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The BICEP3 and BICEP Array polarimeters are small-aperture refracting telescopes located at the South Pole designed to measure primordial gravitational wave signatures in the Cosmic Microwave Background (CMB) polarization, predicted by inflation. Constraining the inflationary signal requires not only excellent sensitivity, but also careful control of instrumental systematics. Both instruments use antenna-coupled orthogonally polarized detector pairs, and the polarized sky signal is reconstructed by taking the difference in each detector pair. As a result, the differential response between detectors within a pair becomes an important systematic effect we must control. Additionally, mapping the intensity and polarization response in regions away from the main beam can inform how sidelobe levels affect CMB measurements. Extensive calibration measurements are taken in situ every austral summer for control of instrumental systematics and instrument characterisation. In this work, we detail the set of beam calibration measurements that we conduct on the BICEP receivers, from deep measurements of main beam response to polarized beam response and sidelobe mapping. We discuss the impact of these measurements for instrumental systematics studies and design choices for future CMB receivers., Comment: 13 pages, 7 figures, 1 table, Proceedings paper SPIE 2024

In practical use cases, polygonal mesh editing can be faster than generating new ones, but it can still be challenging and time-consuming for users. Existing solutions for this problem tend to focus on a single task, either geometry or novel view synthesis, which often leads to disjointed results between the mesh and view. In this work, we propose LEMON, a mesh editing pipeline that combines neural deferred shading with localized mesh optimization. Our approach begins by identifying the most important vertices in the mesh for editing, utilizing a segmentation model to focus on these key regions. Given multi-view images of an object, we optimize a neural shader and a polygonal mesh while extracting the normal map and the rendered image from each view. By using these outputs as conditioning data, we edit the input images with a text-to-image diffusion model and iteratively update our dataset while deforming the mesh. This process results in a polygonal mesh that is edited according to the given text instruction, preserving the geometric characteristics of the initial mesh while focusing on the most significant areas. We evaluate our pipeline using the DTU dataset, demonstrating that it generates finely-edited meshes more rapidly than the current state-of-the-art methods. We include our code and additional results in the supplementary material.

Measurements of B-mode polarization in the CMB sourced from primordial gravitational waves would provide information on the energy scale of inflation and its potential form. To achieve these goals, one must carefully characterize the Galactic foregrounds, which can be distinguished from the CMB by conducting measurements at multiple frequencies. BICEP Array is the latest-generation multi-frequency instrument of the BICEP/Keck program, which specifically targets degree-scale primordial B-modes in the CMB. In its final configuration, this telescope will consist of four small-aperture receivers, spanning frequency bands from 30 to 270 GHz. The 220/270 GHz receiver designed to characterize Galactic dust is currently undergoing commissioning at Stanford University and is scheduled to deploy to the South Pole during the 2024--2025 austral summer. Here, we will provide an overview of this high-frequency receiver and discuss the integration status and test results as it is being commissioned.

All-optical helicity-independent switching (AO-HIS) is of interest for ultrafast and energy efficient magnetic switching in future magnetic data storage approaches. Yet, to achieve high bit density magnetic recording it is necessary to reduce the size of the magnetic bits addressed by laser pulses at well-controlled positions. Metallic nanostructures that support localized surface plasmons enable spatial electromagnetic confinement well below the diffraction limit and rare-earth transition metal alloys such as GdTbCo have demonstrated nanometre-sized stable domains. Here, we deposit plasmonic gold nanostructures on a GdTbCo film and probe the magnetic state using magnetic force microscopy. We observe localized AO-HIS down to a critical dimension of 240 nm after excitation of the gold nanostructures by a single 370 fs long laser pulse with a centre wavelength of 1030 nm. We demonstrate that the strong localization of optical fields through plasmonic nanostructures enables reproducible localized nanoscale AO-HIS at sub-wavelength length scales. We study the influence of the localized electromagnetic field enhancement by the plasmonic nanostructures on the required fluence to switch the magnetization., Comment: 10 pages, 4 figures

