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We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity of interest $G(u^\star)$ with a linear goal functional $G$ and the solution $u^\star$ to a general second-order nonsymmetric linear elliptic partial differential equation. The current state of the analysis of iterative algebraic solvers for nonsymmetric systems lacks the contraction property in the norms that are prescribed by the functional analytic setting. This seemingly prevents their application in the optimality analysis of goal-oriented adaptivity. As a remedy, this paper proposes a goal-oriented adaptive iteratively symmetrized finite element method (GOAISFEM). It employs a nested loop with a contractive symmetrization procedure, e.g., the Zarantonello iteration, and a contractive algebraic solver, e.g., an optimal multigrid solver. The various iterative procedures require well-designed stopping criteria such that the adaptive algorithm can effectively steer the local mesh refinement and the computation of the inexact discrete approximations. The main results consist of full linear convergence of the proposed adaptive algorithm and the proof of optimal convergence rates with respect to both degrees of freedom and total computational cost (i.e., optimal complexity). Numerical experiments confirm the theoretical results and investigate the selection of the parameters.

We present a new algorithm for solving linear-quadratic regulator (LQR) problems with linear equality constraints. This is the first such exact algorithm that is guaranteed to have a runtime that is linear in the number of stages, as well as linear in the number of both state-only constraints as well as mixed state-and-control constraints, without imposing any restrictions on the problem instances. We also show how to easily parallelize this algorithm to run in parallel runtime logarithmic in the number of stages of the problem.

Differentially private stochastic gradient descent (DP-SGD) refers to a family of optimization algorithms that provide a guaranteed level of differential privacy (DP) through DP accounting techniques. However, current accounting techniques make assumptions that diverge significantly from practical DP-SGD implementations. For example, they may assume the loss function is Lipschitz continuous and convex, sample the batches randomly with replacement, or omit the gradient clipping step. In this work, we analyze the most commonly used variant of DP-SGD, in which we sample batches cyclically with replacement, perform gradient clipping, and only release the last DP-SGD iterate. More specifically - without assuming convexity, smoothness, or Lipschitz continuity of the loss function - we establish new R\'enyi differential privacy (RDP) bounds for the last DP-SGD iterate under the mild assumption that (i) the DP-SGD stepsize is small relative to the topological constants in the loss function, and (ii) the loss function is weakly-convex. Moreover, we show that our bounds converge to previously established convex bounds when the weak-convexity parameter of the objective function approaches zero. In the case of non-Lipschitz smooth loss functions, we provide a weaker bound that scales well in terms of the number of DP-SGD iterations.

Linear causal disentanglement is a recent method in causal representation learning to describe a collection of observed variables via latent variables with causal dependencies between them. It can be viewed as a generalization of both independent component analysis and linear structural equation models. We study the identifiability of linear causal disentanglement, assuming access to data under multiple contexts, each given by an intervention on a latent variable. We show that one perfect intervention on each latent variable is sufficient and in the worst case necessary to recover parameters under perfect interventions, generalizing previous work to allow more latent than observed variables. We give a constructive proof that computes parameters via a coupled tensor decomposition. For soft interventions, we find the equivalence class of latent graphs and parameters that are consistent with observed data, via the study of a system of polynomial equations. Our results hold assuming the existence of non-zero higher-order cumulants, which implies non-Gaussianity of variables.

This work aims to extend the well-known high-order WENO finite-difference methods for systems of conservation laws to nonconservative hyperbolic systems. The main difficulty of these systems both from the theoretical and the numerical points of view comes from the fact that the definition of weak solution is not unique: according to the theory developed by Dal Maso, LeFloch, and Murat in 1995, it depends on the choice of a family of paths. A general strategy is proposed here in which WENO operators are not only used to reconstruct fluxes but also the nonconservative products of the system. Moreover, if a Roe linearization is available, the nonconservative products can be computed through matrix-vector operations instead of path-integrals. The methods are extended to problems with source terms and two different strategies are introduced to obtain well-balanced schemes. These numerical schemes will be then applied to the two-layer shallow water equations in one- and two- dimensions to obtain high-order methods that preserve water-at-rest steady states.

