Doron Levy - Publications

  1. Non-Oscillatory Central Schemes for the Incompressible 2D Euler Equations
    Mathematical Research Letters, 4, 1997, pp. 321-340. (with E. Tadmor)

  2. On a Class of a Thermal Blow-up Patterns
    Physics Letters A., 236, 1997, pp. 483-493. (with P. Rosenau)

  3. From Semi-Discrete to Fully-Discrete: The Stability of Runge-Kutta Schemes by the Energy Method
    SIAM Review, 40, no.1, 1998, pp. 40-73. (with E. Tadmor)

  4. On Burgers-type Equations with Non-Monotonic Dissipative Fluxes
    Communications on Pure and Applied Mathematics, 51, no. 5, 1998, pp. 443-473. (with A. Kurganov and P. Rosenau)

  5. High-Resolution Non-Oscillatory Central Schemes with Non-Staggered Grids for Hyperbolic Conservation Laws
    SIAM Journal on Numerical Analysis, 35, no. 6, 1998, pp. 2147-2168. (with G.-S. Jiang, C.-T. Lin, S. Osher and E. Tadmor)

  6. Third-order 2D Central Schemes for Hyperbolic Conservation Laws
    INRIA School on Hyperbolic Systems, Vol. I, 1998, pp. 489-504.

  7. Self-Focusing in the Complex Ginzburg-Landau Limit of the Critical Nonlinear Schrodinger Equation
    Physics Letters A., 249, 1998, pp. 286-294. (with G. Fibich)

  8. Compactons in a Class of Nonlinearly Quintic Equations
    Physics Letters A., 252, 1999, pp. 297-306. (with P. Rosenau)

  9. A Modified Structured Central Scheme for 2D Hyperbolic Conservation Laws
    Applied Math Letters, 12, no. 6, 1999, pp. 89-96. (with T. Katsaounis)

  10. Central WENO Schemes for Hyperbolic Systems of Conservation Laws
    Mathematical Modelling and Numerical Analysis, 33, no. 3, 1999, pp. 547-571. (with G. Puppo and G. Russo)

  11. Dissipative Behavior of Some Fully Non-Linear KdV-Type Equations
    Physica D., 137, 2000, pp. 277-294 (with Y. Brenier)

  12. A Third Order Central WENO Scheme for 2D Conservation Laws
    Applied Numerical Mathematics, 33, 2000, pp. 407-414. (with G. Puppo and G. Russo)

  13. On the Behavior of the Total Variation in CWENO Methods for Conservation Laws
    Applied Numerical Mathematics, 33, 2000, pp. 415-421. (with G. Puppo and G. Russo)

  14. Optimal Prediction for Hamiltonian Partial Differential Equations
    Journal of Computational Physics, 162, 2000, pp. 267-297. (with A.J. Chorin and R. Kupferman)

  15. Compact Central WENO Schemes for MultiDimensional Conservation Laws
    SIAM Journal on Scientific Computing, 22, 2000, pp. 656-672. (with G. Puppo and G. Russo)

  16. A Third-Order Semi-Discrete Central Scheme for Conservation Laws and Convection-Diffusion Equations
    SIAM Journal on Scientific Computing, 22, 2000, pp. 1461-1488. (with A. Kurganov)

  17. Particle Methods for Dispersive Equations
    Journal of Computational Physics, 171, 2001, pp. 708-730. (with A. Chertock)

  18. Central-Upwind Schemes for the Saint-Venant System
    Mathematical Modelling and Numerical Analysis, 36, 2002, pp. 397-425. (with A. Kurganov)

  19. A Fourth Order Central WENO Scheme for Multi-Dimensional Hyperbolic Systems of Conservation Laws
    SIAM Journal on Scientific Computing, 24, 2002, pp. 480-506. (with G. Puppo and G. Russo)

  20. A Particle Method for the KdV Equation
    Journal of Scientific Computing, 17, 2002, pp. 491-499. (with A. Chertock)

  21. High-order Central WENO Schemes for 1D Hamilton-Jacobi Equations
    in F. Brezzi et al. (Eds.), ``Numerical Mathematics and Advanced Applications'',
    Proc. ENUMATH 2001, Ischia, Italy; Springer-Verlag, Italy, 2003, pp.45--54. (with S. Bryson)

  22. High-Order Central WENO Schemes for Multi-dimensional Hamilton-Jacobi Equations
    in T.Y. Hou and E. Tadmor (Eds.), ``Hyperbolic Problems: Theory, Numerics, Applications'',
    Proc. Ninth International Conference on Hyperbolic Problems, CalTech, 2002;
    Springer-Verlag, Berlin, 2003, pp.387--396. (with S. Bryson)

  23. High-Order Semi-Discrete Central-Upwind Schemes for Multi-Dimensional Hamilton-Jacobi Equations
    Journal of Computational Physics, 189, 2003, pp. 63-87. (with S. Bryson)

  24. High-Order Central WENO Schemes for Multi-Dimensional Hamilton-Jacobi Equations
    SIAM Journal of Numerical Analysis, 41, 2003, pp. 1339-1369. (with S. Bryson)

  25. Central Schemes for Multi-Dimensional Hamilton-Jacobi Equations
    SIAM Journal on Scientific Computing, 25, 2003, pp. 769-791. (with S. Bryson)

  26. Local Discontinuous Galerkin Methods for Nonlinear Dispersive Equations
    Journal of Computational Physics, 196, 2004, pp. 751-772. (with C.-W. Shu and J. Yan)

  27. Central Schemes for Hamilton-Jacobi Equations on Unstructured Grids
    in M. Feistauer et al. (Eds.), ``Numerical Mathematics and Advanced Applications'',
    Proceedings of ENUMATH 2003, Prague, Czech Republic; Springer-Verlag, Berlin, 2004, pp.623--630. (with S. Nayak)

  28. On Wavelet-Based Numerical Homogenization
    Multiscale Modeling and Simulation, 3, 2004, pp. 65-88. (with A. Chertock)

  29. Registration-based Morphing of Active Contours for Segmentation of CT Scans
    Mathematical Biosciences and Engineering, 2, 2005, pp. 79-96. (with Y.-N. Young)