Evaluating the output of generative large language models (LLMs) is challenging and difficult to scale. Most evaluations of LLMs focus on tasks such as single-choice question-answering or text classification. These tasks are not suitable for assessing open-ended question-answering capabilities, which are critical in domains where expertise is required, such as health, and where misleading or incorrect answers can have a significant impact on a user's health. Using human experts to evaluate the quality of LLM answers is generally considered the gold standard, but expert annotation is costly and slow. We present a method for evaluating LLM answers that uses ranking signals as a substitute for explicit relevance judgements. Our scoring method correlates with the preferences of human experts. We validate it by investigating the well-known fact that the quality of generated answers improves with the size of the model as well as with more sophisticated prompting strategies.

We present a knowledge-guided machine learning (KGML) framework for modeling multi-scale processes, and study its performance in the context of streamflow forecasting in hydrology. Specifically, we propose a novel hierarchical recurrent neural architecture that factorizes the system dynamics at multiple temporal scales and captures their interactions. This framework consists of an inverse and a forward model. The inverse model is used to empirically resolve the system's temporal modes from data (physical model simulations, observed data, or a combination of them from the past), and these states are then used in the forward model to predict streamflow. In a hydrological system, these modes can represent different processes, evolving at different temporal scales (e.g., slow: groundwater recharge and baseflow vs. fast: surface runoff due to extreme rainfall). A key advantage of our framework is that once trained, it can incorporate new observations into the model's context (internal state) without expensive optimization approaches (e.g., EnKF) that are traditionally used in physical sciences for data assimilation. Experiments with several river catchments from the NWS NCRFC region show the efficacy of this ML-based data assimilation framework compared to standard baselines, especially for basins that have a long history of observations. Even for basins that have a shorter observation history, we present two orthogonal strategies of training our FHNN framework: (a) using simulation data from imperfect simulations and (b) using observation data from multiple basins to build a global model. We show that both of these strategies (that can be used individually or together) are highly effective in mitigating the lack of training data. The improvement in forecast accuracy is particularly noteworthy for basins where local models perform poorly because of data sparsity.

This paper presents a novel methodology for predicting international bilateral trade flows, emphasizing the growing importance of Preferential Trade Agreements (PTAs) in the global trade landscape. Acknowledging the limitations of traditional models like the Gravity Model of Trade, this study introduces a two-stage approach combining explainable machine learning and factorization models. The first stage employs SHAP Explainer for effective variable selection, identifying key provisions in PTAs, while the second stage utilizes Factorization Machine models to analyze the pairwise interaction effects of these provisions on trade flows. By analyzing comprehensive datasets, the paper demonstrates the efficacy of this approach. The findings not only enhance the predictive accuracy of trade flow models but also offer deeper insights into the complex dynamics of international trade, influenced by specific bilateral trade provisions.

This paper proposes a short-range FMCW radar-based facial authentication and out-of-distribution (OOD) detection framework. Our pipeline jointly estimates the correct classes for the in-distribution (ID) samples and detects the OOD samples to prevent their inaccurate prediction. Our reconstruction-based architecture consists of a main convolutional block with one encoder and multi-decoder configuration, and intermediate linear encoder-decoder parts. Together, these elements form an accurate human face classifier and a robust OOD detector. For our dataset, gathered using a 60 GHz short-range FMCW radar, our network achieves an average classification accuracy of 98.07% in identifying in-distribution human faces. As an OOD detector, it achieves an average Area Under the Receiver Operating Characteristic (AUROC) curve of 98.50% and an average False Positive Rate at 95% True Positive Rate (FPR95) of 6.20%. Also, our extensive experiments show that the proposed approach outperforms previous OOD detectors in terms of common OOD detection metrics., Comment: Accepted at ICIP 2024

