A Multiresolution Approach for Modeling of Diffusion Phenomenon

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1 A Multiresolution Approach for Modeling of Diffusion Phenomenon Reza Madankan a Puneet Singla a Tarunraj Singh a Peter D. Scott b rm93@buffalo.edu psingla@buffalo.edu tsingh@buffalo.edu peter@buffalo.edu a Department of Mechanical & Aerospace Engineering b Department of Computer Science & Engineering University at Buffalo, Buffalo, NY-426. Abstract A multiresolution approach is developed to model the spatio-temporal dispersion of geospatial data such as evolution of population, epidemics, dispersion of chemical/biological agents subsequent to natural or man-made disasters, etc. The basic idea is to make use of diffusion equations with spatially and temporally varying diffusion coefficient to model the evolution of geospatial activities. At the heart of our approach is a recently developed powerful averaging process that allows construction of a global family of local independent approimations with a desired order of continuity and quantifying error bounds. The ability to have independent local approimations enables intuitive and effecient approimation with high accuracy and compression ratio. Numerical simulation results are presented to show the efficacy of the presented ideas. Inde Terms Parameter Estimation, Multiresolution Modeling, Diffusion Coefficient. INTRODUCTION The diffusion model has been used for a wide variety of applications [], [2] including spatial modeling of infectious diseases, air quality modeling, transport of amino acids within cells, diffusion of oygen into blood of mammalian lungs, doping of semiconductors, and recently on how information, or products are disseminated and adopted based on social networks. The diffusion coefficient in most of these applications is not a constant across the system of interest, rather it varies spatially and often temporally as well. For instance consider the spread of the emerald ash borer, which is a virulent parasite which feeds on ash trees. Clearly, the spatial distribution of the ash trees will influence the strength of the dispersion of the pest in a specific direction [3]. In applications where one needs to control, or forecast, the dispersing substance, one needs an accurate representation of the diffusion process. This implies a need for an accurate estimate of the generally spatio-temporally varying diffusion coefficient. System identification is the process of determining the system model based on measurements. This work endeavors to develop techniques which can identify spatio-temporally varying diffusion coefficients to permit accurate forecasting of the diffusing substance of interest. This has ramification in better management and deployment of resources for epidemic control, better doping of semi-conductors or estimation of the state evolution of other applications modeled by the diffusion process. The structure of paper is as follows: first a multiresolution framework for the modeling of diffusion coefficient is presented followed by a brief description of the GLO-MAP averaging process. Net, an optimization problem is formulated to find unknown coefficient of diffusion model and finally, numerical simulation are presented. 2. MULTIRESOLUTION MODELING The basic idea is to model the evolution of geospatial activities [4], [5]. The homogeneous form of the proposed diffusion model is u(, t) t = 2 K(, t)u(, t) 2 () where, u(, t) is the spatially distributed quantity of interest, is the spatial, generally vector-valued variable and K(, t) is the spatio-temporally varying diffusion coefficient, generally matri valued. To motivate choice of this model, consider the population dynamics in a rural region surrounding a simple highway network. We assume that domain

2 (a) Initial Population Distribution (b) Diffused Population Distribution Fig.. Modeling of Spatio-Temporal Varying Process knowledge informs us that the coefficient of diffusion is high immediately adjacent to a major road and reduces monotonically with distance from the roads. To numerically illustrate the growth of residential construction given a high potential of jobs at the center of the city, we model the distribution of residential construction as a diffusion process. As the jobs increase, one can use the diffusion process to study how the growth of residential construction evolves to cater to the growing requirement of houses for the increase number of workers. Such dynamics cannot be effectively modeled using a uniform spatio-temporally invariant coefficient field. We model this hypothesized scenario by using an initial high concentration of the diffusing material in the center of the test-space as shown in Fig. (a). Three highways emanate from the center as shown and we associate a large coefficient of diffusion with each of these highways, with the east running highway which has a larger capacity having a higher coefficient of diffusion. Solving the resulting diffusion model, the contours in Figs. (a) & (b) illustrate the predicted growth of residential housing in the vicinity of the center and highways, respectively. The main challenge is to find the spatiotemporally varying diffusion coefficient from measured data ũ(, t). A key consideration is how to model the unknown diffusion coefficient K(, t). If K(, t) is assumed to be a continuous function in and t, then according to Weierstrass approimation theorem [6], K(, t) can be approimated arbitrarily closely by a polynomial series. K(, t) a T (t)φ() (2) where, Φ(.) R N is a finite dimensional vector of polynomial functions and a is a vector of amplitudes corresponding to the selected polynomial basis functions. It is well known that as the number of basis functions increases, i.e., N, the amplitude vector may be selected so that the approimation error goes to zero. However, in practice this is not possible since such a model will have unboundedly many parameters to optimize. An alternative is to have a multiresolution model for learning the diffusion coefficient K(, t) which adequately minimizes the approimation error to the desired level. As the name suggests, multiresolution approimation involves hierarchically decomposing the inputoutput approimation to capture both global and local features of the system dynamics. Conventional spline [7], finite element methods, piecewise linear approimation [8] and wavelet approimation [9] are some eamples of the multiresolution algorithms. Although global-local decomposition of the region of interest leads to improved approimation algorithms, for a specific multiresolution algorithm one cannot use arbitrary sets of basis functions to obtain different local approimations without introducing discontinuity of the coefficient or its derivatives across the boundaries of different local regions []. For eample, in the case of waveletbased approimation, one is restricted to use the same wavelet kernel at different resolution levels.