  30. Semi-Discrete Central-Upwind Schemes with Reduced Dissipation for Hamilton-Jacobi Equations
    IMA Journal of Numerical Analysis, 25, 2005, pp. 113-138. (with S. Bryson, A. Kurganov, and G. Petrova)

  31. High-Order Shock-Capturing Methods for Modeling Dynamics of the Solar Atmosphere
    Physica D., 201, 2005, pp. 1-26. (with S. Bryson and A. Kosovichev)

  32. Quantitation of the a Priori Dosimetric Capabilities of Spatial Points in Inverse Planning and its Significant Implication in Defining IMRT Solution Space
    Physics in Medicine and Biology, 50, 2005, pp. 1469-1482. (with Z. Shou, Y. Yang, C. Cotrutz and L. Xing)

  33. PDE-based Segmentation for Radiation Therapy Treatment Planning
    Mathematical Biosciences and Engineering, 2, 2005, pp. 209-226. (with F. Gibou, C. Cardenas, P. Liu, and A. Boyer)

  34. Approximate Model Equations for Water Waves
    Communications in Mathematical Sciences, 3, 2005, pp. 159-170 (with R. Fetecau)

  35. Post Transplantation Dynamics of the Immune Response to Chronic Myelogenous Leukemia
    Journal of Theoretical Biology, 236, 2005, pp. 39-59 (with R. DeConde, P. Kim, and P. Lee)

  36. A Stable Semi-Discrete Central Scheme for the Two-Dimensional Incompressible Euler Equations
    IMA Journal of Numerical Analysis, 25, 2005, pp. 507-522.

  37. Balanced Central Schemes for the Shallow Water Equations on Unstructured Grids
    SIAM Journal on Scientific Computing, 27, 2005, pp. 532-552. (with S. Bryson)

  38. Multiscale Image Registration
    Mathematical Biosciences and Engineering, 3, 2006, pp. 389-418. (with D. Paquin, E. Schreibmann, and L. Xing)

  39. On the Total Variation of High-Order Semi-Discrete Central Schemes for Conservation Laws
    Journal of Scientific Computing, 27, 2006, pp. 163-175. (with S. Bryson)

  40. Mapped WENO and Weighted Power ENO Reconstructions in Semi-Discrete Central Schemes for Hamilton-Jacobi Equations
    Applied Numerical Mathematics, 56, 2006, pp. 1211-1224. (with S. Bryson)

  41. Central WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes
    SIAM Journal on Scientific Computing, 28, 2006, pp. 2229-2247 (with S. Nayak, C.-W. Shu, and Y.-T. Zhang)

  42. On the Stability Crossing Boundaries of Some Delay Systems Modeling Immune Dynamics in Leukemia
    Proc 17th Int Symp on Mathematical Theory of Networks and Systems, Kyoto, 2006 (with S.-I. Niculescu, P. Kim, and K. Gu)

  43. Modeling Regulatory Mechanisms in the Immune System
    Journal of Theoretical Biology, 246, 2007, pp. 33-69 (with P. Kim and P. Lee)

  44. On Stability of a Combined Gleevec and Immune Model of Chronic Myelogenous Leukemia: Exploiting Delay System Structure,
    To Appear,(with S. Niculescu, P. Kim and P. Lee)

  45. Group Dynamics of Phototaxis: Interacting Stochastic Many-Particle Systems and their Continuum Limit
    To Appear, (with D. Bhaya and T. Requiejo)

  46. new Mini-Transplants for Chronic Myelogenous Leukemia: A Modeling Perspective,
    To Appear (with P. Kim and P. Lee)

  47. new Hybrid Multiscale Landmark and Deformable Registration
    Mathematical Biosciences and Engineering, to appear (with D. Paquin and L. Xing)

  48. new Multiscale Deformable Registration
    Submitted (with D. Paquin and L. Xing)

  49. new Stability Crossing Boundaries of Delay Systems Modeling Immune Dynamics in Leukemia
    Journal of Mathematical Biology, submitted (with K. Gu, P. Kim and S. Niculescu)

  50. new Dynamics and Impact of the Immune Response to Chronic Myelogenous Leukemia
    Submitted, (with C. Chen, P. Kim, H. Maecker and P. Lee)

  51. newModeling Imatinib-Treated Chronic Myelogenous Leukemia: Reducing the Complexity of Agent-Based Models
    Submitted (with P. Kim and P. Lee)

  52. newModeling Group Dynamics of Phototaxis: From Particles to PDEs
    Submitted (with T. Requeijo)

  53. newStochastic Models for Phototaxis
    Submitted (with T. Requeijo)

  54. newOn the Role of Adaptive Regulatory Cells in the Immune System
    In preparation (with P. Kim and P. Lee)

  55. newRegistration of Cone-Beam CT and Spiral CT Images,
    In preparation (with D. Paquin and L. Xing)

Non-Oscillatory Central Schemes for the Incompressible 2D Euler Equations (3.5 MB)
(with E. Tadmor),
Mathematical Research Letters, 4, 1997, pp. 321-340.

We adopt a non-oscillatory central scheme, first presented in the context of Hyperbolic conservation laws in [nessyahu-tadmor] followed by [jiang-tadmor], to the framework of the incompressible Euler equations in their vorticity formulation. The embedded duality in these equations, enables us to toggle between their two equivalent representations - the conservative Hyperbolic-like form vs. the convective form. We are therefore able to apply local methods, to problems with a global nature. This results in a new stable and convergent method which enjoys high-resolution without the formation of spurious oscillations. These desirable properties are clearly visible in the numerical simulations we present.

On a Class of a Thermal Blow-up Patterns (630 KB)
(with P. Rosenau)
Physics Letters A, 236, 1997, pp. 483-493.

We study the patterns of thermal explosion as described via $u_t = (\Delta + 1)u^m , m>1$. These processes, characterized by an intrinsic length-scale, always converge into a very simple, universal, space-time separable, axisymmetric pattern(s) with a compact support - referred to as dissipative compactons . When the initial datum is specified on an axisymmetric annulus, though the evolving pattern seems to preserve this symmetry, at a later stage, it collapses very quickly to the center. In a perturbed annulus, local axisymmetric patches of blow-up form instead of a collapse. For a planar, homogeneous, Dirichlet problem, the space-time separability of the emerging pattern is preserved as well, but the competition between the intrinsic and extrinsic characteristic scales, generates a wider variety of spatial patterns, with the self-localization taking place on large domains. As the width of the domain diminishes, then depending on width-length ratio, the emerging pattern first partially, and then fully, attaches to the boundaries. With further decrease of the domain, the emerging separable pattern instead of exploding decays algebraically in time.