The BICEP/$\textit{Keck}$ (BK) series of cosmic microwave background (CMB) polarization experiments has, over the past decade and a half, produced a series of field-leading constraints on cosmic inflation via measurements of the "B-mode" polarization of the CMB. Primordial B modes are directly tied to the amplitude of primordial gravitational waves (PGW), their strength parameterized by the tensor-to-scalar ratio, $r$, and thus the energy scale of inflation. Having set the most sensitive constraints to-date on $r$, $\sigma(r)=0.009$ ($r_{0.05}<0.036, 95\%$ C.L.) using data through the 2018 observing season ("BK18"), the BICEP/$\textit{Keck}$ program has continued to improve its dataset in the years since. We give a brief overview of the BK program and the "BK18" result before discussing the program's ongoing efforts, including the deployment and performance of the $\textit{Keck Array}$'s successor instrument, BICEP Array, improvements to data processing and internal consistency testing, new techniques such as delensing, and how those will ultimately serve to allow BK reach $\sigma(r) \lesssim 0.003$ using data through the 2027 observing season., Comment: 9 pages, 5 figures. Contribution to the 2024 Cosmology session of the 58th Rencontres de Moriond

We demonstrate a wide-band diplexed focal plane suitable for observing low-frequency foregrounds that are important for cosmic microwave background polarimetry. The antenna elements are composed of slotted bowtie antennas with 60% bandwidth that can be partitioned into two bands. Each pixel is composed of two interleaved 12$\times$12 pairs of linearly polarized antenna elements forming a phased array, designed to synthesize a symmetric beam with no need for focusing optics. The signal from each antenna element is captured in-phase and uniformly weighted by a microstrip summing tree. The antenna signal is diplexed into two bands through the use of two complementary, six-pole Butterworth filters. This filter architecture ensures a contiguous impedance match at all frequencies, and thereby achieves minimal reflection loss between both bands. Subsequently, out-of-band rejection is increased with a bandpass filter and the signal is then deposited on a transition-edge sensor bolometer island. We demonstrate the performance of this focal plane with two distinct bands, 30 and 40 GHz, each with a bandwidth of $\sim$20 and 15 GHz, respectively. The unequal bandwidths between the two bands are caused by an unintentional shift in diplexer frequency from its design values. The end-to-end optical efficiency of these detectors are relatively modest, at 20-30%, with an efficiency loss due to an unknown impedance mismatch in the summing tree. Far-field beam maps show good optical characteristics with edge pixels having no more than $\sim$ 5% ellipticity and $\sim$10-15% peak-to-peak differences for A-B polarization pairs., Comment: 17 pages, 13 figures. Accepted for publication in ApJS

We consider a distributed optimal control problem subject to a parabolic evolution equation as constraint. The control will be considered in the energy norm of the anisotropic Sobolev space $[H_{0;,0}^{1,1/2}(Q)]^\ast$, such that the state equation of the partial differential equation defines an isomorphism onto $H^{1,1/2}_{0;0,}(Q)$. Thus, we can eliminate the control from the tracking type functional to be minimized, to derive the optimality system in order to determine the state. Since the appearing operator induces an equivalent norm in $H_{0;0,}^{1,1/2}(Q)$, we will replace it by a computable realization of the anisotropic Sobolev norm, using a modified Hilbert transformation. We are then able to link the cost or regularization parameter $\varrho>0$ to the distance of the state and the desired target, solely depending on the regularity of the target. For a conforming space-time finite element discretization, this behavior carries over to the discrete setting, leading to an optimal choice $\varrho = h_x^2$ of the regularization parameter $\varrho$ to the spatial finite element mesh size $h_x$. Using a space-time tensor product mesh, error estimates for the distance of the computable state to the desired target are derived. The main advantage of this new approach is, that applying sparse factorization techniques, a solver of optimal, i.e., almost linear, complexity is proposed and analyzed. The theoretical results are complemented by numerical examples, including discontinuous and less regular targets. Moreover, this approach can be applied also to optimal control problems subject to non-linear state equations.