3 Basically, a significant challenge is the lack of rigorous methods to merge different independent local approimations to obtain a desired order globally smooth approimation. Recently, Singla and Junkins [] have developed a multiresolution approach known as the Global-Local Orthogonal MAPping (GLO-MAP) algorithm, which provides a mechanism to blend diverse local models into a statistically unbiased global representation. 2.. GLO-MAP Averaging Process In this section, we briefly summarize the basic idea of the GLO-MAP-based approimation which permits blending locally independent approimations to form a globally continuous function. More information about this approach can be found in Refs. [ [], []]. For illustration purposes, let us assume that X = {, 2, Nl } is a uniform -D grid with spacing h and K = {K (),, K Nl ()} is a set of continuous functions that approimates the global spatio-temporally separable diffusion coefficient field spatial factor K() at points i X. We define the weighted average approimation centered on i th verte i valid over [ i, i+ ]: where, K i () = w( i )K i () + w( i+ )K i+ () (3) i = ( i ) h The weighting function w( i ) is used to blend or average the two adjacent preliminary local approimations K i () and K i+ () and the global function is given by the epression: N l K() = w( i )K i (), i [, ]. (4) i= Notice that the preliminary local approimations K i () K are completely arbitrary, as long as they are smooth and represents the local behavior of global unknown diffusion coefficient K() well. In Refs. [], it is shown that if the weighting functions of Eq. (3) satisfies the following boundary value problem (Eq. 5), then the weighted average approimation in Eq. (3) form an m th -order continuous globally valid model with complete freedom m=2 m= m=.5.5 (a) -D Weighting functions for the first three orders of continuity. (b) 2-D Weighting functions for second order continuity. Fig. 2. Weighting functions w(τ i) in the choice of the local approimations in K. w() =, w() =, d k w d d k = =, k w d k = =, k =,, m w( ) + w( ) =,, (5) Notice that the first two aforementioned conditions guarantee the continuity of the blended approimation, whereas the third condition, that the weights form a partition of unity, guarantees that the blended approimation is unbiased. Assuming the weighting functions to be polynomial in nature, and adopting the procedure listed in Ref. [], the generic epression for m th order continuity weight function can be written as { (2m + )!( ) w( ) = η m+ m (m!) 2 m ( ) k ( } m )η m k 2m k + k where k= [, ], η = (6) Figs. 2(a) and 2(b) show the lowest order weight function (for m = ) assuming it to be a polynomial

