations arising in multiscale diffusion problems, achieving root-exponential convergence with respect to the total number of representation parameters. In this talk, we will discuss the recently proposed low-rank multilevel frame representation of functions, with a focus on the stability of the representation, on the quasi-optimality of the associated approximation algorithms and on the application of the representation for the numerical solution of PDE problems, where the low-rank adaptivity serves to resolve multiscale structure.

This talk begins with novel results on classical time-marching schemes for parabolic problems, including an improved variant of the implicit Euler scheme and interpolation operators tailored to relevant norms. Motivated by the challenge of highly singular solutions, we then explore schemes that enable local mesh refinements in space-time. After examining the limitations of approaches in the standard $L^2H^1_0 \cap H^1H^{-1}$ norm, we introduce a novel variational formulation based on fractional norms. Within this framework, we present a state-of-the-art minimal residual method in dual norms, which yields quasi-optimal approximations. The resulting discretized problem is then efficiently solved using a conjugate gradient method with a (nearly) optimal preconditioner.

Solving the Schrödinger evolution equation numerically poses significant challenges due to the inherent high dimensionality of the solution space. This presentation introduces a novel reformulation of the evolution problem as the minimization of a quadratic space-time functional. This approach enables the computation of solutions using nonlinear ansatz functions. We demonstrate the effectiveness of our method through numerical experiments employing specific ansatz, such as low-rank formats and free Gaussian wave packets.

We review recent work on h, hp and adaptive space-time methods for wave equations in polyhedral domains, formulated as an equivalent (weakly) coercive boundary integral equation. First, for smooth data solutions exhibit singularities at edges and corners. Based on an analysis of the singular behavior of the solution near nonsmooth boundary points, graded meshes are shown to recover quasioptimal approximation rates for the h version. More generally, we discuss the error analysis of p and hp versions on quasi-uniform meshes. As in the time-independent case, the p version converges at twice the rate of the h version. Second, we discuss space-time adaptive mesh refinement procedures. A reliable a posteriori error estimate of residual type is obtained. The error estimate leads to a space-time adaptive Galerkin boundary element method, based on the four steps: Solve - Estimate - Mark - Refine. Numerical experiments are presented, which confirm the theoretical results and show the efficiency of the proposed approaches.

We consider space-time conforming Galerkin discretizations of the acoustic wave equation. Unlike time-stepping methods, the time variable here is treated as an additional dimension. Traditional conforming second-order-in-time space–time discretizations typically require a CFL condition for stability. Recent works by O. Steinbach and M. Zank (2018), and S. Fraschini, G. Loli, A. Moiola, and G. Sangalli (2024) have introduced unconditionally stable schemes using maximal regularity splines in time, incorporating non-consistent penalty terms in the bilinear forms. While stability and error analyses have been rigorously established for low-order spaces, a full theoretical framework for higher-order methods is still lacking, despite their promising numerical results. Our goal is to design numerical schemes that are unconditionally stable, quasi-optimally convergent, and suitable for efficient implementation. Ideally, we aim for a formulation that guarantees these properties under minimal assumptions on the discrete spaces, allowing for broad applicability. In particular, we are interested in methods that go beyond piecewise continuous polynomials, extending also to spline functions of arbitrary regularity. We show that a conforming first-order-in-time formulation, in contrast with second-order-in-time formulations, is unconditionally stable without the need for stabilization terms. The introduction of exponential weights in the $L^2$-scalar products simplifies the analysis, allowing for variational proofs that yield error estimates. This talk is based on joint works with S. Fraschini, G. Loli, I. Perugia and E. Zampa.