We propose, analyze, and test new robust iterative solvers for systems of linear algebraic equations arising from the space-time finite element discretization of reduced optimality systems defining the approximate solution of hyperbolic distributed, tracking-type optimal control problems with both the standard $L^2$ and the more general energy regularizations. In contrast to the usual time-stepping approach, we discretize the optimality system by space-time continuous piecewise-linear finite element basis functions which are defined on fully unstructured simplicial meshes. If we aim at the asymptotically best approximation of the given desired state $y_d$ by the computed finite element state $y_{\varrho h}$, then the optimal choice of the regularization parameter $\varrho$ is linked to the space-time finite element mesh-size $h$ by the relations $\varrho=h^4$ and $\varrho=h^2$ for the $L^2$ and the energy regularization, respectively. For this setting, we can construct robust (parallel) iterative solvers for the reduced finite element optimality systems. These results can be generalized to variable regularization parameters adapted to the local behavior of the mesh-size that can heavily change in the case of adaptive mesh refinements. The numerical results illustrate the theoretical findings firmly.

We propose a multi-objective global pattern search algorithm for the task of locating and quantifying damage in flexible mechanical structures. This is achieved by identifying eigenfrequencies and eigenmodes from measurements and matching them against the results of a finite element simulation model, which leads to a nonsmooth nonlinear bi-objective parameter estimation problem. A derivative-free optimization algorithm is required since the problem is nonsmooth and also because complex mechanical simulation models are often solved using commercial black-box software. Moreover, the entire set of non-dominated solutions is of interest to practitioners. Most solution approaches published to date are based on meta-heuristics such as genetic algorithms. The proposed multi-objective pattern-search algorithm provides a mathematically well-founded alternative. It features a novel sorting procedure that reduces the complexity in our context. Test runs on two experimental structures with multiple damage scenarios are used to validate the approach. The results demonstrate that the proposed algorithm yields accurate damage locations and requires moderate computational resources. From the engineer's perspective it represents a promising tool for structural health monitoring.

We consider the aeroelastic simulation of flexible mechanical structures submerged in subsonic fluid flows at low Mach numbers. The nonlinear kinematics of flexible bodies are described in the total Lagrangian formulation and discretized by finite elements. The aerodynamic loads are computed using the unsteady vortex-lattice method wherein a free wake is tracked over time. Each implicit time step in the dynamic simulation then requires solving a nonlinear equation system in the structural variables with additional aerodynamic load terms. Our focus here is on the efficient numerical solution of this system by accelerating the Newton algorithm. The particular structure of the aeroelastic nonlinear system suggests the structural derivative as an approximation to the full derivative in the linear Newton system. We investigate and compare two promising algorithms based in this approximation, a quasi-Newton type algorithm and a novel inexact Newton algorithm. Numerical experiments are performed on a flexible plate and on a wind turbine. Our computational results show that the approximation can indeed accelerate the Newton algorithm substantially. Surprisingly, the theoretically preferable inexact Newton algorithm is much slower than the quasi-Newton algorithm, which motivates further research to speed up derivative evaluations.

As learned image codecs (LICs) become more prevalent, their low coding efficiency for out-of-distribution data becomes a bottleneck for some applications. To improve the performance of LICs for screen content (SC) images without breaking backwards compatibility, we propose to introduce parameterized and invertible linear transformations into the coding pipeline without changing the underlying baseline codec's operation flow. We design two neural networks to act as prefilters and postfilters in our setup to increase the coding efficiency and help with the recovery from coding artifacts. Our end-to-end trained solution achieves up to 10% bitrate savings on SC compression compared to the baseline LICs while introducing only 1% extra parameters., Comment: 7 pages, 6 figures, 2 tables

Hand-Object Interactions (HOIs) are conditioned on spatial and temporal contexts like surrounding objects, previous actions, and future intents (for example, grasping and handover actions vary greatly based on objects proximity and trajectory obstruction). However, existing datasets for 4D HOI (3D HOI over time) are limited to one subject interacting with one object only. This restricts the generalization of learning-based HOI methods trained on those datasets. We introduce ADL4D, a dataset of up to two subjects interacting with different sets of objects performing Activities of Daily Living (ADL) like breakfast or lunch preparation activities. The transition between multiple objects to complete a certain task over time introduces a unique context lacking in existing datasets. Our dataset consists of 75 sequences with a total of 1.1M RGB-D frames, hand and object poses, and per-hand fine-grained action annotations. We develop an automatic system for multi-view multi-hand 3D pose annotation capable of tracking hand poses over time. We integrate and test it against publicly available datasets. Finally, we evaluate our dataset on the tasks of Hand Mesh Recovery (HMR) and Hand Action Segmentation (HAS).

