Gradient flow of elastic energy converges to elastica.
problem Optimizing closed curves to minimize elastic energy.
method Proving the existence of a unique global solution and convergence via Łojasiewicz--Simon gradient inequality.
result Convergence to elastica established for the H2(ds)-gradient flow of modified elastic energy. Study curves evolving by gradient flow of elastic energy, proving existence, smoothing, and convergence.
problem Evolution of curves with fixed length and clamped boundary conditions.
method Negative L2-gradient flow of elastic energy, existence, parabolic smoothing, constrained Lojasiewicz-Simon gradient inequality. result Convergence to a critical point as time tends to infinity.
Study on dynamic curves with elastic energy and spontaneous curvature.
problem Modeling and analyzing dynamic planar curves with elastic energy.
method Gradient flow of inclination angle, nonlocal quasilinear system, local well-posedness, global existence, convergence.
result Local well-posedness, global existence, convergence of the flow for weak regularity initial data.
The elastic flow of curves converges smoothly to a critical point.
problem Smooth convergence of elastic flow of curves.
method Application of Lojasiewicz-Simon inequality.
result Smooth convergence to a critical point.
Scheme minimizes p-elastic energy of curves over time.
problem Minimizing p-elastic energy of curves over time. method Minimizing movement scheme with approximate normal graphs.
result Short-time existence and lower bound on solution's lifetime.
The paper studies the free elastic flow of closed curves and finds their asymptotic shape converges to a circle.
problem Challenges in studying the asymptotic behavior of the free elastic flow for closed curves.
method Analysis of the free elastic flow as an L2-gradient flow for Euler's elastic energy. result An appropriate rescaling of initial curves geometrically close to circles converges to a unique round circle.
The elastic flow, which is the L2-gradient flow of the elastic energy, has several applications in geometry and elasticity theory. We present stable discretizations for the elastic flow in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples in…
We study the evolution of closed inextensible planar curves under a second order flow that decreases the p-elastic energy. A short time existence result for p∈(1,∞) is obtained via a minimizing movements method. For p=2, that is in the case of the classic elastic energy, long-time existence is retrieve…
Unified theory solves strain compatibility and elasticity of origami metamaterials.
problem Understanding and controlling the morphing paths of origami metamaterials.
method Unified theory for a wide array of origami tessellations, solving strain compatibility and elasticity.
result Origami metamaterials exhibit equal but opposite in-plane and out-of-plane Poisson's ratios and bending energy depends on strain gradient.
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.
Straight lines are a basin of attraction for the elastic flow at least to level 1.9615π.
problem Understanding the basin of attraction for the free boundary free elastic flow.
method Steepest descent gradient flow for elastic energy, numerical evidence.
result Straight lines have a basin of attraction at least to level 1.9615π.
Li-Yau inequality applied to curves in 2D space.
problem Curves in 2D space with low elastic energy.
method Classical Li-Yau inequality applied to curves.
result Analogous results for curves in 2D space with low elastic energy.
We consider parallel asynchronous Markov Chain Monte Carlo (MCMC) sampling for problems where we can leverage (stochastic) gradients to define continuous dynamics which explore the target distribution. We outline a solution strategy for this setting based on stochastic gradient Hamiltonian Monte Carlo sampling (SGHMC) …
Study on semiconcavity of solutions to gradient obstacle problems on compact manifolds.
problem Gradient obstacle problems on compact Riemannian manifolds.
method Uniform semiconcavity estimates and fine convergence results for solutions and free boundaries.
result The elastic and λ-elastic sets of solutions converge to the cut locus and λ-cut locus of the manifold. Study on local elasticity in neural network training, improving detection of class-specific changes.
problem Improving the detection of class-specific changes in neural network training.
method Comprehensive study of local elasticity, proposing a new definition to address limitations.
result New definition of local elasticity more sharply detects class-specific changes in neural network training.
This paper presents a phenomenon in neural networks that we refer to as \textit{local elasticity}. Roughly speaking, a classifier is said to be locally elastic if its prediction at a feature vector $\bx'$ is \textit{not} significantly perturbed, after the classifier is updated via stochastic gradient descent at a (labe…
Study on evolving interfaces with complex curvature and density effects.
problem Understanding the dynamics of evolving heterogeneous elastic interfaces.
method Modeling an evolving curve with a density function, analyzing the associated gradient flow evolution.
result Analysis of the preservation and asymptotic behavior of geometric properties in the evolving system.
