\begin{algorithm}
\caption{ESTIMATE on a single interval $I_m$, i.e. time-stepping based}
\begin{algorithmic}
\Require $\hat{u}_{kh}$ on $I_{m-1}$ and $I_m$ and $\hat{z}_{kh}$ on $I_{m-1}$, $I_m$ and $I_{m+1}$
\State interpolate/reconstruct $\tilde{u}$, $u_k$, $u_{kh}$ as well as 
$\tilde{z}$, $z_k$, $z_{kh}$ at quadrature points.
\State Calculate $\eta_{\star}^m$ and $\eta_\star^{m,*}$, 
where $\star$ denotes $k$ or $kh$
\State Calculate $\eta_{\bullet}^{i,m}$ and $\eta_{\bullet}^{i,m,*}$ for each PU-DoF $i$ on $\mathcal{T}_h^m$, 
where, again, $\bullet$ denotes $h$ or $kh$
\end{algorithmic}
\label{algo_estimate_timestep}
\end{algorithm}

\begin{algorithm}
\caption{ESTIMATE on a single interval $I_m$, i.e. time-stepping based}
\begin{algorithmic}
\Require $\hat{u}_{kh}$ on $I_{m-1}$ and $I_m$ and $\hat{z}_{kh}$ on $I_{m-1}$, $I_m$ and $I_{m+1}$
\State interpolate/reconstruct $\tilde{u}$, $u_k$, $u_{kh}$ as well as
$\tilde{z}$, $z_k$, $z_{kh}$ at quadrature points.
\State Calculate $\eta_{\star}^m$ and $\eta_\star^{m,*}$,
where $\star$ denotes $k$ or $kh$
\State Calculate $\eta_{\bullet}^{i,m}$ and $\eta_{\bullet}^{i,m,*}$ for each PU-DoF $i$ on $\mathcal{T}_h^m$,
where, again, $\bullet$ denotes $h$ or $kh$
\end{algorithmic}
\end{algorithm}
{
    "title": "numerical modeling and open-source implementation of variational   partition-of-unity localizations of space-time dual-weighted residual   estimators for parabolic problems",
    "id": "2207.04764",
    "abstract": "in this work, we consider space-time goal-oriented a posteriori error estimation for parabolic problems. temporal and spatial discretizations are based on galerkin finite elements of continuous and discontinuous type. the main objectives are the development and analysis of space-time estimators, in which the localization is based on a weak form employing a partition-of-unity. the resulting error indicators are used for temporal and spatial adaptivity. our developments are substantiated with several numerical examples.",
    "categories": "math.na cs.na math.oc",
    "doi": "",
    "created": "2022-07-11",
    "updated": "2024-02-04",
    "authors": [
        "jan philipp thiele",
        "thomas wick"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2207.04764"
}
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    "ToxicityFilter": {
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                "toxicity_score": 0.009173225,
                "severe_toxicity_score": 0.0010585785,
                "identity_attack_score": 0.0026453468,
                "insult_score": 0.007474876,
                "profanity_score": 0.012722028,
                "threat_score": 0.0069516995,
                "sexually_explicit_score": 0.009731233,
                "flirtation_score": 0.17961895
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            "main.tex": {
                "toxicity_score": 0.018095128,
                "severe_toxicity_score": 0.0018024445,
                "identity_attack_score": 0.0064006294,
                "insult_score": 0.008823884,
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            "Intro_nonstat_PU.bib": {
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                "profanity_score": 0.018392462,
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                "sexually_explicit_score": 0.015805881,
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            "main.bbl": {
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                "identity_attack_score": 0.003866276,
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                "sexually_explicit_score": 0.010969753,
                "flirtation_score": 0.27629572
            },
            "main.fls": {
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                "severe_toxicity_score": 0.00045776367,
                "identity_attack_score": 0.0011515582,
                "insult_score": 0.0056556147,
                "profanity_score": 0.009903891,
                "threat_score": 0.0058059995,
                "sexually_explicit_score": 0.005248968,
                "flirtation_score": 0.37866667
            },
            "main.out": {
                "toxicity_score": 0.011120965,
                "severe_toxicity_score": 0.001964569,
                "identity_attack_score": 0.0014891637,
                "insult_score": 0.007028373,
                "profanity_score": 0.02030538,
                "threat_score": 0.0061296434,
                "sexually_explicit_score": 0.020995872,
                "flirtation_score": 0.40309274
            }
        },
        "reason": null
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{
    "num_done": {
        "figure": 0,
        "algorithm": 3
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}
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        "white_pixels_ratio": 85.58447442609747,
        "hash": "00092f142707ff7f",
        "most_frequent_color_ratio": 85.58447442609747
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