In this paper, we analyze the discrete inf-sup condition and related error estimates for a modified Hilbert transformation as used in the space-time discretization of time-dependent partial differential equations. It turns out that the stability constant depends linearly on the finite element mesh parameter, but in most cases, we can show optimal convergence. We present a series of numerical experiments which illustrate the theoretical findings.

We present a monolithic finite element formulation for (nonlinear) fluid-structure interaction in Eulerian coordinates. For the discretization we employ an unfitted finite element method based on inf-sup stable finite elements. So-called ghost penalty terms are used to guarantee the robustness of the approach independently of the way the interface cuts the finite element mesh. The resulting system is solved in a monolithic fashion using Newton's method. Our developments are tested on a numerical example with fixed interface.

With the rapid proliferation of textual data, predicting long texts has emerged as a significant challenge in the domain of natural language processing. Traditional text prediction methods encounter substantial difficulties when grappling with long texts, primarily due to the presence of redundant and irrelevant information, which impedes the model's capacity to capture pivotal insights from the text. To address this issue, we introduce a novel approach to long-text classification and prediction. Initially, we employ embedding techniques to condense the long texts, aiming to diminish the redundancy therein. Subsequently,the Bidirectional Encoder Representations from Transformers (BERT) embedding method is utilized for text classification training. Experimental outcomes indicate that our method realizes considerable performance enhancements in classifying long texts of Preferential Trade Agreements. Furthermore, the condensation of text through embedding methods not only augments prediction accuracy but also substantially reduces computational complexity. Overall, this paper presents a strategy for long-text prediction, offering a valuable reference for researchers and engineers in the natural language processing sphere., Comment: AI4TS Workshop@AAAI 2024 accepted publication

We propose, analyze, and test new iterative solvers for large-scale systems of linear algebraic equations arising from the finite element discretization of reduced optimality systems defining the finite element approximations to the solution of elliptic tracking-type distributed optimal control problems with both the standard $L_2$ and the more general energy regularizations. If we aim at an approximation of the given desired state $y_d$ by the computed finite element state $y_h$ that asymptotically differs from $y_d$ in the order of the best $L_2$ approximation under acceptable costs for the control, then the optimal choice of the regularization parameter $\varrho$ is linked to the mesh-size $h$ by the relations $\varrho=h^4$ and $\varrho=h^2$ for the $L_2$ and the energy regularization, respectively. For this setting, we can construct efficient parallel iterative solvers for the reduced finite element optimality systems. These results can be generalized to variable regularization parameters adapted to the local behavior of the mesh-size that can heavily change in case of adaptive mesh refinement. Similar results can be obtained for the space-time finite element discretization of the corresponding parabolic and hyperbolic optimal control problems.

In this paper, we recast the variational formulation corresponding to the single layer boundary integral operator $\operatorname{V}$ for the wave equation as a minimization problem in $L^2(\Sigma)$, where $\Sigma := \partial \Omega \times (0,T)$ is the lateral boundary of the space-time domain $Q := \Omega \times (0,T)$. For discretization, the minimization problem is restated as a mixed saddle point formulation. Unique solvability is established by combining conforming nested boundary element spaces for the mixed formulation such that the related bilinear form is discrete inf-sup stable. We analyze under which conditions the discrete inf-sup stability is satisfied, and, moreover, we show that the mixed formulation provides a simple error indicator, which can be used for adaptivity. We present several numerical experiments showing the applicability of the method to different time-domain boundary integral formulations used in the literature.

