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We address parameter estimation in second-order stochastic differential equations (SDEs), prevalent in physics, biology, and ecology. Second-order SDE is converted to a first-order system by introducing an auxiliary velocity variable raising two main challenges. First, the system is hypoelliptic since the noise affects only the velocity, making the Euler-Maruyama estimator ill-conditioned. To overcome that, we propose an estimator based on the Strang splitting scheme. Second, since the velocity is rarely observed we adjust the estimator for partial observations. We present four estimators for complete and partial observations, using full likelihood or only velocity marginal likelihood. These estimators are intuitive, easy to implement, and computationally fast, and we prove their consistency and asymptotic normality. Our analysis demonstrates that using full likelihood with complete observations reduces the asymptotic variance of the diffusion estimator. With partial observations, the asymptotic variance increases due to information loss but remains unaffected by the likelihood choice. However, a numerical study on the Kramers oscillator reveals that using marginal likelihood for partial observations yields less biased estimators. We apply our approach to paleoclimate data from the Greenland ice core and fit it to the Kramers oscillator model, capturing transitions between metastable states reflecting observed climatic conditions during glacial eras.

Matrix-vector multiplication forms the basis of many iterative solution algorithms and as such is an important algorithm also for hierarchical matrices. However, due to its low computational intensity, its performance is typically limited by the available memory bandwidth. By optimizing the storage representation of the data within such matrices, this limitation can be lifted and the performance increased. This applies not only to hierarchical matrices but for also for other low-rank approximation schemes, e.g. block low-rank matrices.

For estimating the proportion of false null hypotheses in multiple testing, a family of estimators by Storey (2002) is widely used in the applied and statistical literature, with many methods suggested for selecting the parameter $\lambda$. Inspired by change-point concepts, our new approach to the latter problem first approximates the $p$-value plot with a piecewise linear function with a single change-point and then selects the $p$-value at the change-point location as $\lambda$. Simulations show that our method has among the smallest RMSE across various settings, and we extend it to address the estimation in cases of superuniform $p$-values. We provide asymptotic theory for our estimator, relying on the theory of quantile processes. Additionally, we propose an application in the change-point literature and illustrate it using high-dimensional CNV data.

Solving linear systems of equations is an important problem in science and engineering. Many quantum algorithms, such as the Harrow-Hassidim-Lloyd (HHL) algorithm (for quantum-gate computers) and the box algorithm (for quantum-annealing machines), have been proposed for solving such systems. The focus of this paper is on improving the efficiency of the box algorithm. The basic principle behind this algorithm is to transform the linear system into a series of quadratic unconstrained binary optimization (QUBO) problems, which are then solved on annealing machines. The computational efficiency of the box algorithm is entirely determined by the number of iterations, which, in turn, depends on the box contraction ratio, typically set to 0.5. Here, we show through theory that a contraction ratio of 0.5 is sub-optimal and that we can achieve a speed-up with a contraction ratio of 0.2. This is confirmed through numerical experiments where a speed-up between $20 \%$ to $60 \%$ is observed when the optimal contraction ratio is used.

The detection of traversable regions on staircases and the physical modeling constitutes pivotal aspects of the mobility of legged robots. This paper presents an onboard framework tailored to the detection of traversable regions and the modeling of physical attributes of staircases by point cloud data. To mitigate the influence of illumination variations and the overfitting due to the dataset diversity, a series of data augmentations are introduced to enhance the training of the fundamental network. A curvature suppression cross-entropy(CSCE) loss is proposed to reduce the ambiguity of prediction on the boundary between traversable and non-traversable regions. Moreover, a measurement correction based on the pose estimation of stairs is introduced to calibrate the output of raw modeling that is influenced by tilted perspectives. Lastly, we collect a dataset pertaining to staircases and introduce new evaluation criteria. Through a series of rigorous experiments conducted on this dataset, we substantiate the superior accuracy and generalization capabilities of our proposed method. Codes, models, and datasets will be available at //github.com/szturobotics/Stair-detection-and-modeling-project.

