Novel method for shape optimization of non-smooth PDEs.
problem Optimizing shapes governed by non-smooth PDEs.
method Functional variational approach and sensitivity analysis.
result Necessary conditions for locally optimal shapes.
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48 results for “locally optimal shapes”
A new method for 3D surface registration using dynamic programming.
problem Elastic shape registration of 3D surfaces.
method Optimization over a subset of reparametrizations using dynamic programming.
result Proposes an algorithm that produces a solution closer to optimal than gradient-based methods.
A new method shapes reinforcement learning environments by abstracting large state spaces.
problem Learning in large, noisy environments with sparse feedback.
method Environment shaping using state abstraction.
result Agent's policy in shaped environment preserves near-optimal behavior in original environment.
Difficult image segmentation problems, for instance left atrium MRI, can be addressed by incorporating shape priors to find solutions that are consistent with known objects. Nonetheless, a single multivariate Gaussian is not an adequate model in cases with significant nonlinear shape variation or where the prior distri…
Fine shape of local compacta represented by ordinary maps.
problem Representing fine shape of local compacta.
method Constructing a space for each local compactum such that fine shape classes correspond to homotopy classes of maps to .
result Fine shape classes from any locally compact metrizable space to bijectively correspond to homotopy classes of maps from to .
The study finds minimal hypersurfaces in wedge-shaped manifolds with boundary.
problem Finding minimal hypersurfaces in wedge-shaped manifolds with boundary.
method Developed a min-max theory for locally wedge-shaped manifolds with boundary.
result Proved existence of smooth free boundary minimal hypersurfaces in wedge-shaped manifolds.
Local minimizers are convex and close to Wulff shapes.
problem Finding local minimizers in anisotropic isoperimetric problems.
method Showed local minimizers are geodesically convex and small smooth perturbations of tangent Wulff shapes.
result Local minimizers are quantitatively close to Wulff shapes.
Linear Shape Deformation Models with Local Support Using Graph-based Structured Matrix Factorisationcs.CV
Representing 3D shape deformations by linear models in high-dimensional space has many applications in computer vision and medical imaging, such as shape-based interpolation or segmentation. Commonly, using Principal Components Analysis a low-dimensional (affine) subspace of the high-dimensional shape space is determin…
Curvature estimates for stable free boundary minimal hypersurfaces in locally wedge-shaped manifoldsmath.DG
Curvature estimate for stable free boundary minimal hypersurfaces in wedge-shaped manifolds.
problem Estimating curvature of stable free boundary minimal hypersurfaces in wedge-shaped manifolds.
method Compactness theorem and Schoen-Simon-Yau estimates.
result Curvature estimate for free boundary minimal hypersurfaces in wedge-shaped manifolds.
In this work, we prove an optimal Penrose inequality for asymptotically locally hyperbolic manifolds which can be realized as graphs over Kottler space. Such inequality relies heavily on an optimal weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Kottler space.
Fine shape theory extends strong shape to noncompact metrizable spaces.
problem Computational complexity in extending strong shape to noncompact spaces.
method Introducing FDR-embeddings and mapping cylinders to extend SSDR-maps to noncompact spaces.
result Fine shape category can be represented as a left fraction localization.
A new shape space allows optimization of non-smooth shapes in fluid mechanics.
problem Optimizing non-smooth shapes in fluid mechanics.
method Constructing a product manifold to include piecewise-smooth shapes.
result Numerical results show applicability in minimizing viscous energy dissipation.
Optimization of piecewise smooth shapes under uncertainty using the example of Navier-Stokes flowmath.OC
Optimizes shapes in uncertain Navier-Stokes flow problems.
problem Optimizing shapes with geometric constraints and physical uncertainty.
method Multi-shape calculus and stochastic augmented Lagrangian method.
result Successfully optimized shapes in uncertain Navier-Stokes flow.
Design-by-Morphing creates radical airfoil designs without geometric constraints.
problem Design constraints limit airfoil design novelty and small changes.
method Design-by-Morphing (DbM) creates a search space without geometric constraints.
result DbM generates radical airfoils with remarkable lift-over-drag ratio and stall angle tolerance.
Optimizes shapes on non-standard manifolds.
problem Optimization on non-standard infinite-dimensional manifolds.
method Develops gradient descent on weak Riemannian manifolds.
result Establishes foundational properties for optimization on various weak Riemannian manifolds.