We propose HAROOD as a short-range FMCW radar-based human activity classifier and out-of-distribution (OOD) detector. It aims to classify human sitting, standing, and walking activities and to detect any other moving or stationary object as OOD. We introduce a two-stage network. The first stage is trained with a novel loss function that includes intermediate reconstruction loss, intermediate contrastive loss, and triplet loss. The second stage uses the first stage's output as its input and is trained with cross-entropy loss. It creates a simple classifier that performs the activity classification. On our dataset collected by 60 GHz short-range FMCW radar, we achieve an average classification accuracy of 96.51%. Also, we achieve an average AUROC of 95.04% as an OOD detector. Additionally, our extensive evaluations demonstrate the superiority of HAROOD over the state-of-the-art OOD detection methods in terms of standard OOD detection metrics., Comment: Accepted at ICASSP 2024

In this work we consider the hybrid Data-Driven Computational Mechanics (DDCM) approach, in which a smooth constitutive manifold is reconstructed to obtain a well-behaved nonlinear optimization problem (NLP) rather than the much harder discrete-continous NLP (DCNLP) of the direct DDCM approach. The key focus is on the addition of geometric inequality constraints to the hybrid DDCM formulation. Therein, the required constraint force leads to a contact problem in the form of a mathematical program with complementarity constraints (MPCC), a problem class that is still less complex than the DCNLP. For this MPCC we propose a heuristic quick-shot solution approach, which can produce verifiable solutions by solving up to four NLPs. We perform various numerical experiments on three different contact problems of increasing difficulty to demonstrate the potential and limitations of this approach., Comment: 23 pages

The quantum-phase-field concept of matter is revisited with special emphasis on the introverted view of space. Extroverted space surrounds physical objects, while introverted space lies in between physical objects. Space between objects leads to a network structure of matter: a network in which one-dimensional spaces connect individual particles., Comment: 7 pages, 2 figures

The COVID-19 pandemic shifted many events in our daily lives into the virtual domain. While virtual conference systems provide an alternative to physical meetings, larger events require a muted audience to avoid an accumulation of background noise and distorted audio. However, performing artists strongly rely on the feedback of their audience. We propose a concept for a virtual audience framework which supports all participants with the ambience of a real audience. Audience feedback is collected locally, allowing users to express enthusiasm or discontent by selecting means such as clapping, whistling, booing, and laughter. This feedback is sent as abstract information to a virtual audience server. We broadcast the combined virtual audience feedback information to all participants, which can be synthesized as a single acoustic feedback by the client. The synthesis can be done by turning the collective audience feedback into a prompt that is fed to state-of-the-art models such as AudioGen. This way, each user hears a single acoustic feedback sound of the entire virtual event, without requiring to unmute or risk hearing distorted, unsynchronized feedback., Comment: 4 pages, 2 figures

Time-Division Multiplexing is the readout architecture of choice for many ground and space experiments, as it is a very mature technology with proven outstanding low-frequency noise stability, which represents a central challenge in multiplexing. Once fully populated, each of the two BICEP Array high frequency receivers, observing at 150GHz and 220/270GHz, will have 7776 TES detectors tiled on the focal plane. The constraints set by these two receivers required a redesign of the warm readout electronics. The new version of the standard Multi Channel Electronics, developed and built at the University of British Columbia, is presented here for the first time. BICEP Array operates Time Division Multiplexing readout technology to the limits of its capabilities in terms of multiplexing rate, noise and crosstalk, and applies them in rigorously demanding scientific application requiring extreme noise performance and systematic error control. Future experiments like CMB-S4 plan to use TES bolometers with Time Division/SQUID-based readout for an even larger number of detectors., Comment: 10 pages, 7 figures, Submitted to Journal of Low Temperature Physics

As labor shortage increases in the health sector, the demand for assistive robotics grows. However, the needed test data to develop those robots is scarce, especially for the application of active 3D object detection, where no real data exists at all. This short paper counters this by introducing such an annotated dataset of real environments. The captured environments represent areas which are already in use in the field of robotic health care research. We further provide ground truth data within one room, for assessing SLAM algorithms running directly on a health care robot.