From Semi-Discrete to Fully-Discrete: The Stability of Runge-Kutta Schemes by the Energy Method (877 KB)
(with E. Tadmor)
SIAM Review, 40, no.1, 1998, pp. 40-73.

The integration of semi-discrete approximations for time-dependent problems is encountered in a variety of applications. The Runge-Kutta (RK) methods are widely used to integrate the ODE systems which arise in this context, resulting in large ODE systems called methods of lines. These methods of lines are governed by possibly ill-conditioned systems with a growing dimension: consequently, the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading. Instead, we present here a stability analysis of RK methods for well-posed semi-discrete approximations, based on a general energy method. We review the stability question for such RK approximations, and highlight its intricate dependence on the growing dimension of the problem. In particular, we prove the strong-stability of general fully-discrete RK methods governed by coercive approximations. We conclude with two non-trivial examples which demonstrate the versatility of our approach in the context of general systems of convection-diffusion equations with variable coefficients. A straightforward implementation of our results verify the strong-stability of RK methods for local finite-difference schemes as well as global spectral approximations. Since our approach is based on the energy method (which is carried in the physical space), and since it avoids the von-Neumann analysis (which is carried in the dual Fourier space), we are able to easily adapt additional extensions due to non-periodic boundary conditions, general geometries, etc.

On Burgers-type Equations with Non-Monotonic Dissipative Fluxes (661 KB)
(with A. Kurganov and P. Rosenau)
CPAM, 51, no. 5, 1998, pp. 443-473.

We study a formation of patterns in Burgers-type equations endowed with a bounded but non-monotonic dissipative fluxes; $u_{t} + f(u)_{x} = \pm \nu Q(u_{x})_{x}, \, Q(s) = s/(1+s^{2})$. Issues of uniqueness, existence and smoothness of a solution are addressed. Asymptotic regions of solution are discussed and in particular, classical, and non-classical traveling waves with an embedded sub-shock are constructed.

A Modified Structured Central Scheme for 2D Hyperbolic Conservation Laws (109 KB)
(with T. Katsaounis)
Applied Math Letters, 12, no. 6, 1999, pp. 89-96.

We present a new central scheme for approximating solutions of two-dimensional systems of hyperbolic conservation laws. This method is based on a modification of the staggered grid proposed in [jiang-tadmor] which prevents the crossings of discontinuities in the normal direction, while retaining the simplicity of the central framework. Our method satisfies a local maximum principle which is based on a more compact stencil. Unlike the previous method, it enables a natural extension to adaptive methods on structured grids.

Central WENO Schemes for Hyperbolic Systems of Conservation Laws (541 KB)
(with G. Puppo and G. Russo)
Mathematical Modelling and Numerical Analysis, 33, no. 3, 1999, pp. 547-571.

We present a family of high-order, essentially non-oscillatory, central schemes for approximating solutions of hyperbolic systems of conservation laws. These schemes are based on a new centered version of the Weighted Essentially Non-Oscillatory (WENO) reconstruction of point-values from cell-averages, which is then followed by an accurate approximation of the fluxes via a natural continuous extension of Runge-Kutta solvers. We explicitly construct the third and fourth-order scheme and demonstrate their high-resolution properties in several numerical tests.

Self-Focusing in the Complex Ginzburg-Landau Limit of the Critical Nonlinear Schrodinger Equation (143 KB)
(with G. Fibich)
Physics Letters A., 249, 1998, pp. 286-294.

We analyze self-focusing and singularity formation in the complex Ginzburg-Landau equation (CGL) in the regime where it is close to the critical nonlinear Schrodinger equation. Using modulation theory [Fibich and Papanicolaou, Phys. Lett. A 239:167--173, 1998], we derive a reduced system of ordinary differential equations that describes self-focusing in CGL. Analysis of the reduced system shows that in the physical regime of the parameters there is no blowup in CGL. Rather, the solution focuses once and then defocuses. The validity of the analysis is verified by comparison of numerical solutions of CGL with those of the reduced system.

Third-order 2D Central Schemes for Hyperbolic Conservation Laws (60 KB)
INRIA School on Hyperbolic Systems, Vol. I, 1998, pp. 489-504.

We extend a one-dimensional central-scheme first introduced by Liu and Tadmor, to the rather general framework of two-dimensional systems of conservation laws. This scheme is based on a two-dimensional piecewise-quadratic reconstruction computed from the cell-averages at each time-step. Our scheme is implemented for the incompressible two-dimensional Euler equation demonstrating its high-resolution nature.

High-Resolution Non-Oscillatory Central Schemes with Non-Staggered Grids for Hyperbolic Conservation Laws (2.2 MB)
(with G.-S. Jiang, C.-T. Lin, S. Osher and E. Tadmor)
SINUM, 35, no. 6, 1998, pp. 2147-2168.

We present a general procedure to convert schemes which are based on staggered spatial grids into non-staggered schemes. This procedure is then utilized to construct a new family of non-staggered, central schemes for hyperbolic conservation laws, by converting the family of staggered central schemes recently introduced in [nessyahu-tadmor], [liu-tadmor], [jiang-tadmor]. These new non-staggered central schemes retain the desirable properties of simplicity and high-resolution, and in particular, they yield Riemann-solver-free recipes which avoid dimensional splitting. Most importantly, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.

Compactons in a Class of Nonlinearly Quintic Equations (593 KB)
(with P. Rosenau)
Physics Letters A., 252, 1999, pp. 297-306

We introduce a nonlinear dispersive quintic equation. Its travelling waves are governed by a linear equation. We construct a large variety of explicit compact solitary waves with one or many humps. Some of these compactons are very robust, others decompose very quickly. Numerical simulations also reveal the existence of compact travelling breathers.

