We study the query complexity of geodesically convex (g-convex) optimization on a manifold. To isolate the effect of that manifold's curvature, we primarily focus on hyperbolic spaces. In a variety of settings (smooth or not; strongly g-convex or not; high- or low-dimensional), known upper bounds worsen with curvature. It is natural to ask whether this is warranted, or an artifact. For many such settings, we propose a first set of lower bounds which indeed confirm that (negative) curvature is detrimental to complexity. To do so, we build on recent lower bounds (Hamilton and Moitra, 2021; Criscitiello and Boumal, 2022) for the particular case of smooth, strongly g-convex optimization. Using a number of techniques, we also secure lower bounds which capture dependence on condition number and optimality gap, which was not previously the case. We suspect these bounds are not optimal. We conjecture optimal ones, and support them with a matching lower bound for a class of algorithms which includes subgradient descent, and a lower bound for a related game. Lastly, to pinpoint the difficulty of proving lower bounds, we study how negative curvature influences (and sometimes obstructs) interpolation with g-convex functions.
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A general a posteriori error analysis applies to five lowest-order finite element methods for two fourth-order semi-linear problems with trilinear non-linearity and a general source. A quasi-optimal smoother extends the source term to the discrete trial space, and more importantly, modifies the trilinear term in the stream-function vorticity formulation of the incompressible 2D Navier-Stokes and the von K\'{a}rm\'{a}n equations. This enables the first efficient and reliable a posteriori error estimates for the 2D Navier-Stokes equations in the stream-function vorticity formulation for Morley, two discontinuous Galerkin, $C^0$ interior penalty, and WOPSIP discretizations with piecewise quadratic polynomials.
Solving multiphysics-based inverse problems for geological carbon storage monitoring can be challenging when multimodal time-lapse data are expensive to collect and costly to simulate numerically. We overcome these challenges by combining computationally cheap learned surrogates with learned constraints. Not only does this combination lead to vastly improved inversions for the important fluid-flow property, permeability, it also provides a natural platform for inverting multimodal data including well measurements and active-source time-lapse seismic data. By adding a learned constraint, we arrive at a computationally feasible inversion approach that remains accurate. This is accomplished by including a trained deep neural network, known as a normalizing flow, which forces the model iterates to remain in-distribution, thereby safeguarding the accuracy of trained Fourier neural operators that act as surrogates for the computationally expensive multiphase flow simulations involving partial differential equation solves. By means of carefully selected experiments, centered around the problem of geological carbon storage, we demonstrate the efficacy of the proposed constrained optimization method on two different data modalities, namely time-lapse well and time-lapse seismic data. While permeability inversions from both these two modalities have their pluses and minuses, their joint inversion benefits from either, yielding valuable superior permeability inversions and CO2 plume predictions near, and far away, from the monitoring wells.
Transition amplitudes and transition probabilities are relevant to many areas of physics simulation, including the calculation of response properties and correlation functions. These quantities can also be related to solving linear systems of equations. Here we present three related algorithms for calculating transition probabilities. First, we extend a previously published short-depth algorithm, allowing for the two input states to be non-orthogonal. Building on this first procedure, we then derive a higher-depth algorithm based on Trotterization and Richardson extrapolation that requires fewer circuit evaluations. Third, we introduce a tunable algorithm that allows for trading off circuit depth and measurement complexity, yielding an algorithm that can be tailored to specific hardware characteristics. Finally, we implement proof-of-principle numerics for models in physics and chemistry and for a subroutine in variational quantum linear solving (VQLS). The primary benefits of our approaches are that (a) arbitrary non-orthogonal states may now be used with small increases in quantum resources, (b) we (like another recently proposed method) entirely avoid subroutines such as the Hadamard test that may require three-qubit gates to be decomposed, and (c) in some cases fewer quantum circuit evaluations are required as compared to the previous state-of-the-art in NISQ algorithms for transition probabilities.
Iterative refinement (IR) is a popular scheme for solving a linear system of equations based on gradually improving the accuracy of an initial approximation. Originally developed to improve upon the accuracy of Gaussian elimination, interest in IR has been revived because of its suitability for execution on fast low-precision hardware such as analog devices and graphics processing units. IR generally converges when the error associated with the solution method is small, but is known to diverge when this error is large. We propose and analyze a novel enhancement to the IR algorithm by adding a line search optimization step that guarantees the algorithm will not diverge. Numerical experiments verify our theoretical results and illustrate the effectiveness of our proposed scheme.
