math.AP

arXiv:2503.22297v1 Announce Type: new Abstract: We consider second-order PDE problems set in unbounded domains and discretized by Lagrange finite elements on a finite mesh, thus introducing an artificial boundary in the discretization. Specifically, we consider the reaction diffusion equation as well as Helmholtz problems in waveguides with perfectly matched layers. The usual procedure to deal with such problems is to first consider a modeling error due to the introduction of the artificial boundary, and estimate the remaining discretization error with a standard a posteriori technique. A shortcoming of this method, however, is that it is typically hard to obtain sharp bounds on the modeling error. In this work, we propose a new technique that allows to control the whole error by an a posteriori error estimator. Specifically, we propose a flux-equilibrated estimator that is slightly modified to handle the truncation boundary. For the reaction diffusion equation, we obtain fully-computable guaranteed error bounds, and the estimator is locally efficient and polynomial-degree-robust provided that the elements touching the truncation boundary are not too refined. This last condition may be seen as an extension of the notion of shape-regularity of the mesh, and does not prevent the design of efficient adaptive algorithms. For the Helmholtz problem, as usual, these statements remain valid if the mesh is sufficiently refined. Our theoretical findings are completed with numerical examples which indicate that the estimator is suited to drive optimal adaptive mesh refinements.

arXiv:2503.22621v1 Announce Type: new Abstract: We prove optimal convergence rates for certain low-regularity integrators applied to the one-dimensional periodic nonlinear Schr\"odinger and wave equations under the assumption of $H^1$ solutions. For the Schr\"odinger equation we analyze the exponential-type scheme proposed by Ostermann and Schratz in 2018, whereas in the wave case we treat the corrected Lie splitting proposed by Li, Schratz, and Zivcovich in 2023. We show that the integrators converge with their full order of one and two, respectively. In this situation only fractional convergence rates were previously known. The crucial ingredients in the proofs are known space-time bounds for the solutions to the corresponding linear problems. More precisely, in the Schr\"odinger case we use the $L^4$ Strichartz inequality, and for the wave equation a null form estimate. To our knowledge, this is the first time that a null form estimate is exploited in numerical analysis. We apply the estimates for continuous time, thus avoiding potential losses resulting from discrete-time estimates.

arXiv:2407.09301v3 Announce Type: replace-cross Abstract: We prove non asymptotic total variation estimates for the kinetic Langevin algorithm in high dimension when the target measure satisfies a Poincar\'e inequality and has gradient Lipschitz potential. The main point is that the estimate improves significantly upon the corresponding bound for the non kinetic version of the algorithm, due to Dalalyan. In particular the dimension dependence drops from $O(n)$ to $O(\sqrt n)$.

arXiv:2503.10487v1 Announce Type: new Abstract: In this work, we contribute to the broader understanding of inverse problems by introducing a versatile multiscale modeling framework tailored to the challenges of sediment concentration estimation. Specifically, we propose a novel approach for sediment concentration measurement in water flow, modeled as a multiscale inverse medium problem. To address the multiscale nature of the sediment distribution, we treat it as an inhomogeneous random field and use the homogenization theory in deriving the effective medium model. The inverse problem is formulated as the reconstruction of the effective medium model, specifically, the sediment concentration, from partial boundary measurements. Additionally, we develop numerical algorithms to improve the efficiency and accuracy of solving this inverse problem. Our numerical experiments demonstrate the effectiveness of the proposed model and methods in producing accurate sediment concentration estimates, offering new insights into sediment concentration measurement in complex environments.

arXiv:2503.05267v1 Announce Type: cross Abstract: Parabolic equations on evolving domains model a multitude of applications including various industrial processes such as the molding of heated materials. Such equations are numerically challenging as they require large-scale computations and the usage of parallel hardware. Domain decomposition is a common choice of numerical method for stationary domains, as it gives rise to parallel discretizations. In this study, we introduce a variational framework that extends the use of such methods to evolving domains. In particular, we prove that transmission problems on evolving domains are well posed and equivalent to the corresponding parabolic problems. This in turn implies that the standard non-overlapping domain decompositions, including the Robin-Robin method, become well defined approximations. Furthermore, we prove the convergence of the Robin--Robin method. The framework is based on a generalization of fractional Sobolev-Bochner spaces on evolving domains, time-dependent Steklov-Poincar\'e operators, and elements of the approximation theory for monotone maps.

arXiv:2408.15858v2 Announce Type: replace-cross Abstract: In this article, we study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set $\Omega\subset \mathbb{R}^d$, in dimension $d\geq 2$. More precisely, we consider the simple random walk on $\mathbb{Z}^d$ killed upon exiting the (large) bounded domain $\Omega_N = (N\Omega)\cap \mathbb{Z}^d$. We denote by $P_N$ the corresponding transition matrix and we study the properties of its ($L^2$-normalized) principal eigenvector $\phi_N$. With probabilistic arguments (namely gambler's ruin estimates and coupling techniques) and under mild assumptions on the domain, we are able to give regularity estimates on $\phi_N$ (namely on its $k$-th order differences), with a uniform control inside $\Omega_N$. The same holds true for the first eigenfunction $\varphi_1$ of the corresponding continuous spectral Dirichlet problem, this being related to a Brownian motion killed upon exiting $\Omega$. As corollaries, we derive the $L^2$ and uniform convergence of $\phi_N$ to $\varphi_1$ in Lipschitz bounded domains.