The purpose of this work is the formulation of optimality conditions for phase-field optimal control problems. The forward problem is first stated as an abstract nonlinear optimization problem, and then the necessary optimality conditions are derived. The sufficient optimality conditions are also examined. The choice of suitable function spaces to ensure the regularity of the nonlinear optimization problem is a true challenge here. Afterwards the optimal control problem with a tracking type cost functional is formulated. The constraints are given by the previously derived first order optimality conditions of the forward problem. Herein regularity is proven under certain conditions and first order optimality conditions are formulated., Comment: 24 pages

In recent years, the increasing threat of devastating wildfires has underscored the need for effective prescribed fire management. Process-based computer simulations have traditionally been employed to plan prescribed fires for wildfire prevention. However, even simplified process models like QUIC-Fire are too compute-intensive to be used for real-time decision-making, especially when weather conditions change rapidly. Traditional ML methods used for fire modeling offer computational speedup but struggle with physically inconsistent predictions, biased predictions due to class imbalance, biased estimates for fire spread metrics (e.g., burned area, rate of spread), and generalizability in out-of-distribution wind conditions. This paper introduces a novel machine learning (ML) framework that enables rapid emulation of prescribed fires while addressing these concerns. By incorporating domain knowledge, the proposed method helps reduce physical inconsistencies in fuel density estimates in data-scarce scenarios. To overcome the majority class bias in predictions, we leverage pre-existing source domain data to augment training data and learn the spread of fire more effectively. Finally, we overcome the problem of biased estimation of fire spread metrics by incorporating a hierarchical modeling structure to capture the interdependence in fuel density and burned area. Notably, improvement in fire metric (e.g., burned area) estimates offered by our framework makes it useful for fire managers, who often rely on these fire metric estimates to make decisions about prescribed burn management. Furthermore, our framework exhibits better generalization capabilities than the other ML-based fire modeling methods across diverse wind conditions and ignition patterns.

We consider the numerical solution of an abstract operator equation $Bu=f$ by using a least-squares approach. We assume that $B: X \to Y^*$ is an isomorphism, and that $A : Y \to Y^*$ implies a norm in $Y$, where $X$ and $Y$ are Hilbert spaces. The minimizer of the least-squares functional $\frac{1}{2} \, \| Bu-f \|_{A^{-1}}^2$, i.e., the solution of the operator equation, is then characterized by the gradient equation $Su=B^* A^{-1}f$ with an elliptic and self-adjoint operator $S:=B^* A^{-1} B : X \to X^*$. When introducing the adjoint $p = A^{-1}(f-Bu)$ we end up with a saddle point formulation to be solved numerically by using a mixed finite element method. Based on a discrete inf-sup stability condition we derive related a priori error estimates. While the adjoint $p$ is zero by construction, its approximation $p_h$ serves as a posteriori error indicator to drive an adaptive scheme when discretized appropriately. While this approach can be applied to rather general equations, here we consider second order linear partial differential equations, including the Poisson equation, the heat equation, and the wave equation, in order to demonstrate its potential, which allows to use almost arbitrary space-time finite element methods for the adaptive solution of time-dependent partial differential equations.

Instance segmentation has witnessed promising advancements through deep neural network-based algorithms. However, these models often exhibit incorrect predictions with unwarranted confidence levels. Consequently, evaluating prediction uncertainty becomes critical for informed decision-making. Existing methods primarily focus on quantifying uncertainty in classification or regression tasks, lacking emphasis on instance segmentation. Our research addresses the challenge of estimating spatial certainty associated with the location of instances with star-convex shapes. Two distinct clustering approaches are evaluated which compute spatial and fractional certainty per instance employing samples by the Monte-Carlo Dropout or Deep Ensemble technique. Our study demonstrates that combining spatial and fractional certainty scores yields improved calibrated estimation over individual certainty scores. Notably, our experimental results show that the Deep Ensemble technique alongside our novel radial clustering approach proves to be an effective strategy. Our findings emphasize the significance of evaluating the calibration of estimated certainties for model reliability and decision-making.