Although the applications of Non-Homogeneous Poisson Processes to model and study the threshold overshoots of interest in different time series of measurements have proven to provide good results, they needed to be complemented with an efficient and automatic diagnostic technique to establish the location of the change-points, which, when taken into account, make the estimated model fit poorly in regards of the information contained in the real model. For this reason, we propose a new method to solve the segmentation uncertainty of the time series of measurements, where the emission distribution of exceedances of a specific threshold is the focus of investigation. One of the great contributions of the present algorithm is that all the days that overflowed are candidates to be a change-point, so all the possible configurations of overflow days are the possible chromosomes, which will unite to have offspring. Under the heuristics of a genetic algorithm, the solution to the problem of finding such change points will be guaranteed to be non-local and the best possible one, reducing wasted machine time evaluating the least likely chromosomes to be a solution to the problem. The analytical evaluation technique will be by means of the Minimum Description Length (\textit{MDL}) as the objective function, which is the joint posterior distribution function of the parameters of each regime and the change points that determines them and which account as well for the influence of the presence of said times.
The sequential composition of propositional logic programs has been recently introduced. This paper studies the sequential {\em decomposition} of programs by studying Green's relations $\mathcal{L,R,J}$ -- well-known in semigroup theory -- between programs. In a broader sense, this paper is a further step towards an algebraic theory of logic programming.
Combinatorial optimization - a field of research addressing problems that feature strongly in a wealth of scientific and industrial contexts - has been identified as one of the core potential fields of applicability of quantum computers. It is still unclear, however, to what extent quantum algorithms can actually outperform classical algorithms for this type of problems. In this work, by resorting to computational learning theory and cryptographic notions, we prove that quantum computers feature an in-principle super-polynomial advantage over classical computers in approximating solutions to combinatorial optimization problems. Specifically, building on seminal work by Kearns and Valiant and introducing a new reduction, we identify special types of problems that are hard for classical computers to approximate up to polynomial factors. At the same time, we give a quantum algorithm that can efficiently approximate the optimal solution within a polynomial factor. The core of the quantum advantage discovered in this work is ultimately borrowed from Shor's quantum algorithm for factoring. Concretely, we prove a super-polynomial advantage for approximating special instances of the so-called integer programming problem. In doing so, we provide an explicit end-to-end construction for advantage bearing instances. This result shows that quantum devices have, in principle, the power to approximate combinatorial optimization solutions beyond the reach of classical efficient algorithms. Our results also give clear guidance on how to construct such advantage-bearing problem instances.
Linear systems with a tensor product structure arise naturally when considering the discretization of Laplace type differential equations or, more generally, multidimensional operators with separable coefficients. In this work, we focus on the numerical solution of linear systems of the form $$ \left(I\otimes \dots\otimes I \otimes A_1+\dots + A_d\otimes I \otimes\dots \otimes I\right)x=b,$$ where the matrices $A_t\in\mathbb R^{n\times n}$ are symmetric positive definite and belong to the class of hierarchically semiseparable matrices. We propose and analyze a nested divide-and-conquer scheme, based on the technology of low-rank updates, that attains the quasi-optimal computational cost $\mathcal O(n^d (\log(n) + \log(\kappa)^2 + \log(\kappa) \log(\epsilon^{-1})))$ where $\kappa$ is the condition number of the linear system, and $\epsilon$ the target accuracy. Our theoretical analysis highlights the role of inexactness in the nested calls of our algorithm and provides worst case estimates for the amplification of the residual norm. The performances are validated on 2D and 3D case studies.
We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $l$ in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter $L\gg l$ around the support of the charge. We show that the algorithm of Lu, Otto and Wang, suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion of Bella, Giunti and Otto, still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of $l$, $L$ and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multi-scale logarithmic Sobolev inequality, where our main tool is an extension of the semi-group estimates established by the first author. As part of our strategy, we construct sub-linear second-order correctors in this correlated setting which is of independent interest.
When modelling discontinuities (interfaces) using the finite element method, the standard approach is to use a conforming finite-element mesh in which the mesh matches the interfaces. However, this approach can prove cumbersome if the geometry is complex, in particular in 3D. In this work, we develop an efficient technique for a non-conforming finite-element treatment of weak discontinuities by using laminated microstructures. The approach is inspired by the so-called composite voxel technique that has been developed for FFT-based spectral solvers in computational homogenization. The idea behind the method is rather simple. Each finite element that is cut by an interface is treated as a simple laminate with the volume fraction of the phases and the lamination orientation determined in terms of the actual geometrical arrangement of the interface within the element. The approach is illustrated by several computational examples relevant to the micromechanics of heterogeneous materials. Elastic and elastic-plastic materials at small and finite strain are considered in the examples. The performance of the proposed method is compared to two alternative, simple methods showing that the new approach is in most cases superior to them while maintaining the simplicity.
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