Marvel: A Data-Centric Approach for Mapping Deep Learning Operators on Spatial Accelerators

A spatial accelerator’s efficiency depends heavily on both its mapper and cost models to generate optimized mappings for various operators of DNN models. However, existing cost models lack a formal boundary over their input programs (operators) for accurate and tractable cost analysis of the mappings, and this results in adaptability challenges to the cost models for new operators. We consider the recently introduced Maestro Data-Centric (MDC) notation and its analytical cost model to address this challenge because any mapping expressed in the notation is precisely analyzable using the MDC’s cost model. In this article, we characterize the set of input operators and their mappings expressed in the MDC notation by introducing a set of conformability rules . The outcome of these rules is that any loop nest that is perfectly nested with affine tensor subscripts and without conditionals is conformable to the MDC notation. A majority of the primitive operators in deep learning are such loop nests. In addition, our rules enable us to automatically translate a mapping expressed in the loop nest form to MDC notation and use the MDC’s cost model to guide upstream mappers. Our conformability rules over the input operators result in a structured mapping space of the operators, which enables us to introduce a mapper based on our decoupled off-chip/on-chip approach to accelerate mapping space exploration. Our mapper decomposes the original higher-dimensional mapping space of operators into two lower-dimensional off-chip and on-chip subspaces and then optimizes the off-chip subspace followed by the on-chip subspace. We implemented our overall approach in a tool called Marvel , and a benefit of our approach is that it applies to any operator conformable with the MDC notation. We evaluated Marvel over major DNN operators and compared it with past optimizers.

Influence of coupling on the dynamics of three delayed oscillators

The purpose of this study is to construct the asymptotics of the relaxation regimes of a system of differential equations with delay, which simulates three diffusion-coupled oscillators with nonlinear compactly supported delayed feedback under the assumption that the factor in front of the feedback function is large enough. Also, the purpose is to study the influence of the coupling between the oscillators on the nonlocal dynamics of the model. Methods. We construct the asymptotics of solutions of the considered model with initial conditions from a special set. From the asymptotics of the solutions, we obtain an operator of the translation along the trajectories that transforms the set of initial functions into a set of the same type. The main part of this operator is described by a finite-dimensional mapping. The study of its dynamics makes it possible to refine the asymptotics of the solutions of the original model and draw conclusions about its dynamics. Results. It follows from the form of the constructed mapping that for positive coupling parameters of the original model, starting from a certain moment of time, all three generators have the same main part of the asymptotics — the generators are “synchronized”. At negative values of the coupling parameter, both inhomogeneous relaxation cycles and irregular regimes are possible. The connection of these modes with the modes of the constructed finite-dimensional mapping is described. Conclusion. From the results of the work it follows that the dynamics of the model under consideration is fundamentally influenced by the value of the coupling parameter between the generators.

Gridded GDP Projections Compatible With the Five SSPs (Shared Socioeconomic Pathways)

Historical and future spatially explicit population and gross domestic product (GDP) data are essential for the analysis of future climate risks. Unlike population projections that are generally available, GDP projections—particularly for scenarios compatible with shared socioeconomic pathways (SSPs)—are limited. Our objective is to perform a high-resolution and long-term GDP estimation under SSPs utilizing a wide variety of geographic auxiliary information. We estimated the GDP in a 1/12-degree grid scale. The estimation is done through downscaling of historical GDP data for 1850–2010 and SSP future scenario data for 2010–2100. In the downscaling, we first modeled the spatial and economic interactions among cities and projected different future urban growth patterns according to the SSPs. Subsequently, the projected patterns and other auxiliary geographic data were used to estimate the gridded GDP distributions. Finally, the GDP projections were visualized via three-dimensional mapping to enhance the clarity for multiple stakeholders. Our results suggest that the spatial pattern of urban and peri-urban GDP depends considerably on the SSPs; the GDP of the existing major cities grew rapidly under SSP1, moderately grew under SSP 2 and SSP4, slowly grew under SSP3, and dispersed growth under SSP5.

Three-dimensional mapping of distal humerus fracture

Background Distal humerus fractures (DHFs) constitute one-third of elbow fractures approximately. In this study, we aim to define and analyze the fracture lines and morphological features of DHFs using mapping technique. Methods One hundred and two DHFs were retrospectively reviewed. All the computed tomography (CT) data were used to manually reconstruct and virtually reduce the DHF fragments to fit a standard 3D model. Smooth curves were depicted accurately onto the surface of the template to represent the fracture lines. All the curves were overlapped onto the model to create the 3D fracture map and heat map. Results Our analysis was based on 102 CT images of DHFs, contributed by 59 male and 43 female patients (mean age, 46 years; range, 18-93 years), and included 15 type A, 25 type B, and 62 type C fractures. On mapping, the hot zones were located in the radial fossa, coronoid fossa, olecranon fossa, and the external part of the trochlear. Conversely, the cold zones were noted in medial condyle, the medial side of the trochlear, and the anterolateral area on the supracondylar ridge. Conclusions Our study firstly shows the fracture lines and morphological features of distal humeral fractures by three-dimensional mapping technology. Distal humerus fracture lines are characteristic and highly related to the micro-architecture difference of distal humerus, which may provide some guidance for the treatment plan selection and surgical fixation design.

On derivatives of fuzzy multi-dimensional mappings and applications under generalized differentiability

This article identifies and presents the generalized difference (g-difference) of fuzzy numbers, Fréchet and Gâteaux generalized differentiability (g-differentiability) for fuzzy multi-dimensional mapping which consists of a new concept, fuzzy g-(continuous linear) function; Moreover, the relationship between Fréchet and Gâteaux g-differentiability is studied and shown. The concepts of directional and partial g-differentiability are further framed and the relationship of which will the aforementioned concepts are also explored. Furthermore, characterization is pointed out for Fréchet and Gâteaux g-differentiability; based on level-set and through differentiability of endpoints real-valued functions a characterization is also offered and explored for directional and partial g-differentiability. The sufficient condition for Fréchet and Gâteaux g-differentiability, directional and partial g-differentiability based on level-set and through employing level-wise gH-differentiability (LgH-differentiability) is expressed. Finally, to illustrate the ability and reliability of the aforementioned concepts we have solved some application examples.