public 01:06:01

Vakhtang Poutkaradze : Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries

  -   Applied Math and Analysis ( 57 Views )

Physics-Informed Neural Networks (PINNs) have received much attention recently due to their potential for high-performance computations for complex physical systems, including data-based computing, systems with unknown parameters, and others. The idea of PINNs is to approximate the equations and boundary and initial conditions through a loss function for a neural network. PINNs combine the efficiency of data-based prediction with the accuracy and insights provided by the physical models. However, applications of these methods to predict the long-term evolution of systems with little friction, such as many systems encountered in space exploration, oceanography/climate, and many other fields, need extra care as the errors tend to accumulate, and the results may quickly become unreliable. We provide a solution to the problem of data-based computation of Hamiltonian systems utilizing symmetry methods. Many Hamiltonian systems with symmetry can be written as a Lie-Poisson system, where the underlying symmetry defines the Poisson bracket. For data-based computing of such systems, we design the Lie-Poisson neural networks (LPNets). We consider the Poisson bracket structure primary and require it to be satisfied exactly, whereas the Hamiltonian, only known from physics, can be satisfied approximately. By design, the method preserves all special integrals of the bracket (Casimirs) to machine precision. LPNets yield an efficient and promising computational method for many particular cases, such as rigid body or satellite motion (the case of SO(3) group), Kirchhoff's equations for an underwater vehicle (SE(3) group), and others. Joint work with Chris Eldred (Sandia National Lab), Francois Gay-Balmaz (CNRS and ENS, France), and Sophia Huraka (U Alberta). The work was partially supported by an NSERC Discovery grant.

public 01:14:21

Valery A. Kholodnyi : Foreign Exchange Option Symmetry and a Coordinate-Free Description of a Foreign Exchange Option Market

  -   Applied Math and Analysis ( 28 Views )

In spite of the fact that symmetries play one of the major roles in physics, the ir usage in finance is relatively new and, to the best of our knowledge, can be traced to 1995 when Kholodnyi introduced the beliefs-preferences gauge symmetry. In this talk we present another symmetry, foreign exchange option symmetry, int roduced by Kholodnyi and Price in 1996. Foreign exchange option symmetry associa tes financially equivalent options on opposite sides of the foreign exchange mar ket. In a two-currency market, the foreign exchange option symmetry is formalized in terms of the one-dimensional Kelvin transform. In a multiple-currency market the foreign exchange option symmetry is formalized in terms of differential geometr y on graphs, that is, in terms of vector lattice bundles on graphs and connectio ns on these bundles. Foreign exchange option symmetry requires no assumptions on the nature of a prob ability distribution for exchange rates. In fact, it does not even require the a ssumptions of the existence of such a distribution. Furthermore, the symmetry is applicable not only to a foreign exchange market but to any financial market as well. The practical applications of the foreign exchange option symmetry range from th e detection of a new type of true arbitrage to the detection of inconsistent mod els of foreign exchange option markets and the development of algorithms and sof tware to value and analyze portfolios of foreign exchange options.

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public 01:11:27

E. Bruce Pitman : Tubuloglomerular Feedback-Mediated Dynamics in Two Coupled Nephrons

  -   Applied Math and Analysis ( 31 Views )

Previously, we developed a ``minimal'' dynamic model for the tubuloglomerular feedback (TGF) system in a single, short-looped nephron of the mammalian kidney. In that model, a semilinear hyperbolic partial differential equation was used to represent two fundamental processes of mass transport in the nephron's thick ascending limb (TAL): chloride advection by fluid flow through the TAL lumen and transepithelial chloride transport from the lumen to the interstitium. An empirical function and a time delay were used to relate nephron glomerular filtration rate to the chloride concentration at the macula densa of the TAL. Analysis of the model equations indicated that limit-cycle oscillations (LCO) in nephron fluid flow and chloride concentration can emerge for suffficiently large feedback gain and time delay. In this study, the single-nephron model has been extended to two nephrons, which are coupled through their filtration rates. Explicit analytical conditions were obtained for bifurcation loci corresponding to two special cases: (1) identical time-delays, but differing gains, and (2) identical feedback gain magnitudes, but differing time delays. Similar to the case of a single nephron, the analysis indicates that LCO can emerge in coupled nephrons for sufficiently large gains and delays. However, these LCO may emerge at lower values of the feedback gain, relative to a single (i.e., uncoupled) nephron, or at shorter delays, provided the delays are sufficiently close. These results suggest that, in vivo, if two nephrons are sufficiently similar, then coupling will tend to increase the likelihood of LCO. (In collaboration with Roman M. Zaritski, Leon C. Moore and H. E. Layton)

public 57:45

Edward Belbruno : Low Energy Trajectories in Celestial Mechanics and Stability Transition Regions With Applications to Astronomy and Space Travel

  -   Applied Math and Analysis ( 32 Views )