A Third Order Central WENO Scheme for 2D Conservation Laws (148 KB)
(with G. Puppo and G. Russo)
Applied Numerical Mathematics, 33, 2000, pp. 407-414.

We present a new third-order essentially non-oscillatory central scheme for approximating solutions of two-dimensional hyperbolic conservation laws. Our scheme is based on a two-dimensional extension of the centered weighted essentially non-oscillatory (CWENO) reconstruction we presented in [Levy, Puppo, Russo, Central Weno Schemes for Hyperbolic Systems of Conservation Laws, Math. Mode. and Nume. Anal.]. This is a ``true'' 2D method; it is not based on a direction-by-direction approach. Our method is formalized in terms of a black box which needs as an input only the specific flux. The numerical results we present support our expectations for a robust and high-resolution method.

On the Behavior of the Total Variation in CWENO Methods for Conservation Laws (62 KB)
(with G. Puppo and G. Russo)
Applied Numerical Mathematics, 33, 2000, pp. 415-421.

We consider a family of high-order, weighted essentially non-oscillatory central schemes (CWENO) for approximating solutions of one-dimensional hyperbolic systems of conservation laws. We are interested in the behavior of the total variation (TV) of the approximate solution obtained with these methods. Our numerical results suggest that even though CWENO methods are not total variation diminishing (TVD), they do have bounded total variation (TVB). Moreover, the TV of the approximate solution seems to never increase above the theoretical value, and it approaches it as the mesh is refined. These results are hopefully a first step in the quest for proving the convergence of such high-order methods.

Dissipative Behavior of Some Fully Non-Linear KdV-Type Equations (409 KB)
(with Y. Brenier)
Physica D., 137, 2000, pp. 277-294.

The KdV equation can be considered as a special case of the general equation $ u_{t} + f(u)_{x} - \delta g(u_{xx})_x = 0, \delta > 0$, where $f$ is non-linear and $g$ is linear, namely $f(u)=u^2/2$ and $g(v)=v$. As the parameter $\delta$ tends to $0$, the dispersive behavior of the KdV equation has been throughly investigated (see, e.g., [whitham], [lax-levermore], [drazin] and the references therein). We show, through numerical evidence that a completely different, dissipative behavior occurs when $g$ is non-linear, namely when $g$ is an even concave function such as $g(v)=-|v|$ or $g(v)=-v^2$. In particular, our numerical results hint that as $\delta$ tends to zero, the solutions strongly converge to the unique entropy solution of the formal limit equation, in total contrast with the solutions of the KdV equation.

A Third-Order Semi-Discrete Central Scheme for Conservation Laws and Convection-Diffusion Equations (1.6 MB)
(with A. Kurganov)
SIAM Journal on Scientific Computing, 22, 2000, pp. 1461-1488.

We present a new third-order, semi-discrete, central method for approximating solutions to multi-dimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems. Our method is a high-order extension of the recently proposed second-order, semi-discrete method in [kurganov-tadmor]. The method is derived independently of the specific piecewise polynomial reconstruction which is based on the previously computed cell-averages. We demonstrate our results, by focusing on the new third-order CWENO reconstruction presented in [levy-puppo-russo] The numerical results we present, show the desired accuracy, high resolution and robustness of our method.

Optimal Prediction for Hamiltonian partial differential equations (225 KB)
(with A.J. Chorin and R. Kupferman)
Journal of Computational Physics, 162, 2000, pp. 267-297.

Optimal prediction methods compensate for a lack of resolution in the numerical solution of time-dependent differential equations through the use of prior statistical information. We present a new derivation of the basic methodology, show that field-theoretical perturbation theory provides a useful device for dealing with quasi-linear problems, and provide a nonlinear example that illuminates the difference between a pseudo-spectral method and an optimal prediction method with Fourier kernels. Along the way, we explain the differences and similarities between optimal prediction, the representer method in data assimilation, and duality methods for finding weak solutions. We also discuss the conditions under which a simple implementation of the optimal prediction method can be expected to perform well.

Compact Central WENO Schemes for MultiDimensional Conservation Laws (481 KB)
(with G. Puppo and G. Russo)
SIAM Journal on Scientific Computing, 22, 2000, pp. 656-672.

We present a new third-order central scheme for approximating solutions of systems of conservation laws in one and two space dimensions. In the spirit of Godunov-type schemes, our method is based on reconstructing a piecewise-polynomial interpolant from cell-averages which is then advanced exactly in time. In the reconstruction step, we introduce a new third-order, compact, CWENO reconstruction, which is written as a convex combination of interpolants based on different stencils. The heart of the matter is that one of these interpolants is taken as an arbitrary quadratic polynomial and the weights of the convex combination are set as to obtain third-order accuracy in smooth regions. The embedded mechanism in the WENO-like schemes guarantees that in regions with discontinuities or large gradients, there is an automatic switch to a one-sided second-order reconstruction, which prevents the creation of spurious oscillations. In the one-dimensional case, our new third order scheme is based on an extremely compact four stencil. Analogous compactness is retained in more space dimensions. The accuracy, robustness and high-resolution properties of our scheme are demonstrated in a variety of one and two dimensional problems.

Particle Methods for Dispersive Equations (167 KB)
(with A. Chertock)
Journal of Computational Physics, 171, 2001, pp. 708-730.

We introduce a new dispersion-velocity particle method for approximating solutions of linear and nonlinear dispersive equations. This is the first time in which particle methods are being used for solving such equations. Our method is based on an extension of the diffusion-velocity method of Degond and Mustieles [SIAM J. Sci. Stat. Comp., 11 no. 2, (1990), pp.293--310] to the dispersive framework. The main analytical result we provide is the short time existence and uniqueness of a solution to the resulting dispersion-velocity transport equation. We numerically test our new method for a variety of linear and nonlinear problems. In particular we are interested in nonlinear equations which generate structures that have non-smooth fronts. Our simulations show that this particle method is capable of capturing the nonlinear regime of a compacton-compacton type interaction.

Central-Upwind Schemes for the Saint-Venant System (800 KB)
(with A. Kurganov)
Mathematical Modelling and Numerical Analysis, 36, 2002, pp. 397-425.