4 in the independent variable η. Observe that by choosing the weighting functions given by Eq. (6), we are guaranteed global piecewise continuity for all possible continuous local approimations in K. One retains the freedom to choose the local approimations as needed, to fit the local behavior of K(), and to rely upon w( ) to enforce continuity across knot points, i as illustrated in Fig. 3(a). A fundamental theoretical result thus obtained [] K() A K () = + 2 K 2 () = 2 B K 3 () = +sin(2π) C K 4 () = 2 (a) Local approimations K (), K 2(), K 3() and K 4(). K() Globally continuous function from local approima- Fig. 3. tions A K () = + 2 K 2 () = 2 B Local Approimations Resulting Curve C K4 () = 2 K 3 () = +sin(2π) (b) Weighting Function Approimation of K(). states that if the local approimations are unbiased estimations of the input-output data, then: (i) the final blended approimation is un-biased, (ii) the variance of the blended approimation is substantially smaller than the variance of any of the averaged local approimations, (iii) the mathematical structure of various local approimations can be varied as appropriate to capture the local space-variant properties of the diffusion model, and finally, (iv) the blending weight functions guarantee the global piecewise continuous nature of the approimation. In Refs. [] [2], these ideas are developed systematically and etended rigorously to approimation with arbitrary order continuity in an n dimensional space. In general, the final approimation in any hypercube is obtained by averaging 2 n overlapping approimations centered at the vertices of that local hypercube. 3. ESTIMATION OF DIFFUSION COEFFICIENT Now making use of the GLO-MAP averaging procedure, the unknown diffusion coefficient term K(, t) can be written as N l K(, t) = w i (, t)k i (, t) (7) i= where N l is the total number of knot points in general n dimensional space, R n and K i (, t) represent the i th local approimation of the global unknown function K(, t) centered at i th knot point. Furthermore, we can approimate each K i (, t) using a set of independent and separable basis functions, φ j (.). K i (, t) = a i (t)φ i (), i =, 2,, N l (8) where, φ i (.) is a N vector of independent basis functions set, and a i is a vector of corresponding basis function coefficients. Notice that one can choose a basis function set φ i for each local approimation independently to each other. After substituting Eq. (8) in Eq. (7), we can rewrite the unknown nonlinear term K(, t) as N l K(, t) = w i ()a i φ i () (9) i= = A(t) Φ()W() }{{} Ψ() () = A(t)Ψ() () where A is a matri of basis function coefficients for all local approimations. Note that due to the compact support of GLO-MAP weighting functions, only 2 n neighboring approimations need to be considered, depending upon the value of. Now, given measurement of spatio-temporal process ũ(, t), problem reduces to find the optimum value for coefficient matri A. To find the optimum

5 value of A, we minimize the following least square criterion: J = min i=n i= j=n t (ũ( i, t j ) u( i, t j )) 2 (2) j= where, u( i, t j ) is the solution of Eq. () while using current estimated diffusion coefficient, and ũ( i, t j ) denotes spatio-temporal measurement data. N t and N are number of temporal and spatial points, respectively. 4. NUMERICAL SIMULATIONS In this section, we present numerical simulation results to show the efficacy of the proposed ideas. 4.. Test Case As the first eample, we assume the true value of diffusion coefficient to be one, i.e., K(, t) =. We solve the diffusion equation with following boundary and initial conditions to generate measurement data on a grid in (, t): u(, ) = u(, t) = t 2 u(, t) = t 2 (3) We use the MATLAB pde toolbo to solve the resulting diffusion equation. Fig. 4(a) shows the surface plot of the solution u(,t). To estimate the diffusion coefficient K(, t), we minimize the cost function of Eq. (2) with initial guess of diffusion coefficient to be. We use gradient descent algorithm in MATLAB to find an optimal value for the diffusion coefficient and use both Global-Galerkin method and MATLAB pde toolbo to solve the resulting diffusion equation at each iteration. Fig. 4(b) shows the plot of estimated diffusion quantity while making use of MATLAB pde toolbo. Table I lists the optimal value of diffusion coefficient for different order of Galerkin solution and by making use of MATLAB pde toolbo. As epected, estimation error decreases as we increase the order of Galerkin approimation and MATLAB pde toolbo always gives us the true value for diffusion coefficient. TABLE I OBTAINED VALUES OF K galerkin () AND K pde () FOR DIFFERENT ORDER OF GALERKIN SOLUTION Order K galerkin () K pde () K true() Test Case 2 Having demonstrated the ability of the method to converge to the constant value of a space-invariant coefficient, we net consider an eample of spatially varying diffusion coefficient. We assume the true time-invariant diffusion coefficient to be: K() = e 2 Consider the following boundary and initial conditions: u(, ) = u(, t) = t(t ) (4) u(, t) = t(t ) Measurements over grid in (, t) space were generated using the MATLAB pde toolbo. We use the GLO-MAP method as described in the previous section to approimate the spatially varying diffusion coefficient. Once again Gradient descent method in MATLAB was used to minimize cost inde given by Eq. (2). Fig. 5(a) shows the surface plot for the simulated measurement data. Fig. 5(b) shows the plot of estimated diffusion quantity with diffusion coefficient approimated as a weighted average of four independent local approimations. Fig. 6 shows the plot of true and estimated diffusion coefficient for different number of local approimations in the GLO-MAP model. The solid black line corresponds to the true value of diffusion coefficient while crossed blue line and dashed red line correspond to estimated diffusion coefficient with N s = and N s = 4, respectively. Second order polynomials are used to approimate local behavior of diffusion coefficient in each subinterval and GLO-MAP weight functions corresponding to m = 2 are used to construct global approimation for the diffusion coefficient. Fig. 7 shows the plot of estimation error vs. number of local approimations (N s ). As epected, the estimation error decreases with the increase in number of local approimations in the GLO-MAP model.

