Optimization constrained by high-fidelity computational models has potential for transformative impact. However, such optimization is frequently unattainable in practice due to the complexity and computational intensity of the model. An alternative is to optimize a low-fidelity model and use limited evaluations of the high-fidelity model to assess the quality of the solution. This article develops a framework to use limited high-fidelity simulations to update the optimization solution computed using the low-fidelity model. Building off a previous article [22], which introduced hyper-differential sensitivity analysis with respect to model discrepancy, this article provides novel extensions of the algorithm to enable uncertainty quantification of the optimal solution update via a Bayesian framework. Specifically, we formulate a Bayesian inverse problem to estimate the model discrepancy and propagate the posterior model discrepancy distribution through the post-optimality sensitivity operator for the low-fidelity optimization problem. We provide a rigorous treatment of the Bayesian formulation, a computationally efficient algorithm to compute posterior samples, a guide to specify and interpret the algorithm hyper-parameters, and a demonstration of the approach on three examples which highlight various types of discrepancy between low and high-fidelity models.
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The present article aims to design and analyze efficient first-order strong schemes for a generalized A\"{i}t-Sahalia type model arising in mathematical finance and evolving in a positive domain $(0, \infty)$, which possesses a diffusion term with superlinear growth and a highly nonlinear drift that blows up at the origin. Such a complicated structure of the model unavoidably causes essential difficulties in the construction and convergence analysis of time discretizations. By incorporating implicitness in the term $\alpha_{-1} x^{-1}$ and a corrective mapping $\Phi_h$ in the recursion, we develop a novel class of explicit and unconditionally positivity-preserving (i.e., for any step-size $h>0$) Milstein-type schemes for the underlying model. In both non-critical and general critical cases, we introduce a novel approach to analyze mean-square error bounds of the novel schemes, without relying on a priori high-order moment bounds of the numerical approximations. The expected order-one mean-square convergence is attained for the proposed scheme. The above theoretical guarantee can be used to justify the optimal complexity of the Multilevel Monte Carlo method. Numerical experiments are finally provided to verify the theoretical findings.
We consider the statistical linear inverse problem of making inference on an unknown source function in an elliptic partial differential equation from noisy observations of its solution. We employ nonparametric Bayesian procedures based on Gaussian priors, leading to convenient conjugate formulae for posterior inference. We review recent results providing theoretical guarantees on the quality of the resulting posterior-based estimation and uncertainty quantification, and we discuss the application of the theory to the important classes of Gaussian series priors defined on the Dirichlet-Laplacian eigenbasis and Mat\'ern process priors. We provide an implementation of posterior inference for both classes of priors, and investigate its performance in a numerical simulation study.
Recent surge in large-scale generative models has spurred the development of vast fields in computer vision. In particular, text-to-image diffusion models have garnered widespread adoption across diverse domain due to their potential for high-fidelity image generation. Nonetheless, existing large-scale diffusion models are confined to generate images of up to 1K resolution, which is far from meeting the demands of contemporary commercial applications. Directly sampling higher-resolution images often yields results marred by artifacts such as object repetition and distorted shapes. Addressing the aforementioned issues typically necessitates training or fine-tuning models on higher resolution datasets. However, this undertaking poses a formidable challenge due to the difficulty in collecting large-scale high-resolution contents and substantial computational resources. While several preceding works have proposed alternatives, they often fail to produce convincing results. In this work, we probe the generative ability of diffusion models at higher resolution beyond its original capability and propose a novel progressive approach that fully utilizes generated low-resolution image to guide the generation of higher resolution image. Our method obviates the need for additional training or fine-tuning which significantly lowers the burden of computational costs. Extensive experiments and results validate the efficiency and efficacy of our method. Project page: //yhyun225.github.io/DiffuseHigh/
Although recently several foundation models for satellite remote sensing imagery have been proposed, they fail to address major challenges of real/operational applications. Indeed, embeddings that don't take into account the spectral, spatial and temporal dimensions of the data as well as the irregular or unaligned temporal sampling are of little use for most real world uses.As a consequence, we propose an ALIgned Sits Encoder (ALISE), a novel approach that leverages the spatial, spectral, and temporal dimensions of irregular and unaligned SITS while producing aligned latent representations. Unlike SSL models currently available for SITS, ALISE incorporates a flexible query mechanism to project the SITS into a common and learned temporal projection space. Additionally, thanks to a multi-view framework, we explore integration of instance discrimination along a masked autoencoding task to SITS. The quality of the produced representation is assessed through three downstream tasks: crop segmentation (PASTIS), land cover segmentation (MultiSenGE), and a novel crop change detection dataset. Furthermore, the change detection task is performed without supervision. The results suggest that the use of aligned representations is more effective than previous SSL methods for linear probing segmentation tasks.