We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems, of which the inner one can be solved either exactly or cheaply, while the outer one can be recast as an unconstrained optimization task over a smooth real Riemannian manifold. We observe that this paradigm applies to many matrix nearness problems of practical interest appearing in the literature, thus revealing that they are equivalent in this sense to a Riemannian optimization problem. We also show that the objective function to be minimized on the Riemannian manifold can be discontinuous, thus requiring regularization techniques, and we give conditions for this to happen. Finally, we demonstrate the practical applicability of our method by implementing it for a number of matrix nearness problems that are relevant for applications and are currently considered very demanding in practice. Extensive numerical experiments demonstrate that our method often greatly outperforms its predecessors, including algorithms specifically designed for those particular problems.

We consider the parallel-in-time solution of hyperbolic partial differential equation (PDE) systems in one spatial dimension, both linear and nonlinear. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that inter-variable coupling for characteristic variables is weak relative to intra-variable coupling, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small. For an $\ell$-dimensional system of PDEs, applying the preconditioner consists of solving a sequence of $\ell$ scalar linear(ized)-advection-like problems, each being associated with a different characteristic wave-speed in the underlying linear(ized) PDE. We approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media, and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions.

Reduced basis methods for approximating the solutions of parameter-dependant partial differential equations (PDEs) are based on learning the structure of the set of solutions - seen as a manifold ${\mathcal S}$ in some functional space - when the parameters vary. This involves investigating the manifold and, in particular, understanding whether it is close to a low-dimensional affine space. This leads to the notion of Kolmogorov $N$-width that consists of evaluating to which extent the best choice of a vectorial space of dimension $N$ approximates ${\mathcal S}$ well enough. If a good approximation of elements in ${\mathcal S}$ can be done with some well-chosen vectorial space of dimension $N$ -- provided $N$ is not too large -- then a ``reduced'' basis can be proposed that leads to a Galerkin type method for the approximation of any element in ${\mathcal S}$. In many cases, however, the Kolmogorov $N$-width is not so small, even if the parameter set lies in a space of small dimension yielding a manifold with small dimension. In terms of complexity reduction, this gap between the small dimension of the manifold and the large Kolmogorov $N$-width can be explained by the fact that the Kolmogorov $N$-width is linear while, in contrast, the dependency in the parameter is, most often, non-linear. There have been many contributions aiming at reconciling these two statements, either based on deterministic or AI approaches. We investigate here further a new paradigm that, in some sense, merges these two aspects: the nonlinear compressive reduced basisapproximation. We focus on a simple multiparameter problem and illustrate rigorously that the complexity associated with the approximation of the solution to the parameter dependant PDE is directly related to the number of parameters rather than the Kolmogorov $N$-width.

The Lippmann--Schwinger--Lanczos (LSL) algorithm has recently been shown to provide an efficient tool for imaging and direct inversion of synthetic aperture radar data in multi-scattering environments [17], where the data set is limited to the monostatic, a.k.a. single input/single output (SISO) measurements. The approach is based on constructing data-driven estimates of internal fields via a reduced-order model (ROM) framework and then plugging them into the Lippmann-Schwinger integral equation. However, the approximations of the internal solutions may have more error due to missing the off diagonal elements of the multiple input/multiple output (MIMO) matrix valued transfer function. This, in turn, may result in multiple echoes in the image. Here we present a ROM-based data completion algorithm to mitigate this problem. First, we apply the LSL algorithm to the SISO data as in [17] to obtain approximate reconstructions as well as the estimate of internal field. Next, we use these estimates to calculate a forward Lippmann-Schwinger integral to populate the missing off-diagonal data (the lifting step). Finally, to update the reconstructions, we solve the Lippmann-Schwinger equation using the original SISO data, where the internal fields are constructed from the lifted MIMO data. The steps of obtaining the approximate reconstructions and internal fields and populating the missing MIMO data entries can be repeated for complex models to improve the images even further. Efficiency of the proposed approach is demonstrated on 2D and 2.5D numerical examples, where we see reconstructions are improved substantially.

Homogenisation empowers the efficient macroscale system level prediction of physical problems with intricate microscale structures. Here we develop an innovative powerful, rigorous and flexible framework for asymptotic homogenisation of dynamics at the finite scale separation of real physics, with proven results underpinned by modern dynamical systems theory. The novel systematic approach removes most of the usual assumptions, whether implicit or explicit, of other methodologies. By no longer assuming averages the methodology constructs so-called multi-continuum or micromorphic homogenisations systematically based upon the microscale physics. The developed framework and approach enables a user to straightforwardly choose and create such homogenisations with clear physical and theoretical support, and of highly controllable accuracy and fidelity.