arXiv:2503.05437v1 Announce Type: new Abstract: To verify theoretical results it is sometimes important to use a numerical example where the solution has a particular regularity. The paper describes one approach to construct such examples. It is based on the regularity theory for elliptic boundary value problems.

arXiv:2501.12279v1 Announce Type: cross Abstract: We analyze the robustness of optimally controlled evolution equations with respect to spatially localized perturbations. We prove that if the involved operators are domain-uniformly stabilizable and detectable, then these localized perturbations only have a local effect on the optimal solution. We characterize this domain-uniform stabilizability and detectability for the transport equation with constant transport velocity, showing that even for unitary semigroups, optimality implies exponential damping. Finally, we extend our result to the case of a space-dependent transport velocity. Numerical examples in one space dimension complement the theoretical results.

arXiv:2501.11944v1 Announce Type: new Abstract: In this work, we establish that discontinuous Galerkin methods are capable of producing reliable approximations for a broad class of nonlinear variational problems. In particular, we demonstrate that these schemes provide essential flexibility by removing inter-element continuity while also guaranteeing convergent approximations in the quasiconvex case. Notably, quasiconvexity is the weakest form of convexity pertinent to elasticity. Furthermore, we show that in the non-convex case discrete minimisers converge to minimisers of the relaxed problem. In this case, the minimisation problem corresponds to the energy defined by the quasiconvex envelope of the original energy. Our approach covers all discontinuous Galerkin formulations known to converge for convex energies. This work addresses an open challenge in the vectorial calculus of variations: developing and rigorously justifying numerical schemes capable of reliably approximating nonlinear energy minimization problems with potentially singular solutions, which are frequently encountered in materials science.

arXiv:2501.12116v1 Announce Type: new Abstract: We present a machine learning framework to facilitate the solution of nonlinear multiscale differential equations and, especially, inverse problems using Physics-Informed Neural Networks (PINNs). This framework is based on what is called multihead (MH) training, which involves training the network to learn a general space of all solutions for a given set of equations with certain variability, rather than learning a specific solution of the system. This setup is used with a second novel technique that we call Unimodular Regularization (UR) of the latent space of solutions. We show that the multihead approach, combined with the regularization, significantly improves the efficiency of PINNs by facilitating the transfer learning process thereby enabling the finding of solutions for nonlinear, coupled, and multiscale differential equations.

arXiv:2310.15581v4 Announce Type: replace Abstract: In this paper we consider PIDEs with gradient-independent Lipschitz continuous nonlinearities and prove that deep neural networks with ReLU activation function can approximate solutions of such semilinear PIDEs without curse of dimensionality in the sense that the required number of parameters in the deep neural networks increases at most polynomially in both the dimension $ d $ of the corresponding PIDE and the reciprocal of the prescribed accuracy $\epsilon $.

arXiv:2401.14894v2 Announce Type: replace Abstract: This paper is focused on the convergence analysis of an adaptive stochastic collocation algorithm for the stationary diffusion equation with parametric coefficient. The algorithm employs sparse grid collocation in the parameter domain alongside finite element approximations in the spatial domain, and adaptivity is driven by recently proposed parametric and spatial a posteriori error indicators. We prove that for a general diffusion coefficient with finite-dimensional parametrization, the algorithm drives the underlying error estimates to zero. Thus, our analysis covers problems with affine and nonaffine parametric coefficient dependence.

arXiv:2402.00406v2 Announce Type: replace-cross Abstract: In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular domains via semigroup theory and computer-assisted proofs. Once a numerical candidate for the solution is obtained via a finite dimensional projection, Chebyshev series expansions are used to solve the linearized equations about the approximation from which a solution map operator is constructed. Using the solution operator (which exists from semigroup theory), we define an infinite dimensional contraction operator whose unique fixed point together with its rigorous bounds provide the local inclusion of the solution. Applying this technique for multiple time steps leads to constructive proofs of existence of solutions over long time intervals. As applications, we study the 3D/2D Swift-Hohenberg, where we combine our method with explicit constructions of trapping regions to prove global existence of solutions of initial value problems converging asymptotically to nontrivial equilibria. A second application consists of the 2D Ohta-Kawasaki equation, providing a framework for handling derivatives in nonlinear terms.

