Statistical Mechanics of Money, Income, Debt, and Energy Consumption
By analogy with the probability distribution of energy in physics, entropy maximization results in the exponential Boltzmann-Gibbs probability distribution of money among the agents in a closed economic system. Analysis of empirical data shows that income distributions in USA, European Union, and other countries have a well-defined two-class structure. The majority of the population (about 97%) belongs to the lower class characterized by the exponential ("thermal") distribution. The upper class (about 3% of the population) is characterized by the Pareto power-law ("superthermal") distribution, and its share of the total income expands and contracts dramatically during bubbles and busts in financial markets. Globally, inequality in energy consumption per capita around the world has decreased in the last 30 years and now approaches to the exponential probability distribution, in agreement with the maximal entropy principle. All papers are available at http://physics.umd.edu/~yakovenk/econophysics/. This work is currently supported by the Institute for New Economic Thinking, http://ineteconomics.org/grants/
Coagulation-Fragmentation model without detailed balance for animal group-size statistics
In this talk, I will present some mathematical results for a
coagulation-fragmentation model used to study animal group-size
statistics. There is no detailed balance for this
coagulation-fragmentation model and hence the current mathematical
theory can not be used to study this problem. Based on the complete
Bernstein function theory, we establish global existence, convergence to
the steady solution, asymptotic behavior, and complete monotonicity
property of the steady solution for this coagulation-fragmentation
This is a joint work with Pierre Degond and Bob Pego.
Prof. Bo Li, Department of Mathematics and Center for Theoretical Biological Physics, University of California San Diego
Variational Implicit Solvation of Biomolecules
The structure and dynamics of biomolecules such as DNA and proteins determine the functions of underlying biological systems. Modeling biomolecules is, however, extremely challenging due to their enormous complexity. Recent years have seen the initial success of variational implicit-solvent models (VISM) for biomolecules. Central in VISM is an effective free-energy functional of all possible solute-solvent interfaces, coupling together the solute surface energy, solute-solvent van der Waals interactions, and electrostatic contributions. Numerical relaxation by the level-set method of such a functional determines biomolecular equilibrium conformations and minimum free energies. Comparisons with experiments and molecular dynamics simulations demonstrate that the level-set VISM can capture the hydrophobic hydration, multiple dry and wet states, and many other important solvation properties. This talk begins with a description of the level-set VISM and continues to present new developments around the VISM. These include: (1) the coupling of solute molecular mechanical interactions in the VISM; (2) the effective dielectric boundary forces; and (3) the solvent fluid fluctuations. Mathematical theory and numerical methods are discussed, and applications are presented. This is joint work mainly with J. Andrew McCammon, Li-Tien Cheng, Joachim Dzubiella, Jianwei Che, Zhongming Wang, Shenggao Zhou, and Zuojun Guo.
Green's functions for time-dependent Fokker-Planck equations
We construct explicit approximate Green's functions for
time-dependent, linear Fokker-Planck equations in terms of Dyson series, Taylor
expansions, and exact commutator formulas. Under a uniform parabolicity condition on the operator, the associated approximate solution to the initial-value problem is accurate in terms of Sobolev norms to arbitrary order in time in the short-time limit. Bootstrap allows to extend the construction to large time.
The algorithm works well also for certain types of degenerate equations with vanishing and unbounded coefficients, such as those arising in pricing of contingent claims. The advantage of this method is that it requires only numerical integration and appears very stable numerically.
This is joint work with Victor Nistor and Wen Cheng.
Malignant brain tumors are rapidly progressive and fatal. Even though they do not metastasize outside the brain, following neurosurgical resection, radiotherapy and chemotherapy, tumors inevitably recur. Glioma genomes are highly unstable, with elevated genomic mutations and chromosomal abnormalities. However, genomic subtype classifications have been slow to provide increased understanding or effective treatments. We propose a new approach to understanding glioma tumors. Namely, by characterizing glioma growth patterns we aim to determine how glioma tumors can grow in a tissue with no extracellular space while causing minor symptoms for long times, and how individual growth patterns determine response to treatments.
Glioma cells can grow on blood vessels, myelin fibers, subpially, perineuronally and through interstitial space. The molecular basis of such growth patterns remains unknown. We are currently characterizing growth patterns of individual rodent and human glioma cells, and glioma stem cell like cells. We identified a series of cells (both rodent and human) which grow exclusively along blood vessels. We have characterized their detailed growth patterns, and determined the microenvironmental basis which determines glioma growth. Understanding glioma cell behavior has allowed us to uncover how cellular growth patterns determine the progression, microenvironmental interactions with the innate and adaptive immune systems, and the responses to antiangiogenic treatment.
In the long run we are interested in building abstract mathematical models to capture the behavioral specificity and variability of glioma growth, and use mathematical models to help us predict tumor progression and treatment response.
Prof. Richard Tsai, Department of Mathematics and Institute for Computational Engineering and Sciences, University of Texas at Austin
Boundary integral methods using signed distance function and the closest point mapping
We present new numerical algorithms for solving Poisson's and Helmholtz equations on domains with smooth boundaries.