We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semi-discrete central schemes for systems of hyperbolic conservation laws to balance laws. Special attention is paid to the discretization of the source term such as to preserve stationary steady-state solutions. We also prove that the second-order version of our schemes preserves the nonnegativity of the height of the water. This important feature allows one to compute solutions for problems that include dry areas.

A Fourth Order Central WENO Scheme for Multi-Dimensional Hyperbolic Systems of Conservation Laws (1.4 MB)
(with G. Puppo and G. Russo)
SIAM Journal on Scientific Computing, 22, 2002, pp. 480-506.

We present the first fourth-order, central scheme for two-dimensional hyperbolic systems of conservation laws. Our new method is based on a Central Weighted Non-Oscillatory (CWENO) approach. The heart of our method is the reconstruction step, in which a genuinely two-dimensional interpolant is reconstructed from cell-averages by taking a convex combination of building blocks in the form of bi-quadratic polynomials. Similarly to other central schemes, our new method enjoys the simplicity of the black-box approach. All that is required in order to solve a problem is to supply the flux function and an estimate on the speed of propagation. The high-resolution properties of the scheme as well as its resistance to mesh orientation, and the effectiveness of the component-wise approach, are demonstrated in a variety of numerical examples.

A Particle Method for the KdV Equation
(with A. Chertock)
Journal of Scientific Computing, 17, 2002, pp. 491-499.

We extend the dispersion-velocity particle method that we recently introduced to advection models in which the velocity does not depend linearly on the solution or its derivatives. An example is the Korteweg de Vries (KdV) equation for which we derive a particle method and demonstrate numerically how it captures soliton-soliton interactions.

High-Order Semi-Discrete Central-Upwind Schemes for Multi-Dimensional Hamilton-Jacobi Equations (720 KB)
(with S. Bryson)
Journal of Computational Physics, 189, 2003, pp. 63-87.

We present the first fifth-order, semi-discrete central-upwind method for approximating solutions of multi-dimensional Hamilton-Jacobi equations. Unlike most of the commonly used high-order upwind schemes, our scheme is formulated as a Godunov-type scheme. The scheme is based on the fluxes of Kurganov-Tadmor and Kurganov-Noelle-Petrova, and is derived for an arbitrary number of space dimensions. A theorem establishing the monotonicity of these fluxes is provided. The spatial discretization is based on a weighted essentially non-oscillatory reconstruction of the derivative. The accuracy and stability properties of our scheme are demonstrated in a variety of examples. A comparison between our method and other fifth-order schemes for Hamilton-Jacobi equations shows that our method exhibits smaller errors without any increase in the complexity of the computations.

High-Order Central WENO Schemes for Multi-Dimensional Hamilton-Jacobi Equations (1.2 MB)
(with S. Bryson)
Siam Journal on Numerical Analysis, 41, 2003, pp. 1339-1369.

We present new third- and fifth-order Godunov-type central schemes for approximating solutions of the Hamilton-Jacobi (HJ) equation in an arbitrary number of space dimensions. These are the first central schemes for approximating solutions of the HJ equations with an order of accuracy that is greater than two. In two space dimensions we present two versions for the third-order scheme: one scheme that is based on a genuinely two-dimensional Central WENO reconstruction, and another scheme that is based on a simpler dimension-by-dimension reconstruction. The simpler dimension-by-dimension variant is then extended to a multi-dimensional fifth-order scheme. Our numerical examples in one, two and three space dimensions verify the expected order of accuracy of the schemes.

Central Schemes for Multi-Dimensional Hamilton-Jacobi Equations (750 KB)
(with S. Bryson)
Siam Journal on Scientific Computing, 25, 2003, pp. 769-791.

We present new, efficient central schemes for multi-dimensional Hamilton-Jacobi equations. These non-oscillatory, non-staggered schemes are first- and second-order accurate and are designed to scale well with an increasing dimension. Efficiency is obtained by carefully choosing the location of the evolution points and by using a one-dimensional projection step. First- and second-order accuracy is verified for a variety of multi-dimensional, convex and non-convex problems.

Local Discontinuous Galerkin Methods for Nonlinear Dispersive Equations (260 KB)
(with C.-W. Shu and J. Yan)
Journal of Computational Physics, 196, 2004, pp. 751-772.

We develop local discontinuous Galerkin (DG) methods for solving nonlinear dispersive partial differential equations that have compactly supported traveling waves solutions, the so-called "compactons". The schemes we present extend the previous works of Yan and Shu on approximating solutions for linear dispersive equations and for certain KdV-type equations. We present two classes of DG methods for approximating solutions of such PDEs. First, we generate nonlinearly-stable numerical schemes with a stability condition that is induced from a conservation law of the PDE. An alternative approach is based on constructing linearly-stable schemes, i.e. schemes that are linearly stable to small perturbations. The numerical simulations we present verify the desired properties of the methods including their expected order of accuracy. In particular, we demonstrate the potential advantages of using DG methods over pseudo-spectral methods in situations where discontinuous fronts and rapid oscillations co-exist in a solution.

Central Schemes for Hamilton-Jacobi Equations on Unstructured Grids
(with S. Nayak)
in M. Feistauer et al. (Eds.), ``Numerical Mathematics and Advanced Applications'', Proceedings of ENUMATH 2003, Prague, Czech Republic;
Springer-Verlag, Berlin, 2004, pp.623--630.

We present a new semi-discrete central scheme for approximating solutions of Hamilton-Jacobi equations on unstructured meshes. This scheme extends the numerical Hamiltonians of Kurganov et al. to unstructured grids. Similarly to the previous works on structured grids, a semi-discrete formulation of central schemes is made possible due to estimates of the local speeds of propagation. The consistency of the method is obtained following Abgrall's calculations for the consistency of an upwind Lax-Friedrichs scheme on unstructured grids. We conclude with comments on high-order reconstructions.

On Wavelet-Based Numerical Homogenization (1.8 MB)
(with A. Chertock)
Multiscale Modeling and Simulation, 3, 2004, pp. 65-88.