arXiv:2409.12483v2 Announce Type: replace-cross Abstract: Numerical methods of the ADER family, in particular finite-element ADER-DG and finite-volume ADER-WENO methods, are among the most accurate numerical methods for solving quasilinear PDE systems. The internal structure of ADER-DG and ADER-WENO numerical methods contains a large number of basic linear algebra operations related to matrix multiplications. The main interface of software libraries for matrix multiplications for high-performance computing is BLAS. This paper presents an effective method for integration the standard functions of the BLAS interface into the implementation of these numerical methods. The calculated matrices are small matrices; at the same time, the proposed implementation makes it possible to effectively use existing JIT technologies. The proposed approach immediately operates on AoS, which makes it possible to efficiently calculate flux, source and non-conservative terms without need to carry out transposition. The obtained computational costs demonstrated that the effective implementation, based on the use of the JIT functions of the BLAS, outperformed both the implementation based on the general BLAS functions and the vanilla implementations by several orders of magnitude. At the same time, the complexity of developing an implementation based on the approach proposed in this work does not exceed the complexity of developing a vanilla implementation.

arXiv:2408.01227v2 Announce Type: replace Abstract: The study of parameter-dependent partial differential equations (parametric PDEs) with countably many parameters has been actively studied for the last few decades. In particular, it has been well known that a certain type of parametric holomorphy of the PDE solutions allows the application of deep neural networks without encountering the curse of dimensionality. This paper aims to propose a general framework for verifying the desired parametric holomorphy by utilizing the bounds on parametric derivatives. The framework is illustrated with examples of parametric elliptic eigenvalue problems (EVPs), encompassing both linear and semilinear cases. As the results, it will be shown that the ground eigenpairs have the desired holomorphy. Furthermore, under the same conditions, improved bounds for the mixed derivatives of the ground eigenpairs are derived. These bounds are well known to take a crucial role in the error analysis of quasi-Monte Carlo methods.

arXiv:2501.06373v1 Announce Type: cross Abstract: We conduct an analysis of a one-dimensional linear problem that describes the vibrations of a connected suspension bridge. In this model, the single-span roadbed is represented as a thermoelastic Shear beam without rotary inertia. We incorporate thermal dissipation into the transverse displacement equation, following Green and Naghdi's theory. Our work demonstrates the existence of a global solution by employing classical Faedo-Galerkin approximations and three a priori estimates. Furthermore, we establish exponential stability through the application of the energy method. For numerical study, we propose a spatial discretization using finite elements and a temporal discretization through an implicit Euler scheme. In doing so, we prove discrete stability properties and a priori error estimates for the discrete problem. To provide a practical dimension to our theoretical findings, we present a set of numerical simulations.

arXiv:2501.03719v1 Announce Type: new Abstract: The Taylor expansion of wave fields with respect to shape parameters has a wide range of applications in wave scattering problems, including inverse scattering, optimal design, and uncertainty quantification. However, deriving the high order shape derivatives required for this expansion poses significant challenges with conventional methods. This paper addresses these difficulties by introducing elegant recurrence formulas for computing high order shape derivatives. The derivation employs tools from exterior differential forms, Lie derivatives, and material derivatives. The work establishes a unified framework for computing the high order shape perturbations in scattering problems. In particular, the recurrence formulas are applicable to both acoustic and electromagnetic scattering models under a variety of boundary conditions, including Dirichlet, Neumann, impedance, and transmission types.

arXiv:2408.16556v2 Announce Type: replace-cross Abstract: We prove piecewise Sobolev regularity of vector fields that have piecewise regular curl and divergence, but may fail to be globally continuous. The main idea behind our approach is to employ recently developed parametrices for the curl-operator and the regularity theory of Poisson transmission problems. We conclude our work by applying our findings to the heterogeneous time-harmonic Maxwell equations with either a) impedance, b) natural or c) essential boundary conditions and providing wavenumber-explicit piecewise regularity estimates for these equations.

arXiv:2501.01726v1 Announce Type: new Abstract: Working from an observability characterization based on output energy sensitivity to changes in initial conditions, we derive both analytical and empirical observability Gramian tools for a class of continuum material systems. Using these results, optimal sensor placement is calculated for an Euler-Bernoulli cantilever beam for the following cases: analytical observability for the continuum system and analytical observability for a finite number of modes. Error covariance of an Unscented Kalman Filter is determined for both cases and compared to randomly placed sensors to demonstrate effectiveness of the techniques.

arXiv:2501.00345v1 Announce Type: new Abstract: In this work, we develop a novel neural network (NN) approach to solve the discrete inverse conductivity problem of recovering the conductivity profile on network edges from the discrete Dirichlet-to-Neumann map on a square lattice. The novelty of the approach lies in the fact that the sought-after conductivity is not provided directly as the output of the NN but is instead encoded in the weights of the post-trainig NN in the second layer. Hence the weights of the trained NN acquire a clear physical meaning, which contrasts with most existing neural network approaches, where the weights are typically not interpretable. This work represents a step toward designing NNs with interpretable post-training weights. Numerically, we observe that the method outperforms the conventional Curtis-Morrow algorithm for both noisy full and partial data.