Surrogate neural network-based partial differential equation (PDE) solvers have the potential to solve PDEs in an accelerated manner, but they are largely limited to systems featuring fixed domain sizes, geometric layouts, and boundary conditions. We propose Specialized Neural Accelerator-Powered Domain Decomposition Methods (SNAP-DDM), a DDM-based approach to PDE solving in which subdomain problems containing arbitrary boundary conditions and geometric parameters are accurately solved using an ensemble of specialized neural operators. We tailor SNAP-DDM to 2D electromagnetics and fluidic flow problems and show how innovations in network architecture and loss function engineering can produce specialized surrogate subdomain solvers with near unity accuracy. We utilize these solvers with standard DDM algorithms to accurately solve freeform electromagnetics and fluids problems featuring a wide range of domain sizes.

Many complex systems can be accurately modeled as a set of coupled time-dependent partial differential equations (PDEs). However, solving such equations can be prohibitively expensive, easily taxing the world's largest supercomputers. One pragmatic strategy for attacking such problems is to split the PDEs into components that can more easily be solved in isolation. This operator splitting approach is used ubiquitously across scientific domains, and in many cases leads to a set of ordinary differential equations (ODEs) that need to be solved as part of a larger "outer-loop" time-stepping approach. The SUNDIALS library provides a plethora of robust time integration algorithms for solving ODEs, and the U.S. Department of Energy Exascale Computing Project (ECP) has supported its extension to applications on exascale-capable computing hardware. In this paper, we highlight some SUNDIALS capabilities and its deployment in combustion and cosmology application codes (Pele and Nyx, respectively) where operator splitting gives rise to numerous, small ODE systems that must be solved concurrently.

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a variety of PDEs, such as fluid flows. However, the FNO uses the Fast Fourier transform (FFT), which is limited to rectangular domains with uniform grids. In this work, we propose a new framework, viz., geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. The FNO model with the FFT is applied in the latent space. The resulting geo-FNO model has both the computation efficiency of FFT and the flexibility of handling arbitrary geometries. Our geo-FNO is also flexible in terms of its input formats, viz., point clouds, meshes, and design parameters are all valid inputs. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems. Geo-FNO is $10^5$ times faster than the standard numerical solvers and twice more accurate compared to direct interpolation on existing ML-based PDE solvers such as the standard FNO.

Transport phenomena (e.g., fluid flows) are governed by time-dependent partial differential equations (PDEs) describing mass, momentum, and energy conservation, and are ubiquitous in many engineering applications. However, deep learning architectures are fundamentally incompatible with the simulation of these PDEs. This paper clearly articulates and then solves this incompatibility. The local-dependency of generic transport PDEs implies that it only involves local information to predict the physical properties at a location in the next time step. However, the deep learning architecture will inevitably increase the scope of information to make such predictions as the number of layers increases, which can cause sluggish convergence and compromise generalizability. This paper aims to solve this problem by proposing a distributed data scoping method with linear time complexity to strictly limit the scope of information to predict the local properties. The numerical experiments over multiple physics show that our data scoping method significantly accelerates training convergence and improves the generalizability of benchmark models on large-scale engineering simulations. Specifically, over the geometries not included in the training data for heat transferring simulation, it can increase the accuracy of Convolutional Neural Networks (CNNs) by 21.7 \% and that of Fourier Neural Operators (FNOs) by 38.5 \% on average.

When solving systems of banded Toeplitz equations or calculating their inverses, it is necessary to determine the invertibility of the matrices beforehand. In this paper, we equate the invertibility of an $n$-order banded Toeplitz matrix with bandwidth $2k+1$ to that of a small $k*k$ matrix. By utilizing a specially designed algorithm, we compute the invertibility sequence of a class of banded Toeplitz matrices with a time complexity of $5k^2n/2+kn$ and a space complexity of $3k^2$ where $n$ is the size of the largest matrix. This enables efficient preprocessing when solving equation systems and inverses of banded Toeplitz matrices.