6 PDE toolbo solution for known points PDE toolbo solution for known points.5.5 u(,t) u(,t) Time (sec) Time (sec) (a) True Measurement Data (b) Estimated Measurement Data (with MATLAB pde Toolbo) Fig. 4. Simulation Results for Test Case PDE solution at known points PDE solution at known points u(,t) u(,t) Time (sec) Time (sec) (a) True Measurement Data (b) Estimated Measurement Data (with MATLAB pde Toolbo) Fig. 5. Simulation Results for Test Case K true =e 2 K GLO MAP, N s = 4 K GLO MAP, N s = 4.3. Test Case 3 As the last case, we consider another eample of spatially varying diffusion coefficient. We assume true time-invariant diffusion coefficient to be: k().6 K() = 2 + sin(2π) Fig. 6. True and Estimated K() for Test Case 2. To generate corresponding measurement data, we consider boundary and initial conditions as same as Test Case (2): u(, ) = u(, t) = t(t ) u(, t) = t(t ) (5)

7 5. CONCLUSIONS Fig. 7. Estimation Error vs. N s Similar to Case (2), the MATLAB pde toolbo was used to solve diffusion equation over a grid in (t, ) space. We use the GLO-MAP method as described in the previous section to approimate the spatially varying diffusion coefficient. Fig. 8 shows the plot of true and estimated diffusion coefficient. The solid blue line corresponds to the true value of diffusion coefficient while crossed black line and dashed red line correspond to estimated diffusion coefficient with N s = and N s = 5, respectively. Second order polynomials are used to approimate local behavior of diffusion coefficient in each subinterval. Furthermore, second order (m = 2) GLO-MAP weight functions are used to construct global approimation for the diffusion coefficient. It is clear from these plots that with five local approimations, we are able to approimate the diffusion coefficient well. K() K true = 2+sin(2π) K GLO MAP, N s = 5 K GLO MAP, N s = Fig. 8. True and Estimated K() for Test Case 3 In this paper, a multiresolution method for diffusion coefficient estimation is presented to support the modeling and identification of spatiotemporal dispersion phenomenon. The basic idea of the proposed approach is to make use of diffusion equations with spatially and temporally varying diffusion coefficient to model the evolution of geospatial activities. Recently developed GLO- MAP averaging process is used to construct a global approimation of the diffusion coefficient by blending a family of local independent diffusion coefficient approimations with a desired order of continuity. Least-square criterion is used to find an optimal value of unknown parameters of GLO-MAP approimation from diffusion data on a spatio-temporal grid. Preliminary simulation results forms a basis for optimism for the proposed ideas. 6. ACKNOWLEDGEMENT This work was supported by the ONR Contract No. HM The authors gratefully acknowledge the support of NURI grant. REFERENCES [] L. Ross and J. Radcliffe, Spatial Deterministic Epidemics. American Mathematical Society, 23. [2] R. B. Banks, Growth and Diffusion Phenomena. Springer-Verlag, 994. [3] T. K. BenDor and S. S. Metcalf, The spatial dynamics of invasive species spread, System Dynamics Review, vol. 22, no., pp. 27 5, 26. [4] S. Wright, Evolution in mendelian populations, Genetics, vol. 6, pp , 93. [5] P. Narain, The conditioned diffusion equation and its use in population genetics, Journal of the Royal Statistical Society, vol. 36. [6] J. J. Benedetto, Harmonic Analysis and Applications. CRC Press Inc., 996. [7] C. D. Boor, A Practical Guide to Splines. Springer, 978. [8] O. Nelles, Nonlinear System Identification. New York, NY: Springer, 2. [9] R. J. Prazenica and A. J. Kurdila, Volterra Kernel Identification Using Triangular Wavelets, Journal of Vibration and Control, vol., no. 4, pp , 24. [] P. Singla and J. L. Junkins, Multi-Resolution Methods for Modeling and Control of Dynamical Systems. CRC Press, Aug. 28. [] P. Singla, Multi-resolution methods for high fidelity modeling and control allocation in large-scale dynamical systems, Ph.D. dissertation, Teas A&M University, College Station, TX, 26.

8 [2] J. L. Junkins, P. Singla, T. D. Griffith, and T. Henderson, Orthogonal global/local approimation in n-dimensions: Applications to input/output approimation, in 6th International conference on Dynamics and Control of Systems and Structures in Space, Cinque-Terre, Italy, 24.