Recently, a wavelet-based method was introduced for the systematic derivation of subgrid scale models in the numerical solution of partial differential equations. Starting from a discretization of the multiscale differential operator, the discrete operator is represented in a wavelet space and projected onto a coarser subspace. The coarse (homogenized) operator is then replaced by a sparse approximation to increase the efficiency of the resulting algorithm. In this work we show how to improve the efficiency of this numerical homogenization method by choosing a different compact representation of the homogenized operator. In two dimensions our approach for obtaining a sparse representation is significantly simpler than the alternative sparse representations. $L^{\infty}$ error estimates are derived for a sample elliptic problem. An additional improvement we propose is a natural fine-scales correction that can be implemented in the final homogenization step. This modification of the scheme improves the resolution of the approximation without any significant increase in the computational cost. We apply our method to a variety of test problems including one- and two-dimensional elliptic models as well as wave propagation problems in materials with subgrid inhomogeneities.

Registration-Based Morphing of Active Contours for Segmentation of CT Scans (2.1 MB)
(with Y.-N. Young)
Mathematical Biosciences and Engineering, 2, 2005, pp. 79-96.

We present a new algorithm for segmenting organs in CT scans for radiotherapy treatment planning. Given a contour of an organ that is segmented in one image, our algorithm proceeds to segment contours that identify the same organ in the consecutive images. Our technique combines partial differential equations-based morphing active contours with algorithms for joint segmentation and registration. The coupling between these different techniques is done in order to deal with the complexity of segmenting "real" images, where boundaries are not always well-defined, and the initial contour is not an isophote of the image.

Semi-Discrete Central-Upwind Schemes with Reduced Dissipation for Hamilton-Jacobi Equations (254 KB)
(with S. Bryson, A. Kurganov, and G. Petrova)
IMA Journal of Numerical Analaysis, 25, 2005, pp. 113-138.

We introduce a new family of Godunov-type semi-discrete central schemes for multidimensional Hamilton-Jacobi equations. These schemes are a less dissipative generalization of the central-upwind schemes that have been recently proposed in [Kurganov, Noelle and Petrova, SIAM J. Sci. Comput., 23 (2001), pp. 707-740]. We provide the details of the new family of methods in one, two and three space dimensions, and then verify their expected low-dissipative property in a variety of examples.

High-Order Shock-Capturing Methods for Modeling Dynamics of the Solar Atmosphere (1.6 MB)
(with S. Bryson and A. Kosovichev)
Physica D., 201, 2005, pp. 1-26.

We use one-dimensional high-order central shock capturing numerical methods to study the response of various model solar atmospheres to forcing at the solar surface. The dynamics of the atmosphere is modeled with the Euler equations in a variable-sized flux tube in the presence of gravity. We study dynamics of the atmosphere suggestive of spicule formation and coronal oscillations. These studies are performed on observationally-derived model atmospheres above the quiet sun and above sunspots. To perform these simulations, we provide a new extension of existing second- and third-order shock-capturing methods to irregular grids. We also solve the problem of numerically maintaining initial hydrostatic balance via the introduction of new variables in the model equations and a careful initialization mechanism. We find several striking results: all model atmospheres respond to a single impulsive perturbation with several strong shock waves consistent with the rebound-shock model. These shock waves lift material and the transition region well into the initial corona, and the sensitivity of this lift to the initial impulse depends non-linearly on the details of the atmosphere model. We also reproduce an observed 3-minute coronal oscillation above sunspots as well as 5-minute oscillations above the quiet sun.

Quantitation of the a Priori Dosimetric Capabilities of Spatial Points in Inverse Planning and its Significant Implication in Defining IMRT Solution Space (909 KB)
(with Z. Shou, Y. Yang, C. Cotrutz and L. Xing)
Physics in Medicine and Biology, 50, 2005, pp. 1469-1482

In inverse planning, the likelihood for the points in a target or sensitive structure to meet their dosimetric goals is generally heterogeneous and represents the a priori knowledge of the system once the patient and beam configuration are chosen. Because of this intrinsic heterogeneity, in some extreme cases, a region in a target may never meet the prescribed dose without seriously deteriorating the doses in other areas. Conversely, the prescription in a region may be easily met without violating the tolerance of any sensitive structure. In this work, we introduce the concept of dosimetric capability to quantify the a priori information and develop a strategy to integrate the data into the inverse planning process. An iterative algorithm is implemented to numerically compute the capability distribution on a case specific basis. A method of incorporating the capability data into inverse planning is developed by heuristically modulating the importance of the individual voxels according to the a priori capability distribution. The formalism is applied to a few specific examples to illustrate the technical details of the new inverse planning technique. Our study indicates that the dosimetric capability is a useful concept to better understand the complex inverse planning problem and an effective use of the information allows us to construct a clinically more meaningful objective function to improve IMRT dose optimization techniques.

PDE-based Segmentation for Radiation Therapy Treatment Planning (961 KB)
(with F. Gibou, C. Cardenas, P. Liu, and A. Boyer)
Mathematical Biosciences and Engineering, 2, 2005, pp. 209-226.

The purpose of this study is to develop automatic algorithms for the segmentation phase of radiotherapy treatment planning. We develop new image processing techniques that are based on solving a partial differential equation for the evolution of the curve that identifies the segmented organ. The velocity function is based on the piecewise Mumford-Shah functional. Our method incorporates information about the target organ into classical segmentation algorithms. This information, which is given in terms of a three-dimensional wire-frame representation of the organ, serves as an initial guess for the segmentation algorithm. We check the performance of the new algorithm on eight data sets of three different organs: rectum, bladder, and kidney. The results of the automatic segmentation were compared with a manual segmentation of each data set by radiation oncology faculty and residents. The quality of the automatic segmentation was measured with the ``kappa-statistics'', and with a count of over and under-segmented frames, and was shown in most cases to be very close to the manual segmentation of the same data. A typical segmentation of an organ with 60 slices takes less than 10 seconds on a Pentium IV laptop.

Approximate Model Equations for Water Waves (238 KB)
(with R. Fetecau)
Communications in Mathematical Sciences, 3, 2005, pp. 159-170.

