Modern simulation scenarios require real-time or many-query responses from a simulation model. This is the driving force for increased efforts in model order reduction for high-dimensional dynamical systems or partial differential equations. This demand for fast simulation models is even more critical for parameterized problems. Several snapshot-based methods for basis construction exist for parameterized model order reduction, for example, proper orthogonal decomposition or reduced basis methods. They require the careful choice of samples for generation of the reduced model. In this article we address two types of grid-based adaptivity that can be beneficial in such basis generation procedures. First, we describe an approach for training set adaptivity. Second, we introduce an approach for multiple bases on adaptive parameter domain partitions. Due to the modularity, both methods also can easily be combined. They result in efficient reduction schemes with accelerated training times, improved approximation properties and control on the reduced basis size. We demonstrate the applicability of the approaches for instationary partial differential equations and parameterized dynamical systems.
An Interactive Genetic Algorithm is proposed to progressively sketch the desired side-view of a car profile. It adopts a Fourier decomposition of a 2D profile as the genotype, and proposes a cross-over mechanism. In addition, a formula function of two genes' discrepancies is fitted to the perceived dissimilarity between two car profiles. This similarity index is intensively used, throughout a series of user tests, to highlight the added value of the IGA compared to a systematic car shape exploration, to prove its ability to create superior satisfactory designs and to stimulate designer's creativity. These tests have involved six designers with a design goal defined by a semantic attribute. The results reveal that if "friendly" is diversely interpreted in terms of car shapes, "sportive" denotes a very conventional representation which may be a limitation for shape renewal.
The Kalman filter is a widely known tool in control theory for estimating the state of a linear system disturbed by noise. However, when applying the Kalman filter on systems described by parametrerized partial differential equations (PPDEs) the calculation of state estimates can take an excessive amount of time and real-time state estimation may be infeasible. In this work we derive a low dimensional representation of a parameter dependent Kalman filter for PPDEs via the reduced basis method. Thereby rapid state estimation, and in particular the rapid estimation of a linear output of interest, will be feasible. We will also derive a posteriori error bounds for evaluating the quality of the output estimations. Furthermore we will show how to verify the stability of the filter using an observability condition. We will demonstrate the performance of the reduced order Kalman filter and the error bounds with a numerical example modeling the heat transfer in a plate.
PDE-constrained parameter optimization problems suffer from the high dimensionality of the corresponding discretizations, which results in long optimization runtimes. One possible approach to solve such large scale optimization problems more rapidly is to replace the PDE constraint by a low-dimensional model constraint obtained via model reduction. We present a general technique for certification of such surrogate optimization results by a-posteriori error estimation based on Reduced Basis (RB) models. We allow arbitrary PDEs and optimization functionals, in particular cover nonlinear optimization problems. Experiments on a stationary heat-conduction problem demonstrate the applicability of the error bound.
-So as to create innovative car silhouettes, we propose in this paper a model based on an Interactive Genetic Algorithm using an encoding of a design solution by a Fourier analysis approach. This model permits the designer to browse through generations of car profiles from an initial population of existing silhouettes. By qualitatively assessing each individual, the designer converges towards solutions complying with his/her requirements and preferences, possibly creating novelty and generating surprise. We describe here tests for assessing the efficiency of this innovative design platform. These tests are mainly based on a similarity index, a similarity measure being the perceived distance between two cars silhouettes. The results highlight a good convergence toward a satisfactory solution. In addition, this design process turns out to be very flexible because of the local and intuitive modifications allowed on a given individual solution at any moment of the design process.
An Interactive Genetic Algorithm system is proposed for designing a car silhouette while involving the style designer in the evaluation process of a population of individuals. This IGA is based on the principle of an indirect encoding of a closed curve genotype using a primary Fourier decomposition. A crossing over operator is proposed for mixing the parents’ genes by a random weighted average into a new child’s genotype. A perceived similarity index between two genotypes is built to check that our IGA is able to converge toward a target individual starting from the genes of an initial population.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.