Model‐reduction techniques aim to reduce the computational complexity of simulating dynamical systems by applying a (Petrov–)Galerkin projection process that enforces the dynamics to evolve in a ...low‐dimensional subspace of the original state space. Frequently, the resulting reduced‐order model (ROM) violates intrinsic physical properties of the original full‐order model (e.g., global conservation, Lagrangian structure, state‐variable bounds) because the projection process does not generally ensure preservation of these properties. However, in many applications, ensuring the ROM preserves such intrinsic properties can enable the ROM to retain physical meaning and lead to improved accuracy and stability properties. In this work, we present a general constrained‐optimization formulation for projection‐based model reduction that can be used as a template to enforce the ROM to satisfy specific properties on the kinematics and dynamics. We introduce constrained‐optimization formulations at both the time‐continuous (i.e., ODE) level, which leads to a constrained Galerkin projection, and at the time‐discrete level, which leads to a least‐squares Petrov–Galerkin projection, in the context of linear multistep schemes. We demonstrate the ability of the proposed formulations to equip ROMs with desired properties such as global energy conservation and bounds on the total variation.
The Equal Earth map projection Šavrič, Bojan; Patterson, Tom; Jenny, Bernhard
International journal of geographical information science : IJGIS,
03/2019, Volume:
33, Issue:
3
Journal Article
Peer reviewed
The Equal Earth map projection is a new equal-area pseudocylindrical projection for world maps. It is inspired by the widely used Robinson projection, but unlike the Robinson projection, retains the ...relative size of areas. The projection equations are simple to implement and fast to evaluate. Continental outlines are shown in a visually pleasing and balanced way.
We review the introduction of several types of projection filters. Projection structures coming from information geometry are used to obtain a finite dimensional filter in the form of a stochastic ...differential equation (SDE), starting from the exact infinite-dimensional stochastic partial differential equation (SPDE) for the optimal filter. We start with the Stratonovich projection filters based on the Hellinger distance as introduced and developed in Brigo et al. (IEEE Trans Autom Control 43(2):247–252, 1998, Bernoulli 5(3):495–534, 1999), where the SPDE is put in Stratonovich form before projection, hence the term “Stratonovich projection”. The correction step of the filtering algorithm can be made exact by choosing a suitable exponential family as manifold, there is equivalence with assumed density filters and numerical examples have been studied. Other authors further developed these projection filters and we present a brief literature review. A second type of Stratonovich projection filters was introduced in Armstrong and Brigo (Math Control Signals Syst 28(1):1–33, 2016) where a direct
L
2
metric is used for projection. Projecting on mixtures of densities as a manifold coincides with Galerkin methods. All the above projection filters lack optimality, as the single vector fields of the Stratonovich SPDE are projected optimally but the SPDE solution as a whole is not approximated optimally by the projected SDE solution according to a clear criterion. This led to the optimal projection filters in Armstrong et al. (Proc Lond Math Soc 119(1):176–213, 2019, Projection of SDEs onto submanifolds. “Information Geometry”, 2023 special issue on half a century of information geometry, 2018), based on the Ito vector and Ito jet projections, where several types of mean square distances between the optimal filter SPDE solution and the sought finite dimensional SDE approximations are minimized, with numerical examples. After reviewing the above developments, we conclude with the remaining challenges.
Random-projection ensemble classification Cannings, Timothy I.; Samworth, Richard J.
Journal of the Royal Statistical Society. Series B, Statistical methodology,
September 2017, Volume:
79, Issue:
4
Journal Article
Peer reviewed
Open access
We introduce a very general method for high dimensional classification, based on careful combination of the results of applying an arbitrary base classifier to random projections of the feature ...vectors into a lower dimensional space. In one special case that we study in detail, the random projections are divided into disjoint groups, and within each group we select the projection yielding the smallest estimate of the test error. Our random-projection ensemble classifier then aggregates the results of applying the base classifier on the selected projections, with a data-driven voting threshold to determine the final assignment. Our theoretical results elucidate the effect on performance of increasing the number of projections. Moreover, under a boundary condition that is implied by the sufficient dimension reduction assumption, we show that the test excess risk of the random-projection ensemble classifier can be controlled by terms that do not depend on the original data dimension and a term that becomes negligible as the number of projections increases. The classifier is also compared empirically with several other popular high dimensional classifiers via an extensive simulation study, which reveals its excellent finite sample performance.
The literature usually calls downscaled versions of basic conformal map projections “secant”, referring to conceptual developable map surfaces that intersect the reference frame. However, recent ...studies pointed out on the examples of various mappings of the sphere that this model may lead to incorrect conclusions. In this study, we examine the paradigm of secant surfaces for two popular map projections of the ellipsoid, the UTM (Universal Transverse Mercator) and the UPS (Universal Polar Stereographic) projections. Results will show that ellipsoidal map projections can exhibit further anomalies. To support the shift to a paradigm avoiding developable map surfaces, this study recommends the new term reduced map projection with a proper and simple definition to be used instead of secant map projections.
Compound Rank- k Projections for Bilinear Analysis Chang, Xiaojun; Nie, Feiping; Wang, Sen ...
IEEE transaction on neural networks and learning systems,
2016-July, 2016-07-00, 2016-7-00, 20160701, Volume:
27, Issue:
7
Journal Article
In many real-world applications, data are represented by matrices or high-order tensors. Despite the promising performance, the existing 2-D discriminant analysis algorithms employ a single ...projection model to exploit the discriminant information for projection, making the model less flexible. In this paper, we propose a novel compound rank-k projection (CRP) algorithm for bilinear analysis. The CRP deals with matrices directly without transforming them into vectors, and it, therefore, preserves the correlations within the matrix and decreases the computation complexity. Different from the existing 2-D discriminant analysis algorithms, objective function values of CRP increase monotonically. In addition, the CRP utilizes multiple rank-k projection models to enable a larger search space in which the optimal solution can be found. In this way, the discriminant ability is enhanced. We have tested our approach on five data sets, including UUIm, CVL, Pointing'04, USPS, and Coil20. Experimental results show that the performance of our proposed CRP performs better than other algorithms in terms of classification accuracy.
This work proposes a method for model reduction of finite-volume models that guarantees the resulting reduced-order model is conservative, thereby preserving the structure intrinsic to finite-volume ...discretizations. The proposed reduced-order models associate with optimization problems characterized by a minimum-residual objective function and nonlinear equality constraints that explicitly enforce conservation over subdomains. Conservative Galerkin projection arises from formulating this optimization problem at the time-continuous level, while conservative least-squares Petrov–Galerkin (LSPG) projection associates with a time-discrete formulation. We equip these approaches with hyper-reduction techniques in the case of nonlinear flux and source terms, and also provide approaches for handling infeasibility. In addition, we perform analyses that include deriving conditions under which conservative Galerkin and conservative LSPG are equivalent, as well as deriving a posteriori error bounds. Numerical experiments performed on a parameterized quasi-1D Euler equation demonstrate the ability of the proposed method to ensure not only global conservation, but also significantly lower state-space errors than nonconservative reduced-order models such as standard Galerkin and LSPG projection.
•Two model-reduction approaches for finite-volume models that are conservative.•Supporting techniques for handling infeasibility and flux/source hyper-reduction.•Analysis including deriving feasibility/equivalence conditions and error bounds.•Numerical experiments on a parameterized compressible flow problem.
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