We present two new model equations for the unidirectional propagation of long waves in dispersive media for the specific purpose of modeling water waves. The derivation of the new equations uses a Pade (2,2) approximation of the phase velocity that arises in the linear water wave theory. Unlike the Korteweg-deVries (KdV) equation and similarly to the Benjamin-Bona-Mahony (BBM) equation, our models have a bounded dispersion relation. At the same time, the equations we propose provide the best approximation of the phase velocity for small wave numbers that can be obtained with third-order equations. We note that the new model equations can be transformed into previously studied models, such as the BBM and the Burgers-Poisson equations. It is therefore straightforward to establish the existence and uniqueness of solutions to the new equations. We also show that the distance between the solutions of one of the new equations, the KdV equation, and the BBM equation, is of the small order that is formally neglected by all models.

Post Transplantation Dynamics of the Immune Response to Chronic Myelogenous Leukemia (600 KB)
(with R. DeConde, P. Kim, and P. Lee)
Journal of Theoretical Biology, 236, 2005, pp. 39-59

We model the immune dynamics between T cells and cancer cells in leukemia patients after bone marrow transplants. We use a system of six delay differential equations to track the various cell-populations. Our approach incorporates time delays and accounts for the progression of cells through different modes of behavior. We explore possible mechanisms behind a successful cure, whether mediated by a blood-restricted immune response or a cancer-specific graft-versus-leukemia (GVL) effect. Characteristic features of this model include sustained proliferation of T cells after initial stimulation, saturated T cell proliferation rate, and the possible elimination of cancer cells, independent of fixed-point stability. In addition, we use numerical simulations to examine the effects of varying initial cell concentrations on the likelihood of a successful transplant. Among the observed trends, we note that higher initial concentrations of donor-derived, anti-host T cells slightly favor the chance of success, while higher initial concentrations of general host blood cells more significantly favor the chance of success. These observations lead to the hypothesis that anti-host T cells benefit from stimulation by general host blood cells, which induce them to proliferate to sufficient levels to eliminate cancer.

A Stable Semi-Discrete Central Scheme for the Two-Dimensional Incompressible Euler Equations (1.2 MB)
IMA Journal of Numerical Analaysis, 25, 2005, pp. 507-522

We derive a second-order, semi-discrete central-upwind scheme for the incompressible two-dimensional Euler equations in the vorticity formulation. The reconstructed velocity field preserves an exact discrete incompressibility relation. We state a local maximum principle for a fully-discrete version of the scheme and prove it using a convexity argument. We then show how similar convexity arguments can be used to prove that the scheme maps certain Orlicz spaces into themselves. The consequences of this result on the convergence of the scheme are discussed. Numerical simulations support the expected properties of the scheme.

Balanced Central Schemes for the Shallow Water Equations on Unstructured Grids (4.5 MB)
(with S. Bryson)
SIAM Journal on Scientific Computating, 27, 2005, pp. 532-552.

We present a two-dimensional, well-balanced, central-upwind scheme for approximating solutions of the shallow water equations in the presence of a stationary bottom topography on triangular meshes. Our starting point is the recent central scheme of Kurganov and Petrova (KP) for approximating solutions of conservation laws on triangular meshes. In order to extend this scheme from systems of conservation laws to systems of balance laws one has to find an appropriate discretization of the source terms. We first show that for general triangulations there is no discretization of the source terms that corresponds to a well-balanced form of the KP scheme. We then derive a new variant of a central scheme that can be balanced on triangular meshes. We note in passing that it is straightforward to extend the KP scheme to general unstructured conformal meshes. This extension allows us to recover our previous well-balanced scheme on Cartesian grids. We conclude with several simulations, verifying the second-order accuracy of our scheme as well as its well-balanced properties.

Multiscale Image Registration (1.7 MB)
(with D. Paquin, E. Schreibmann, and L. Xing)
Mathematical Biosciences and Engineering, 3, 2006, pp. 389-418.

A multiscale image registration technique is presented for the registration of medical images that contain significant levels of noise. An overview of the medical image registration problem is presented, and various registration techniques are discussed. Experiments using mean squares, normalized correlation, and mutual information optimal linear registration are presented that determine the noise levels at which registration using these techniques fails. Further experiments in which classical denoising algorithms are applied prior to registration are presented, and it is shown that registration fails in this case for significantly high levels of noise, as well. The hierarchical multiscale image decomposition of E. Tadmor, S. Nezzar, and L. Vese \cite{tadmor} is presented, and accurate registration of noisy images is achieved by obtaining a hierarchical multiscale decomposition of the images and registering the resulting components. This approach enables successful registration of images that contain noise levels well beyond the level at which ordinary optimal linear registration fails. Image registration experiments demonstrate the accuracy and efficiency of the multiscale registration technique, and for all noise levels, the multiscale technique is as accurate as or more accurate than ordinary registration techniques.

On the Total Variation of High-Order Semi-Discrete Central Schemes for Conservation Laws (1.8 MB)
(with S. Bryson)
Journal of Scientific Computing, 27, 2006, pp. 163-175. (Proc. ICOSAHOM 2004)

We discuss a new fifth-order, semi-discrete, central-upwind scheme for solving one-dimensional systems of conservation laws. This scheme combines a fifth-order WENO reconstruction, a semi-discrete central-upwind numerical flux, and a strong stability preserving Runge-Kutta method. We test our method with various examples, and give particular attention to the evolution of the total variation of the approximations.

Mapped WENO and Weighted Power ENO Reconstructions in Semi-Discrete Central Schemes for Hamilton-Jacobi Equations (575 KB)
(with S. Bryson)
Applied Numerical Mathematics, 56, 2006, pp. 1211-1224.

We incorporate new high-order WENO-type reconstructions into Godunov-type central schemes for Hamilton-Jacobi equations. We study schemes that are obtained by combining the Kurganov-Noelle-Petrova flux with the Weighted Power ENO and the Mapped WENO reconstructions. We also derive new variants of these reconstructions by composing the Weighted Power ENO and the Mapped WENO reconstructions with each other. While all schemes are, formally, fifth-order accurate, we show that the quality of the approximation does depend on the particular reconstruction that is being used. In certain cases, it is shown that the approximate solution may not converge to the viscosity solution at all.

