Related papers: On the Minimax Bifurcation Formula
A variational method is presented for directly finding the bifurcation point of nonlinear equations as the saddle-node point of the extended nonlinear Rayleigh quotient. The method is applied for solving an open problem on the existence of…
We develop a novel method for finding bifurcations for nonlinear systems of equations based on directly finding bifurcations through saddle points of extended quotients. The method is applied to find the saddle-node bifurcation point for…
This paper provides a direct method of establishing the existence and uniqueness of saddle-node bifurcations for nonlinear equations in general domains. The method employs the scaled extended quotient whose saddle points correspond to the…
We introduce the nonlinear generalized Collatz-Wielandt formula $$ \lambda^*= \sup_{x\in Q}\min_{i:h_i(x) \neq 0} \frac{g_i(x)}{ h_i(x)}, ~~Q \subset \mathbb{R}^n,$$ and prove that its solution $(x^*,\lambda^*)$ yields the maximal…
We develop a new paradigm for finding bifurcations of solutions of nonlinear problems, which is based on the detection of extreme values of new type of variational functional associated with the considering problem. The variational…
With the dual variational principle and the saddle point reduction we use the abstract bifurcation theory recently developed by author in previous work to prove many new bifurcation results for solutions of four types of Hamiltonian…
In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the…
A minimax variational principle for saddle-point solutions with prescribed energy levels is introduced. The approach is based on the development of the linking theorem to the energy level nonlinear generalized Rayleigh quotients. An…
A Langevin equation whose deterministic part undergoes a saddle-node bifurcation is investigated theoretically. It is found that statistical properties of relaxation trajectories in this system exhibit divergent behaviors near a saddle-node…
We analyze, mainly using bifurcation methods, an elliptic superlinear problem in one-dimension with periodic boundary conditions. One of the main novelties is that we follow for the first time a bifurcation approach, relying on a…
This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a…
Using min-max inequality we investigate the existence of solutions and thier dependence on parameters for some second order discrete boundary value problem. The approach is based on variational methods and solutions are obtained as saddle…
In this paper we propose an algorithm for the detection of edges in images that is based on topological asymptotic analysis. Motivated from the Mumford--Shah functional, we consider a variational functional that penalizes oscillations…
We develop primal-dual coordinate methods for solving bilinear saddle-point problems of the form $\min_{x \in \mathcal{X}} \max_{y\in\mathcal{Y}} y^\top A x$ which contain linear programming, classification, and regression as special cases.…
In this work, we propose and analyze a residual-minimization strategy for the numerical solution of nonlinear PDEs posed in Banach spaces. Given a finite-dimensional trial space and a suitably enriched discrete test space (of higher…
Deciding whether saddle points exist or are approximable for nonconvex-nonconcave problems is usually intractable. This paper takes a step towards understanding a broad class of nonconvex-nonconcave minimax problems that do remain…
We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet-Raviart formulation for the biharmonic problem to…
Bifurcations mark qualitative changes of long-term behavior in dynamical systems and can often signal sudden ("hard") transitions or catastrophic events (divergences). Accurately locating them is critical not just for deeper understanding…
In this paper, we present a unified analysis of methods for such a wide class of problems as variational inequalities, which includes minimization problems and saddle point problems. We develop our analysis on the modified Extra-Gradient…
In this paper we obtain, for a semilinear elliptic problem in R^N, families of solutions bifurcating from the bottom of the spectrum of $-\Delta$. The problem is variational in nature and we apply a nonlinear reduction method which allows…