Using either signed distance function or the closest point mapping to the domain boundaries, we formulate and solve the corresponding integral equations which are defined in a thin tubular neighborhoods of the domain boundaries. Our algorithms are suitable for computing interfaces whose dynamics are derived from the solutions of such PDEs in the enclosed set.
We shall also present new algorithms for integration over open curves and surfaces using point clouds sampled from these geometrical objects.
Nonlocal-interaction equations: phenomena and structures
Nonlocal-interaction equations serve as one of the basic models of biological aggregation.
The interaction between individuals is typically attractive at large distances and repulsive
at short distances. We will discuss several phenomena appearing in such systems: the variety of patterns that stable steady states exhibit, rolling traveling waves in heterogeneous environments, and phase separation (flock / empty space) in systems with nonlinear diffusion. In addition new structures appearing in systems where the long range interaction is attenuated by the crowdedness will be discussed.
We will introduce the gradient flow structure of the above systems and indicate how it can be used to prove well-posedness of equations, study the nonlinear stability of steady states, establish the interfacial behavior and study new models that take the crowdedness into account.
An Eulerian-Lagrangian Method for Optimization Problems Governed by Multidimensional Nonlinear Hyperbolic PDEs
In this talk, I will present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. The approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method, which utilizes advantages of both the Eulerian finite-volume scheme (for solving the balance law) and the Lagrangian discrete characteristics method (for solving the adjoint transport equation). Experimental convergence rates as well as numerical results for optimization problems with both linear and nonlinear constraints and a duct design problem will also be presented and discussed.
Continuum descriptions for dynamics of self-propelled particles
Dynamics of particles with self-propulsion, in the simplest case expressed by the condition of constant speed, has received a lot of attention in recent years due to applications in emergent self-organized behavior such as flocking and swarming. In situations when the number of particles becomes large a continuum description becomes possible, in which a system may be described by a coarse-grained density and direction fields. The lack of the momentum conservation in such systems presents specific difficulties in applying known approaches from kinetic theory. I will discuss results of analysis and numerics concerning the problem of validation of hydrodynamic equations for systems of self-propelled particles with various types of interactions.
The purpose of this talk is to present some recent results on the development of semilagrangian high order method for some kinetic equations. Two particular applications are considered, namely Vlasov-Poisson system and BGK model.
For the VP system, high order semilagrangian methods are obtained by tracing back the characteristics form each grid node in phase space, by solving (backward) their evolution equation in a self-consistent electric field. The solution at the foot of the characteristic at time tn is reconstructed by WENO interpolation in space and velocity.
For the BGK model, high order semilagrangian schemes are obtained by integrating the equation along the characteristics (in the forward direction). Since there is no drift, the characteristics are known, and the solution at time tn at the foot of the characteristic is obtained by WENO interpolation in space. Implicit schemes are used to avoid restriction on the time step in case of small relaxation time. Because of the special structure of the collision operator of BGK, the implicit equation can be explicitly solved. Two family of schemes are considered and compared: implicit Runge-Kutta and BDF.
Both approaches described above (for VP and BGK) are non-conservative in nature.
The third part of the talk is devoted to a general technique that can be used to make the method conservative. The approach is a conservative correction that can be applied to a non-conservative method. Examples of conservative schemes constructed by this techniques are illustrated in various contexts. When applied to semilagrangian schemes, the technique suffers from CFL-type stability restriction. Stability analysis is performed to understand the instability due to time discretization, however the contribution of space discretization has not yet been analyzed.
NO SEMINAR, Thanksgiving
2:00 PM **Math 3206** Note Location
PDE/Applied Math Seminar
Dr. Jacob Bedrossian, Courant Institute, New York University
Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations
We prove asymptotic stability of shear flows close to the planar, periodic Couette flow in the 2D incompressible Euler equations. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically mixed to small scales by an almost linear evolution and in general enstrophy is lost in the weak limit. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. Joint work with Nader Masmoudi.
Beyond incoherence and beyond sparsity: A new framework for compressed sensing
Compressed sensing is an area of applied mathematics, engineering and computer science that deals with the recovery of objects - e.g. images and signals - from seemingly highly incomplete sets of data. To achieve this, it relies on three key principles: (i) sparsity of the object to be recovered in some appropriate basis or frame (ii) incoherence between the sparsity system and the sampling system and (iii) uniform random subsampling of the measurement space. Subject to these conditions, the now well-established theory of compressed sensing shows that one can reconstruct objects using near-optimal numbers of measurements.
Unfortunately, in many practical problems (e.g. MRI) incoherence is lacking. Whilst compressed sensing techniques have often been applied successfully in such areas, there is no theory to explain why. In this talk I will present a mathematical framework for compressed sensing that explains such empirical results. In this theory, the concepts of sparsity, incoherence and uniform random subsampling are relaxed to three new principles: asymptotic sparsity, asymptotic incoherence and multilevel random subsampling. As I demonstrate, these new concepts are more realistic in applications. In particular, sparsity is too crude a model to explain the reconstruction quality seen in practice in such applications. The new theory shows that compressed sensing is possible under these more general conditions, and in several important settings (including MRI) it is possible to get near-optimal recovery guarantees. Finally, I will discuss a number of interesting consequences of this new theory.
This is joint work with Anders C. Hansen, Clarice Poon and Bogdan Roman.