Gradient | EPFL Graph Search

In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent. In coordinate-free terms, the gradient of a function f(\mathbf{r}) may be defined by: df=\nabla f \cdot d\mathbf{r} where df is the total infinitesimal c

Biomolecule gradients on surface initiated graft polymers

Ruben Casanova

Controlling the fate of stem cells in vitro is a key challenge towards using these cells in clinical applications. Adult stem cells (ASC) are known to reside in complex microenvironments called niches in vivo. These niches regulate stem cell fate providing a variety of signals that control in a balanced way self-renewal and differentiation. In vitro ASC culture is challenging using standard culture methods because these are not able to recreate an adequate environment mimicking niche signals. The use of bioengineered polymer surfaces may thus be a solution for controlling stem cell fate in a predictable and scalable manner. Surface initiated polymerisation (SIP) is a method allowing controlled free radical polymerisation. This technique can be used to achieve highly controlled polymer architectures. Combining this versatile technology with a microfluidic gradient maker allowed the patterning of surface gradient arrays of desired tethered biomolecules. We have used monomers carrying functional groups that can be used in highly specific and high-yielding reactions such as biotin methacrylate and alkyne methacrylate to achieve surface gradients of Streptavidin as well as azide-functionalized RGD peptides. Surface characterisation was carried out by X-ray photoelectron spectroscopy (XPS) as well as fluorescence microscopy to verify surface modification steps and gradient patterns. The quantification of immobilized biomolecules was carried out using Europium labelling assays. As a proof of concept gradients of RGD peptides were patterned on polymer brushes expecting dose responding attachment of adherent cells in the presence of these strong attachment motifs. The results showed a dose dependent attachment of fibroblast on gradients of azide-RGD. However this outcome could not be achieved with biotinylated RGD as a result of a failure in binding between the biotinylated peptide and streptavidin. Nevertherless, these experiments demonstrate the suitability of the overall strategy as a mechanism for surface patterning of specific molecules. Improvements of the method established here have the potential to result in a sophisticated platform for the screening of cell-material interactions.

2011

An Accelerated First-Order Method for Non-convex Optimization on Manifolds

Nicolas Boumal, Christopher Arnold Criscitiello

We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate first-order critical points faster than regular gradient descent. A randomized version also finds approximate second-order critical points. Both the algorithms and their analyses build extensively on existing work in the Euclidean case. The basic operation consists in running the Euclidean accelerated gradient descent method (appropriately safe-guarded against non-convexity) in the current tangent space, then moving back to the manifold and repeating. This requires lifting the cost function from the manifold to the tangent space, which can be done for example through the Riemannian exponential map. For this approach to succeed, the lifted cost function (called the pullback) must retain certain Lipschitz properties. As a contribution of independent interest, we prove precise claims to that effect, with explicit constants. Those claims are affected by the Riemannian curvature of the manifold, which in turn affects the worst-case complexity bounds for our optimization algorithms.

SPRINGER2022

Sur quelques équations aux dérivées partielles en lien avec le lemme de Poincaré

David Valentin Strütt

In this thesis, we study two distinct problems. The first problem consists of studying the linear system of partial differential equations which consists of taking a k-form, and applying the exterior derivative 'd' to it and add the wedge product with a 1-form 'a'. The study of this differential operator is linked to the study of the multiplication by a two form, that is the system of linear equations where we take a k-form and apply the exterior wedge product by 'da', the exterior derivative of 'a'. We establish links between the partial differential equation and the linear system. The second problem is a generalization of the symmetric gradient and the curl equation. The equation of a symmetric gradient consists of taking a vector field, apply the gradient and then add the transpose of the gradient, whereas in the curl equation we subtract the transpose of the gradient. Both can be seen as an equation of the form A * grad u + (grad u)t * A, where A is a symmetric matrix for the case of the symmetric gradient and skew symmetric for the curl equation. We generalize to the case where A verifies no symmetry assumption and more significantly add a Dirichlet condition on the boundary.

EPFL2018

Biomolecule gradients on surface initiated graft polymers

Ruben Casanova

2011