PointGMM learns hGMMs from point clouds for 3D shape representation.
problem Lack of shape priors and non-local information in point cloud representations.
method Neural network that learns hierarchical Gaussian mixture models (hGMMs) for 3D shapes.
result Generative model learns meaningful latent space for interpolations and novel shape synthesis.
NCG methods improve shape optimization efficiency.
problem Shape optimization problems
method Nonlinear conjugate gradient methods
result NCG methods are efficient for shape optimization
Enhanced data efficiency using deep neural networks and Gaussian processes for aerodynamic design optimizationphysics.comp-ph
Enhanced aerodynamic design using machine learning and Gaussian processes.
problem High computational costs and local optima in adjoint-based aerodynamic optimization.
method Surrogate-based framework combining deep neural networks and Gaussian processes.
result Improves accuracy and reduces computational cost compared to adjoint-based methods.
Proves local maximizers for higher Ekeland-Hofer capacities in 4D star-shaped domains.
problem Finding local maximizers for higher Ekeland-Hofer capacities in specific domains.
method Analogous to 4D local Viterbo conjecture, proving maximizers for rational ellipsoids.
result Local maximizers of the k-th Ekeland-Hofer capacities are symplectomorphic to rational ellipsoids.
The paper proves a Whitehead theorem for fine shape spaces.
problem Proving a Whitehead theorem for fine shape spaces.
method Using Steenrod-Sitnikov homotopy groups and ind-groups.
result Fine shape morphisms are equivalences if they induce isomorphisms on π_i.
Method flattens complex surfaces with consistent density and shape.
problem Shape deformations and local geometric distortions in density-equalizing maps for multiply-connected surfaces.
method Formulates density diffusion as a quasiconformal flow, solving an energy minimization problem involving the Beltrami coefficient to ensure bijectivity and control distortion.
result Achieves optimal parameterization of multiply-connected surfaces with bijective and controlled geometric distortions.
The paper characterizes dynamic return and star-shaped risk measures via BSDEs.
problem Characterizing dynamic return and star-shaped risk measures.
method Characterization of star-shaped functionals and BSDEs.
result Existence of convex BSDEs with non-empty set of supersolutions.
The paper shows how to use fine shape to understand infinite-dimensional spaces.
problem Understanding infinite-dimensional metrizable spaces and their homology theories.
method Obtained results indicating fine shape is tractable and can be used for Polish spaces.
result Every Polish space is fine shape equivalent to the limit of an inverse sequence of simplicial maps.
We introduce a method called multi-scale local shape analysis, or MLSA, for extracting features that describe the local structure of points within a dataset. The method uses both geometric and topological features at multiple levels of granularity to capture diverse types of local information for subsequent machine lea…
A major challenge in current optimization research for deep learning is to automatically find optimal step sizes for each update step. The optimal step size is closely related to the shape of the loss in the update step direction. However, this shape has not yet been examined in detail. This work shows empirically that…
Flat minimal hypersurfaces found in wedge-shaped domains.
problem Finding minimal surfaces in wedge-shaped domains.
method Proving stability and flatness of -to-edge minimal hypersurfaces.
result Stable minimal hypersurfaces are flat in wedge-shaped domains.
Optical scatterometry is a method to measure the size and shape of periodic micro- or nanostructures on surfaces. For this purpose the geometry parameters of the structures are obtained by reproducing experimental measurement results through numerical simulations. We compare the performance of Bayesian optimization to …
In typical applications of Bayesian optimization, minimal assumptions are made about the objective function being optimized. This is true even when researchers have prior information about the shape of the function with respect to one or more argument. We make the case that shape constraints are often appropriate in at…
Space mapping speeds up shape optimization for PDEs.
problem Efficiently solving shape optimization problems constrained by PDEs.
method Combines fine and coarse model optimizations using Riemannian metrics.
result Space mapping methods are highly efficient for complex shape optimization problems.
Information Cascades Model captures dynamical properties of user activity in a social network. In this work, we develop a novel framework for activity shaping under the Continuous-Time Information Cascades Model which allows the administrator for local control actions by allocating targeted resources that can alter the…
Bézier-GAN optimizes airfoil design by reducing shape complexity.
problem High computational cost in aerodynamic shape optimization.
method Generative adversarial networks (GANs) to learn compact shape representations.
result Empirically accelerates optimization convergence by at least two times.
Unified treatment of elastic metrics for curves in any dimension.
problem Defining metrics on spaces of Euclidean curves for statistical analysis.
method Developing a unified approach to elastic metrics, extending results on existence of solutions and algorithms for computing distances and geodesics.
result Unified treatment of elastic metrics for all parameter choices, extending previous work.