Study curves evolving on hypersurfaces with free boundaries, preserving length.
problem Evolution of curves on hypersurfaces with free boundaries.
method Nonlocal evolution equation with nonlinear boundary conditions, short-time existence, uniqueness, and parabolic energy estimates.
result Global existence and convergence to critical points proved.
Paper introduces elastic consistency for distributed SGD, enabling convergence analysis.
problem Training large-scale machine learning models in distributed environments.
method Introduces elastic consistency as a general consistency model for distributed SGD.
result Derives convergence bounds for various distributed SGD methods.
The paper studies the convergence of elastic flows of curves into manifolds, proving smooth convergence under certain conditions.
problem The convergence of elastic flows of curves into manifolds.
method Parabolic estimates and Lojasiewicz-Simon gradient inequality.
result Smooth convergence of the flow to critical points under specific conditions.
Metric anomalies arising from a distribution of point defects (intrinsic interstitials, vacancies, point stacking faults), thermal deformation, biological growth, etc. are well known sources of material inhomogeneity and internal stress. By emphasizing the geometric nature of such anomalies we seek their representation…
EAGC boosts GCD by regulating gradient entanglement, improving known and novel category separability.
problem Gradient entanglement distorts supervised gradients and overlaps known and novel class representations.
method EAGC uses AGA and EEP to align and project gradients, reducing entanglement and overlap.
result EAGC consistently boosts GCD performance, setting new state-of-the-art results.
Study models deep learning training dynamics using locally elastic SDEs to reveal feature separability.
problem Understanding how deep learning models separate features from different classes during training.
method Modeling deep learning training using locally elastic SDEs with a drift term reflecting backpropagation impact.
result Local elasticity in SDEs leads to linear separability of features, resulting in vanishing training loss.
Study of phase separation and geometry on a closed elastic curve, including dynamics and free energy minimization.
problem Free energy and dynamics of a closed elastic filament coupled to a scalar concentration field.
method Analytical and numerical simulations of coupled Willmore flow and Cahn--Hilliard gradient flow on differential geometry.
result Qualitative changes in free energy landscape due to closure constraint, leading to metastable and stable multi-domain morphologies.
Proposes a CL technique to improve accuracy and reduce forgetting.
problem Sequential task learners struggle with forgetting information from previous tasks.
method Extracts modular parts of neural networks and estimates task relatedness.
result Remarkable performance gain in robustness to forgetting for EWC and GEM methods.
The paper studies how curves evolve under area constraints and converges to a critical point.
problem Evolution of plane curves with fixed area under elastic energy gradient.
method Local and global existence of the flow, simplicity assumption, Łojasiewicz--Simon inequality.
result The evolving curve's length remains bounded and converges to a critical point.
A new method relaxes Boolean Matrix Factorization to make it more efficient.
problem High computational cost of solving NP-hard combinatorial optimization problems in Boolean Matrix Factorization.
method Proposes a proximal gradient algorithm using an elastic-binary regularizer to relax BMF.
result Demonstrates improved runtime and better recall, loss, and interpretability on real-world data.
We study the pull-back of the 2-parameter family of quotient elastic metrics introduced in Mio-Srivastava-Joshi on the space of arc-length parameterized loops. This point of view has the advantage of concentrating on the manifold of arc-length parameterized curves, which is a very natural manifold when the analysis of …
Physics-informed GANs estimate elastic moduli from mechanical tests.
problem Estimating spatially-varying elastic moduli from measured deformations.
method Physics-informed Generative Adversarial Networks (PI-GANs) with PDE constraints.
result Generated stiffness samples match true distribution statistics.
Enhanced ECCD speeds up elastic net model training.
problem Efficiently solving generalized linear models with elastic net constraints.
method Redesigned cyclic coordinate descent with Taylor expansion and batched computations.
result Empirically shows consistent 3imes performance improvement over state-of-the-art solvers. Analyzes properties of stiffness tensors for elastic wave imaging.
problem Characterizing stiffness tensor fields for elastic wave imaging.
method Finsler-geometric methods applied to anisotropic stiffness tensor fields.
result Conditions for Finsler-geometric methods to be applicable.