In the past two decades a new type of chaotic dynamics has been noticed in the three and four body problems which has not been understood. In 1986, using a numerical algorithm, an interesting region supporting chaotic motion was discovered about the moon, under the perturbation of the earth. This region is now termed the weak stability boundary. New types of dynamics were subsequently discovered near this boundary. These dynamics have the property that they give rise to very low energy trajectories with many important applications. In 1991, a new type of low energy trajectory to the moon was discovered which was used to place a Japanese spacecraft, Hiten, in orbit about the moon in October of that year. This was the first application of this type of dynamics to space travel. These low energy trajectories, so called WSB transfers, are now being planned by NASA, Europe and Japan for several new missions to the moon, Europa, Mars. Motion near this boundary also gives rise to an interesting resonance transition dynamics, and work by the speaker with Brian Marsden at Harvard is discussed in its relevance to short period comets, and Kuiper belt objects. An analytic representation for this boundary is also presented and its connections with heteroclinic intersections of hyperbolic invariant manifolds is discussed. If there is time, a new type of periodic motion for Hill's problem is looked at.

public 01:03:36

Thomas Y. Hou : Singularity Formation in 3-D Vortex Sheets

  -   Applied Math and Analysis ( 30 Views )

One of the classical examples of hydrodynamic instability occurs when two fluids are separated by a free surface across which the tangential velocity has a jump discontinuity. This is called Kelvin-Helmholtz Instability. Kelvin-Helmholtz instability is a fundamental instability of incompressible fluid flow at high Reynolds number. The idealization of a shear layered flow as a vortex sheet separating two regions of potential flow has often been used as a model to study mixing properties, boundary layers and coherent structures of fluids. In a joint work with G. Hu and P. Zhang, we study the singularity of 3-D vortex sheets using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D vortex sheet. This leading order equation captures the most singular contribution of the integral equation. Moreover, after applying a transformation to the physical variables, we found that this leading order 3-D vortex sheet equation de-generates into a two-dimensional vortex sheet equation in the direction of the tangential velocity jump. This rather surprising result confirms that the tangential velocity jump is the physical driving force of the vortex sheet singularities. It also shows that the singularity type of the three-dimensional problem is similar to that of the two-dimensional problem. Detailed numerical study will be provided to support the analytical results, and to reveal the generic form and the three-dimensional nature of the vortex sheet singularity.

public 01:19:02

Peter K. Moore : An Adaptive H-Refinement Finite Element For Solving Systems of Parabolic Partial Differential Equations in Three Space Dimensions

  -   Applied Math and Analysis ( 29 Views )

Adaptive methods for solving systems of partial differential equations have become widespread. Robust adaptive software for solving parabolic systems in one and two space dimensions is now widely available. Three spatial adaptive strategies and combinations thereof are frequently employed: mesh refinement (h-refinement); mesh motion (r-refinement); and order variation (p-refinement). These adaptive strategies are driven by a priori and a posteriori error estimates. I will present an adaptive h-refinement finite element code in three dimensions on structured grids. These structured grids contain irregular nodes. Solution values at these nodes are determined by continuity requirements across element boundaries rather than by the differential equations. The differential-algebraic system resulting from the spatial discretization is integrated using Linda Petzold's multistep DAE code DASPK. The large linear systems resulting from Newton's method applied to nonlinear system of differential algebraic equations is solved using preconditioned GMRES. In DASPK the matrix-vector products needed by GMRES are approximated by a ``directional derivative''. Thus, the Jacobian matrix need not be assembled. However, this approach is inefficient. I have modified DASPK to compute the matrix-vector product using stored Jacobian matrix. As in the earlier version of DASPK, DASSL, this matrix is kept for several time steps before being updated. I will discuss appropriate preconditioning strategies, including fast-banded preconditioners. In three dimensions when using multistep methods for time integration it is crucial to use a ``warm restart'', that is, to restart the dae solver at the current time step and order. This requires interpolation of the history information. The interpolation must be done in such a way that mode irregularity is enforced on the new grid. A posteriori error estimates on uniform grids can easily be generalized from two-dimensional results (Babuska and Yu showed that in the case of odd order elements, jumps across elemental boundaries give accurate estimates, and in the case of even order elements, local parabolic systems must be solved to obtain accurate estimates). Babuska's work can even be generalized to meshes with irregular modes but now they no longer converge to the true error (in the case of odd order elements). I have developed a new set of estimates that extend the work of Babuska to irregular meshes and finite difference methods. These estimates provide a posteriori error indicators in the finite element context. Several examples that demonstrate the effectiveness of the code will be given.

public 01:02:03

Zhilin Li : The Immersed Interface Method:A Numerical Approach for Interface Problems

  -   Applied Math and Analysis ( 26 Views )

Many physical problems involve interfaces. Examples include phase transition problems where the interface separates the solid and liquid regions, bubble simulation, Hele-Shaw flow, composite materials, and many other important physical phenomena. Mathematically, interface problems usually lead to differential equations whose input data and solutions have discontinuities or non-smoothness across interfaces. As a result, many standard numerical schemes do not work or work poorly for interface problems. This is an introductory talk about the interface problems and our immersed interface method. Through some simple examples, I will try to explain the problems of our interest and related background information. Then I will present our method for some typical model problems in two dimensions. Our method can handle both discontinuous coefficients and singular sources. The main idea is to incorporate the known jumps in the solution and its derivatives into the finite difference scheme, obtaining a modified scheme on the uniform grid for quite arbitrary interfaces. The second part of the talk will focus on applications of the methods combined with the the level set method for moving interface problems: including the Stokes flow with different surface tension, the simulation of Hele-Shaw flow, and computation of crystal growth.