In this paper, we use the inverse mean curvature flow to establish an optimal Minkowski type inquality, weighted Alexandrov-Fenchel inequality for the mean convex star shaped hypersurfaces in Reissner-Nordström-anti-deSitter manifold and Penrose type inequality for asymptotically locally hyperbolic manifolds in which c…
A new measure -variance captures local distributional shape.
problem Summarizing distributional shape with local information.
method Random bipartite matchings and stochastic approximation.
result Easily approximated -variance measures capture local distributional properties.
NVA combines variational posteriors, annealing, and natural-gradient learning for multimodal optimization.
problem Finding multiple global and local modes in nonconvex objectives.
method NVA integrates variational posteriors, annealing, and natural-gradient learning.
result NVA outperforms gradient descent and evolution strategies on simulations and real-world problems.
Study on shape optimization for specific eigenvalue problems on domains.
problem Shape optimization of eigenvalue problems for fourth order Steklov.
method Asymptotic expansion and sharp upper bound derivation.
result Derivation of eigenvalue spectra and shape optimization conclusions.
Improves CNN robustness by reducing texture bias.
problem CNNs' reliance on local texture over global shape.
method Inspired by human vision, InfoDrop decorrelates model output from local texture.
result Enhanced robustness across various scenarios.
Paper connects surface shape analysis and unbalanced optimal transport.
problem Computing the SRNF shape distance on piecewise linear surfaces.
method Characterizes SRNF shape distance as WFR distance pullback, proposes new algorithm for WFR distance computation.
result Direct computation of SRNF shape distance on piecewise linear surfaces.
New method accelerates smooth games using spectral shape analysis.
problem Accelerating optimization in smooth games with complex numerical challenges.
method Matrix iteration theory and spectral shape analysis to characterize and manipulate acceleration.
result Identified a continuum of optimization strategies from convex minimization to gradient descent.
Proves a tree of shapes for n-D images in optimal time.
problem Proving the mathematical structure of tree of shapes in n-D images.
method Optimal quasi-linear time algorithm for computing the tree of shapes.
result The tree of shapes is the self-dual morphological hierarchical structure of n-D gray-level images.
Study examines preservation of curvature-adaptedness during mean curvature flow.
problem Preservation of curvature-adaptedness during mean curvature flow.
method Investigates curvature-adaptedness in locally symmetric spaces.
result Curvature-adaptedness is preserved along mean curvature flow.
Estimates mean curvature, scalar curvature, shape operator in warped products.
problem Estimating geometric properties in warped product spaces.
method Local and global upper estimates for curvature and shape operator.
result Results on pseudo-hyperbolic spaces and space forms.
Study uses outer metrics for PDE-constrained shape optimization over diffeomorphism group.
problem Optimizing shapes governed by PDEs over the diffeomorphism group.
method Outer metrics on diffeomorphism group, Riemannian steepest descent method.
result Riemannian approach outperforms other metrics in solving PDE-constrained shape optimization problems.
Programming evolution of geometry in shape-morphing sheets via spatiotemporal activationcond-mat.soft
New method allows sheets to morph into multiple shapes via spatially varying stimuli.
problem Limitation of current shape-programmed sheets to achieve only one target geometry.
method Patterning the stimulus itself for spatiotemporal control over local deformation magnitudes.
result A single physical sample can be induced to traverse a continuous family of target geometries.
Differentiable pipeline replaces non-differentiable CAE components for shape optimization.
problem Gradient-based optimization is limited by non-differentiable components in CAE workflows.
method Surrogate models replace non-differentiable pipeline components, enabling gradient-based optimization.
result Gradient-based shape optimization possible without differentiable solvers.
New approach reduces shape optimization anomalies and improves design quality.
problem Improving global optimization efficiency and avoiding geometrical anomalies in shape optimization.
method Reducing design variables, modeling generative process via probabilistic models, penalizing anomalous designs.
result Abnormal designs are penalized, leading to high-quality designs and improved convergence.
In this paper, an unsupervised machine learning method for geometric constellation shaping is investigated. By embedding a differentiable fiber channel model within two neural networks, the learning algorithm is optimizing for a geometric constellation shape. The learned constellations yield improved performance to sta…
Paper tackles shape graph registration using neural networks.
problem Constrained registration of shape graphs with varying nodes and edges.
method Shape-Graph Matching Network (SGM-net) with an elastic shape metric loss function.
result State-of-the-art matching performance and reduced computational cost.