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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.

Mediation analysis is widely used in health science research to evaluate the extent to which an intermediate variable explains an observed exposure-outcome relationship. However, the validity of analysis can be compromised when the exposure is measured with error. Motivated by the Health Professionals Follow-up Study (HPFS), we investigate the impact of exposure measurement error on assessing mediation with a survival outcome, based on the Cox proportional hazards outcome model. When the outcome is rare and there is no exposure-mediator interaction, we show that the uncorrected estimators of the natural indirect and direct effects can be biased into either direction, but the uncorrected estimator of the mediation proportion is approximately unbiased as long as the measurement error is not large or the mediator-exposure association is not strong. We develop ordinary regression calibration and risk set regression calibration approaches to correct the exposure measurement error-induced bias when estimating mediation effects and allowing for an exposure-mediator interaction in the Cox outcome model. The proposed approaches require a validation study to characterize the measurement error process. We apply the proposed approaches to the HPFS (1986-2016) to evaluate extent to which reduced body mass index mediates the protective effect of vigorous physical activity on the risk of cardiovascular diseases, and compare the finite-sample properties of the proposed estimators via simulations.

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.

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.

In this paper we introduce a multilevel Picard approximation algorithm for semilinear parabolic partial integro-differential equations (PIDEs). We prove that the numerical approximation scheme converges to the unique viscosity solution of the PIDE under consideration. To that end, we derive a Feynman-Kac representation for the unique viscosity solution of the semilinear PIDE, extending the classical Feynman-Kac representation for linear PIDEs. Furthermore, we show that the algorithm does not suffer from the curse of dimensionality, i.e. the computational complexity of the algorithm is bounded polynomially in the dimension $d$ and the reciprocal of the prescribed accuracy $\varepsilon$. We also provide a numerical example in up to 10'000 dimensions to demonstrate its applicability.

Speech emotion conversion is the task of converting the expressed emotion of a spoken utterance to a target emotion while preserving the lexical content and speaker identity. While most existing works in speech emotion conversion rely on acted-out datasets and parallel data samples, in this work we specifically focus on more challenging in-the-wild scenarios and do not rely on parallel data. To this end, we propose a diffusion-based generative model for speech emotion conversion, the EmoConv-Diff, that is trained to reconstruct an input utterance while also conditioning on its emotion. Subsequently, at inference, a target emotion embedding is employed to convert the emotion of the input utterance to the given target emotion. As opposed to performing emotion conversion on categorical representations, we use a continuous arousal dimension to represent emotions while also achieving intensity control. We validate the proposed methodology on a large in-the-wild dataset, the MSP-Podcast v1.10. Our results show that the proposed diffusion model is indeed capable of synthesizing speech with a controllable target emotion. Crucially, the proposed approach shows improved performance along the extreme values of arousal and thereby addresses a common challenge in the speech emotion conversion literature.

We present a training method with linguistic speech regularization that improves the robustness of spontaneous speech synthesis methods with filled pause (FP) insertion. Spontaneous speech synthesis is aimed at producing speech with human-like disfluencies, such as FPs. Because modeling the complex data distribution of spontaneous speech with a rich FP vocabulary is challenging, the quality of FP-inserted synthetic speech is often limited. To address this issue, we present a method for synthesizing spontaneous speech that improves robustness to diverse FP insertions. Regularization is used to stabilize the synthesis of the linguistic speech (i.e., non-FP) elements. To further improve robustness to diverse FP insertions, it utilizes pseudo-FPs sampled using an FP word prediction model as well as ground-truth FPs. Our experiments demonstrated that the proposed method improves the naturalness of synthetic speech with ground-truth and predicted FPs by 0.24 and 0.26, respectively.

This paper introduces a comprehensive framework to adjust a discrete test statistic for improving its hypothesis testing procedure. The adjustment minimizes the Wasserstein distance to a null-approximating continuous distribution, tackling some fundamental challenges inherent in combining statistical significances derived from discrete distributions. The related theory justifies Lancaster's mid-p and mean-value chi-squared statistics for Fisher's combination as special cases. However, in order to counter the conservative nature of Lancaster's testing procedures, we propose an updated null-approximating distribution. It is achieved by further minimizing the Wasserstein distance to the adjusted statistics within a proper distribution family. Specifically, in the context of Fisher's combination, we propose an optimal gamma distribution as a substitute for the traditionally used chi-squared distribution. This new approach yields an asymptotically consistent test that significantly improves type I error control and enhances statistical power.

While several previous studies have devised methods for segmentation of polyps, most of these methods are not rigorously assessed on multi-center datasets. Variability due to appearance of polyps from one center to another, difference in endoscopic instrument grades, and acquisition quality result in methods with good performance on in-distribution test data, and poor performance on out-of-distribution or underrepresented samples. Unfair models have serious implications and pose a critical challenge to clinical applications. We adapt an implicit bias mitigation method which leverages Bayesian epistemic uncertainties during training to encourage the model to focus on underrepresented sample regions. We demonstrate the potential of this approach to improve generalisability without sacrificing state-of-the-art performance on a challenging multi-center polyp segmentation dataset (PolypGen) with different centers and image modalities.

Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized. This paper reviews and organizes the recent advances in deep learning theory. The literature is categorized in six groups: (1) complexity and capacity-based approaches for analyzing the generalizability of deep learning; (2) stochastic differential equations and their dynamic systems for modelling stochastic gradient descent and its variants, which characterize the optimization and generalization of deep learning, partially inspired by Bayesian inference; (3) the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems; (4) the roles of over-parameterization of deep neural networks from both positive and negative perspectives; (5) theoretical foundations of several special structures in network architectures; and (6) the increasingly intensive concerns in ethics and security and their relationships with generalizability.