This paper introduces an elasticity reconstruction method based on local displacement observations of elastic bodies. Sparse reconstruction theory is applied to formulate the underdetermined inverse problems of elasticity reconstruction including unobserved areas. An online local clustering scheme called a superelement…
Elastic Cash adjusts money supply to stabilize interest rates.
problem Stabilizing interest rates in a decentralized system.
method Modifies supply to keep interest rate fixed by public market.
result Improves elasticity of US Dollar and new cryptocurrencies.
This work introduces methods to compute optimal Monge maps and learn elastic costs for efficient data mapping.
problem Efficiently mapping one probability distribution to another using elastic costs.
method Proposes numerical methods to compute optimal Monge maps and a learning loss for parameterized regularizers.
result Proves the optimality of computed Monge maps and learns the parameters of elastic costs.
This paper puts forth a new formulation and algorithm for the elastic matching problem on unparametrized curves and surfaces. Our approach combines the frameworks of square root normal fields and varifold fidelity metrics into a novel framework, which has several potential advantages over previous works. First, our var…
Characterizes null Lagrangians in Cosserat elasticity.
problem Understanding null Lagrangians in micropolar elasticity.
method Applying Olver and Sivaloganathan's theorem to characterize null Lagrangians.
result Complete characterization of null Lagrangians for three-dimensional bodies and shells.
Approximate 3D elastic curves with exact constraints
problem Designing and approximating 3D elastic curves
method Numerically stable method for recovering 11 parameters
result Fast and stable approximation of arbitrary curves
This work presents a general unified theory for coupled nonlinear elastic and inelastic deformations of curved thin shells. The coupling is based on a multiplicative decomposition of the surface deformation gradient. The kinematics of this decomposition is examined in detail. In particular, the dependency of various ki…
The paper studies rigidity and continuity in nonlinear elasticity on manifolds and hypersurfaces.
problem Rigidity and continuity properties of elastic bodies in non-Euclidean settings.
method Geometric rigidity estimates, asymptotic rigidity of elastic membranes, simplified geometric proof of continuous dependence.
result Established geometric rigidity estimate and proved asymptotic rigidity of elastic membranes.
Study preserves planar and graphical properties of curves under elastic flow.
problem Maintaining planar and graphical properties of non-compact curves under elastic flow.
method Extended recent work on adapted elastic energy to derive thresholds for planar and graphical embeddedness.
result Derived new Li--Yau type inequality for complete planar curves.
Motivated by the problem of finding an explicit description of a developable narrow Moebius strip of minimal bending energy, which was first formulated by M. Sadowsky in 1930, we will develop the theory of elastic strips. Recently E.L. Starostin and G.H.M. van der Heijden found a numerical description for an elastic Mo…
Study on migrating elastic flows of curves across half-planes.
problem Migrating elastic flows of curves from upper to lower half-planes.
method Analytical and numerical construction of migrating elastic flows.
result Construction of various migrating elastic flows.
Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.
problem Recover stiffness tensor and density from Dirichlet-to-Neumann map.
method Analyze invariance under coordinate transformations and gauge freedoms.
result Present gauge freedoms in the Dirichlet-to-Neumann map for Riemannian elastic wave equation.
We address the issue of speeding up the training of convolutional networks. Here we study a distributed method adapted to stochastic gradient descent (SGD). The parallel optimization setup uses several threads, each applying individual gradient descents on a local variable. We propose a new way to share information bet…
Solves curve migration problem with elastic flows.
problem Curve migration problem with natural boundary conditions.
method Constructing migrating elastic flows.
result Extends previous work to purely local flow.
MIBoost boosts variable selection with multiple imputation.
problem Missing data complicates variable selection in statistical models.
method Gradient boosting with multiple imputation, MIBoost.
result MIBoost yields comparable predictive performance to other methods.
New insights into stability of special curves on spheres.
problem Stability of closed p-elastic curves on spheres. method Analytical proof and construction of curves.
result All closed spherical p-elastic curves for p∈(0,1) are unstable. Due to the advantage of achieving a better performance under weak regularization, elastic net has attracted wide attention in statistics, machine learning, bioinformatics, and other fields. In particular, a variation of the elastic net, adaptive elastic net (AEN), integrates the adaptive grouping effect. In this paper,…