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In this article, we present data-driven reduced-order modeling for nonautonomous dynamical systems in multiscale media using Koopman operators. Different from the case of autonomous dynamical systems, the Koopman operator family of nonautonomous dynamical systems significantly depend on a time pair. In order to effectively estimate the time-dependent Koopman operators, a moving time window is used to decompose the snapshot data, and the extended dynamic mode decomposition method is applied to computing the Koopman operators in each local temporal domain. Many physical properties in multiscale media often vary in very different scales. In order to capture multiscale information well, the dimension of collected data may be high. To accurately construct the models of dynamical systems in multiscale media, we use high spatial dimension of observation data. It is challenging to compute the Koopman operators using the very high dimensional data. Thus, the strategy of reduced-order modeling is proposed to treat the difficulty. The proposed reduced-order modeling includes two stages: offline stage and online stage. In offline stage, a block-wise low rank decomposition is used to reduce the spatial dimension of initial snapshot data. For the nonautonomous dynamical systems, real-time observation data may be required to update the Koopman operators. The online reduced-order modeling is proposed to correct the offline reduced-order modeling. Three methods are developed for the online reduced-order modeling: fully online, semi-online and adaptive online. The adaptive online method automatically selects the fully online or semi-online and can achieve a good trade-off between modeling accuracy and efficiency., Comment: 34 pages

In this paper, an online multiscale model reduction method is presented for stochastic partial differential equations (SPDEs) with multiplicative noise, where the diffusion coefficient is spatially multiscale and the noise perturbation nonlinearly depends on the diffusion dynamics. It is necessary to efficiently compute all possible trajectories of the stochastic dynamics for quantifying model's uncertainty and statistic moments. The multiscale diffusion and nonlinearity may cause the computation intractable. To overcome the multiscale difficulty, a constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) is used to localize the computation and obtain an effective coarse model. However, the nonlinear terms are still defined on a fine scale space after the Galerkin projection of CEM-GMsFEM is applied to the nonlinear SPDEs. This significantly impacts on the simulation efficiency by CEM-GMsFEM. To this end, a stochastic online discrete empirical interpolation method (DEIM) is proposed to treat the stochastic nonlinearity. The stochastic online DEIM incorporates offline snapshots and online snapshots. The offline snapshots consist of the nonlinear terms at the approximate mean of the stochastic dynamics and are used to construct an offline reduced model. The online snapshots contain some information of the current new trajectory and are used to correct the offline reduced model in an increment manner. The stochastic online DEIM substantially reduces the dimension of the nonlinear dynamics and enhances the prediction accuracy for the reduced model. Thus, the online multiscale model reduction is constructed by using CEM-GMsFEM and the stochastic online DEIM. A priori error analysis is carried out for the nonlinear SPDEs. We present a few numerical examples with diffusion in heterogeneous porous media and show the effectiveness of the proposed model reduction., Comment: to be published in Multiscale Modeling and Simulation

Rangahau Matauranga o Aotearoa | New Zealand Council for Educational Research (NZCER) began working with Te Ara Ahunga Ora Retirement Commission in mid-2019 to evaluate Sorted in Schools, Te whai hua -- kia ora. As well as 3 years of evaluation, NZCER has undertaken research to support and develop a sustainable professional learning and development (PLD) framework and research that put a Pacific lens on the programme. As part of this portfolio of research and evaluation, Te Ara Ahunga Ora wanted to hear the voices of students/akonga to understand what they think about Sorted in Schools and learning about financial capability. In this study, junior secondary (Years 9 and 10) students/akonga from four English-medium schools were asked about their views about money and financial capability, and their experiences of the Sorted in Schools programme and resources. They spoke about their financial habits and goals, how they currently learn about financial capability, and what and how they would like to learn about it in the future. The ideas and feedback in this report can be used to guide future development of programmes, resources, and support students' ongoing learning about financial capability. The data that inform this report were collected during November and December 2022.

The development and validation of 3D multiphase computational fluid dynamics (M-CFD) models and physics-informed data-driven modeling require data of high-quality and high-resolution. Considering the difficulties in acquiring the corresponding experimental data in prototypical conditions, two-phase boiling simulations by Interface Tracking Method (ITM) based models can be used to generate high-resolution numerical data in a consistent and relatively economical manner. A boiling model is developed in one of the ITM-based multiphase-flow solvers, named PHASTA, to investigate the nucleate boiling phenomenon. The interaction between bubbles forming at adjacent nucleation sites is investigated with this ITM boiling model. Nucleate pool boiling simulations with multiple nucleation sites are presented in this paper and influences of site distance, neighboring bubble size and contact angle effect are investigated. The presented boiling model can conduct boiling simulation on 3D unstructured computational meshes. These simulation results improve our understanding of the physical mechanisms of the nucleate boiling phenomenon and provide high-resolution numerical data for M-CFD validation and advanced boiling model development., Comment: This paper has been accepted by Nuclear Engineering and Design

"Sorted in Schools, Te whai hua -- kia ora" is a financial capability programme for secondary school students in English-medium (EME) and Maori-medium (MME) education settings. The programme aims to equip all young New Zealanders for their financial future. The first resources were piloted in 2018, and schools and kura started to use the Years 9 and 10 resources in 2019. Senior secondary resources for Years 11-13 were launched in June 2020. The programme is led by Te Ara Ahunga Ora Retirement Commission (formerly the Commission for Financial Capability), with resources developed in collaboration with CORE Education, teachers and kaiako. The New Zealand Council for Educational Research (NZCER) began evaluating the implementation and early outcomes of the programme in 2019, with Te whai hua -- kia ora and Sorted in Schools treated as two separate strands within the same evaluation. This manifests in this report by the way we present data and key findings in separate sections but weave the findings together in a conclusion to answer the overarching evaluation questions across the programme as a whole. As part of this phase of the evaluation in 2022, the evaluation framework was reviewed, which led to refreshed evaluation questions, revised evaluative criteria, and a new programme logic. Overall, we rated the performance of "Sorted in Schools, Te whai hua -- kia ora" as very good in 2022. More information on the evaluation rubric is provided in the report below, along with detailed key findings and our methodology.

In this paper, we present a Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for parabolic equations with multiscale coefficients, arising from applications in porous media. We will present the construction of CEM-GMsFEM and rigorously analyze its convergence for the parabolic equations. The convergence rate is characterized by the coarse grid size and the eigenvalue decay of local spectral problems, but is independent of the scale length and contrast of the media. The analysis shows that the method has a first order convergence rate with respect to coarse grid size in the energy norm and second order convergence rate with respect to coarse grid size in $L^2$ norm under some appropriate assumptions. For the temporal discretization, finite difference techniques are used and the convergence analysis of full discrete scheme is given. Moreover, a posteriori error estimator is derived and analyzed. A few numerical results for porous media applications are presented to confirm the theoretical findings and demonstrate the performance of the approach.