We consider the problem of approximating an unknown function in a nonlinear model class from point evaluations. When obtaining these point evaluations is costly, minimising the required sample size becomes crucial. Recently, an increasing focus has been on employing adaptive sampling strategies to achieve this. These strategies are based on linear spaces related to the nonlinear model class, for which the optimal sampling measures are known. However, the resulting optimal sampling measures depend on an orthonormal basis of the linear space, which is known rarely. Consequently, sampling from these measures is challenging in practice. This manuscript presents a sampling strategy that iteratively refines an estimate of the optimal sampling measure by updating it based on previously drawn samples. This strategy can be performed offline and does not require evaluations of the sought function. We establish convergence and illustrate the practical performance through numerical experiments. Comparing the presented approach with standard Monte Carlo sampling demonstrates a significant reduction in the number of samples required to achieve a good estimation of an orthonormal basis.
We propose a topological mapping and localization system able to operate on real human colonoscopies, despite significant shape and illumination changes. The map is a graph where each node codes a colon location by a set of real images, while edges represent traversability between nodes. For close-in-time images, where scene changes are minor, place recognition can be successfully managed with the recent transformers-based local feature matching algorithms. However, under long-term changes -- such as different colonoscopies of the same patient -- feature-based matching fails. To address this, we train on real colonoscopies a deep global descriptor achieving high recall with significant changes in the scene. The addition of a Bayesian filter boosts the accuracy of long-term place recognition, enabling relocalization in a previously built map. Our experiments show that ColonMapper is able to autonomously build a map and localize against it in two important use cases: localization within the same colonoscopy or within different colonoscopies of the same patient. Code: //github.com/jmorlana/ColonMapper.
In this paper, we study an optimal control problem for a coupled non-linear system of reaction-diffusion equations with degenerate diffusion, consisting of two partial differential equations representing the density of cells and the concentration of the chemotactic agent. By controlling the concentration of the chemical substrates, this study can guide the optimal growth of cells. The novelty of this work lies on the direct and dual models that remain in a weak setting, which is uncommon in the recent literature for solving optimal control systems. Moreover, it is known that the adjoint problems offer a powerful approach to quantifying the uncertainty associated with model inputs. However, these systems typically lack closed-form solutions, making it challenging to obtain weak solutions. For that, the well-posedness of the direct problem is first well guaranteed. Then, the existence of an optimal control and the first-order optimality conditions are established. Finally, weak solutions for the adjoint system to the non-linear degenerate direct model, are introduced and investigated.
We prove the convergence of a damped Newton's method for the nonlinear system resulting from a discretization of the second boundary value problem for the Monge-Ampere equation. The boundary condition is enforced through the use of the notion of asymptotic cone. The differential operator is discretized based on a discrete analogue of the subdifferential.
In this work we propose a discretization of the second boundary condition for the Monge-Ampere equation arising in geometric optics and optimal transport. The discretization we propose is the natural generalization of the popular Oliker-Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.
In the finite difference approximation of the fractional Laplacian the stiffness matrix is typically dense and needs to be approximated numerically. The effect of the accuracy in approximating the stiffness matrix on the accuracy in the whole computation is analyzed and shown to be significant. Four such approximations are discussed. While they are shown to work well with the recently developed grid-over finite difference method (GoFD) for the numerical solution of boundary value problems of the fractional Laplacian, they differ in accuracy, economics to compute, performance of preconditioning, and asymptotic decay away from the diagonal line. In addition, two preconditioners based on sparse and circulant matrices are discussed for the iterative solution of linear systems associated with the stiffness matrix. Numerical results in two and three dimensions are presented.
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