Central WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes (1.1 MB)
(with S. Nayak, C.-W. Shu, and Y.-T. Zhang)
SIAM Journal on Scientific Computing, 28, 2006, pp. 2229-2247

We develop the first semi-discrete central schemes for Hamilton-Jacobi equations on triangular meshes. High-order schemes are then obtained by combining our new numerical fluxes with high-order WENO reconstructions on triangular meshes. The numerical fluxes are shown to be monotone in certain cases. The accuracy and high-resolution properties of our scheme are demonstrated in a variety of numerical examples.

On the Stability Crossing Boundaries of Some Delay Systems Modeling Immune Dynamics in Leukemia (380 KB)
(with S.-I. Niculescu, P. Kim, and K. Gu)
Proc 17th Int Symp on Mathematical Theory of Networks and Systems, Kyoto, 2006.

This paper focuses on the characterization of delay effects on the asymptotic stability of some continuous-time delay systems encountered in modeling the post-transplantation dynamics of the immune response to chronic myelogenous leukemia. More explicitly, we shall discuss the stability of the crossing boundaries of the corresponding linearized models in the delay-parameter space. Weak, and strong cell interactions are discussed, and analytic characterizations are proposed. An illustrative example completes the presentation.

Modeling Regulatory Mechanisms in the Immune System (1.5 MB)
(with P. Kim and P. Lee)
Journal of Theoretical Biology, 246, 2007, pp. 33-69

We develop a mathematical framework for modeling regulatory mechanisms in the immune system. The model describes dynamics of key components of the immune network within two compartments: lymph node and tissue. We demonstrate using numerical simulations that our system can eliminate virus-infected cells, which are characterized by a tendency to increase without control (in absence of an immune response), while tolerating normal cells, which are characterized by a tendency to approach a stable equilibrium population. We experiment with different combinations of T cell reactivities that lead to effective systems and conclude that slightly self-reactive T cells can exist within the immune system and are controlled by regulatory cells. We observe that CD8+ T cell dynamics has two phases. In the first phase, CD8+ cells remain sequestered within the lymph node during a period of proliferation. In the second phase, the CD8+ population emigrates to the tissue and destroys its target population. We also conclude that a self-tolerant system must have a mechanism of central tolerance to ensure that self-reactive T cells are not too self-reactive. Furthermore, the effectiveness of a system depends on a balance between the reactivities of the effector and regulatory T cell populations, where the effectors are slightly more reactive than the regulatory cells.

Multiscale Deformable Registration
(with D. Paquin and L. Xing)
IEEE Transactions on Medical Imaging, submitted

Multiscale image registration techniques are presented for the deformable registration of medical images. These techniques are shown to be particularly effective for registration problems in which one or both of the images to be registered contains significant levels of noise. Experiments using B-splines deformable registration models demonstrate that ordinary deformable registration techniques fail to produce accurate results in the presence of significant levels of noise. Further experiments in which classical denoising algorithms are applied prior to registration show that registration fails in this case for high levels of noise, as well. The hierarchical multiscale image decomposition of E. Tadmor, S. Nezzar, and L. Vese, A multiscale image representation using hierarchical (BV,L^2) decompositions, Multiscale Modeling and Simulations, vol. 2, no. 4, pp. 554--579, 2004, is presented, and multiscale image registration algorithms are developed based on the multiscale decomposition. An accurate registration of noisy images is achieved by obtaining a hierarchical multiscale decomposition of the images and registering the resulting components. This approach enables a successful registration of images that contain noise well beyond the level at which ordinary deformable registration fails. Further, an iterated multiscale registration algorithm is presented which improves the accuracy and computational efficiency of other registration methods. The accuracy and efficiency of the multiscale registration techniques are demonstrated using both digital phantom and clinical case studies in two and three dimensions.

Stability Crossing Boundaries of Delay Systems Modeling Immune Dynamics in Leukemia (300 KB)
(with K.Gu, P. Kim and S. Niculescu)
Journal of Mathematical Biology, submitted

This paper focuses on the characterization of delay effects on the asymptotic stability of some continuous-time delay systems encountered in modeling the post-transplantation dynamics of the immune response to chronic myelogenous leukemia. Such models include multiple delays in some large range, from one minute to several days. The main objective of the paper is to discuss the stability of the crossing boundaries of the corresponding linearized models in the delay-parameter space by taking into account the interactions between small and large delays. Weak, and strong cell interactions are discussed, and analytic characterizations are proposed. An illustrative example together with related discussions complete the presentation.

Hybrid Multiscale Landmark and Deformable Registration
(with D. Paquin and L. Xing)
Mathematical Biosciences and Engineering , to appear

A novel hybrid image registration algorithm is presented based on combining landmark registration techniques, deformable registration techniques, and a multiscale decomposition of the images to be registered. The construction of the algorithm is motivated by the idea that the accuracy of standard deformable registration algorithms can be improved by incorporating more information, such as the location of corresponding features, about the images to be registered. The images are decomposed into coarse and fine scales, and the coarse scales are registered with one another using a landmark-based registration algorithm. The resulting transformation is then used as an initial guess to deformably register the original images with each other. Using four-dimensional computed tomography (4D CT) images, the hybrid algorithm is shown to be more accurate and robust than standard deformable registration algorithms, standard landmark-based registration algorithms, combined landmark-based and deformable registration algorithms, and our previous multiscale deformable registration algorithms. Numerous image registration experiments are presented in two and three dimensions. The hybrid algorithm is shown to be particularly robust with respect to the presence of noise in the images to be registered, as well as with respect to the location and number of landmarks used in the coarse scale landmark-based registration stage of the algorithm. The main advantages of our hybrid algorithm over traditional image registration techniques are its improved accuracy, efficiency, and robustness.

Group Dynamics of Phototaxis: Interacting Stochastic Many-Particle Systems and their Continuum Limit (640 KB)
(with D. Bhaya and T. Requiejo)
to appear

In this paper we introduce new models for describing the motion of phototaxis, i.e. bacteria that move towards light. Following experimental observations, the first model describes the locations of bacteria, an internal property of the bacteria that is related to the group dynamics, and the interaction between the bacteria and the medium in which it resides. The second model is a new multi-particle system following the same quantities as in the first model. The main theorem shows how to obtain a new system of PDEs, which we refer to as the phototaxis system as the limit dynamics of the multi-particle system. Numerical simulations are provided.