Advection-diffusion equation on a half-line with boundary Lévy noise

Previous Article
Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion
DCDS-B Home
This Issue
Next Article
Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps

February 2019, 24(2): 637-655. doi: 10.3934/dcdsb.2018200

Advection-diffusion equation on a half-line with boundary Lévy noise

Lena-Susanne Hartmann and Ilya Pavlyukevich

Friedrich Schiller University Jena, School of Mathematics and Computer Science, Institute for Mathematics, Ernst-Abbe-Platz 2, 07743 Jena, Germany

Received May 2017 Revised February 2018 Published June 2018

In this paper we study a one-dimensional linear advection-diffusion equation on a half-line driven by a Lévy boundary noise. The problem is motivated by the study of contaminant transport models under random sources (P. P. Wang and C. Zheng, Ground water, 43 (2005), [34]). We determine the closed form formulae for mild solutions of this equation with Dirichlet and Neumann noise and study approximations of these solutions by classical solutions obtained with the help of Wong-Zakai approximations of the driving Lévy process.

Keywords: Advection-diffusion equation, boundary noise, Lévy noise, Wong-Zakai approximations, $ M_1 $-convergence.

Mathematics Subject Classification: Primary: 60H15; Secondary: 35K10, 60F17, 60G51, 86A05.

Citation: Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200

References:

[1]	E. Alòs and S. Bonaccorsi, Stability for stochastic partial differential equations with Dirichlet white-noise boundary conditions, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 5 (2002), 465-481. doi: 10.1142/S0219025702000948.
[2]	E. Alòs and S. Bonaccorsi, Stochastic partial differential equations with Dirichlet white-noise boundary conditions, Ann. Inst. H. Poincaré Probab. Statist, 38 (2002), 125-154. doi: 10.1016/S0246-0203(01)01097-4.
[3]	A. V. Balakrishnan, Applied Functional Analysis, vol. 3 of Applications of Mathematics, 2nd edition, Springer, New York, 1981.
[4]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
[5]	P. Brune, J. Duan and B. Schmalfuss, Random dynamics of the Boussinesq system with dynamical boundary conditions, Stochastic Analysis and Applications, 27 (2009), 1096-1116. doi: 10.1080/07362990902976546.
[6]	Z. Brzeźniak and F. Flandoli, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations, Stochastic Processes and Their Applications, 55 (1995), 329-358. doi: 10.1016/0304-4149(94)00037-T.
[7]	Z. Brzeźniak, B. Goldys, S. Peszat and F. Russo, Second order PDEs with Dirichlet white noise boundary conditions, Journal of Evolution Equations, 15 (2015), 1-26. doi: 10.1007/s00028-014-0246-2.
[8]	Z. Brzeźniak and S. Peszat, Hyperbolic equations with random boundary conditions, in Recent Development in Stochastic Dynamics and Stochastic Analysis (eds. J. Duan, S. Luo and C. Wang), vol. 8 of Interdisciplinary Mathematical Sciences, World Scientific, Singapore, 2010, 1-21. doi: 10.1142/9789814277266_0001.
[9]	H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solid, The Clarendon Press, Oxford University Press, New York, 1988.
[10]	A. Chaudhuri and M. Sekhar, Stochastic modeling of solute transport in 3-D heterogeneous porous media with random source condition, Stochastic Environmental Research and Risk Assessment, 21 (2006), 159-173. doi: 10.1007/s00477-006-0053-6.
[11]	A. Chojnowska-Michalik, On processes of Ornstein-Uhlenbeck type in Hilbert space, Stochastics, 21 (1987), 251-286. doi: 10.1080/17442508708833459.
[12]	I. Chueshov and B. Schmalfuss, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential and Integral Equations, 17 (2004), 751-780.
[13]	G. Da Prato and J. Zabczyk, Evolution equations with white-noise boundary conditions, Stochastics and Stochastics Reports, 42 (1993), 167-182. doi: 10.1080/17442509308833817.
[14]	G. Fabbri and B. Goldys, An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise, SIAM Journal on Control and Optimization, 48 (2009), 1473-1488. doi: 10.1137/070711529.
[15]	D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, EMS Textbooks in Mathematics, European Mathematical Society, Zürich, 2008.
[16]	E. Hausenblas and P. A. Razafimandimby, Controllability and qualitative properties of the solutions to SPDEs driven by boundary Lévy noise, Stochastic Partial Differential Equations: Analysis and Computations, 3 (2015), 221-271. doi: 10.1007/s40072-015-0047-9.
[17]	W. A. Jury and H. Flühler, Transport of chemicals through soil: Mechanisms, models, and field applications, Advances in agronomy, 47 (1992), 141-201. doi: 10.1016/S0065-2113(08)60490-3.
[18]	A. Kreft and A. Zuber, On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions, Chemical Engineering Science, 33 (1978), 1471-1480. doi: 10.1016/0009-2509(78)85196-3.
[19]	J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, vol. 181 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1972.
[20]	C. Man and C. W. Tsai, Stochastic partial differential equation-based model for suspended sediment transport in surface water flows, Journal of Engineering Mechanics, 133 (2007), 422-430. doi: 10.1061/(ASCE)0733-9399(2007)133:4(422).
[21]	F. Masiero, A stochastic optimal control problem for the heat equation on the halfline with Dirichlet boundary-noise and boundary-control, Applied Mathematics & Optimization, 62 (2010), 253-294. doi: 10.1007/s00245-010-9103-z.
[22]	S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation on the haf-line, Transactions of the American Mathematical Society, 353 (2000), 1635-1659. doi: 10.1090/S0002-9947-00-02665-9.
[23]	J. C. Parker and M. T. van Genuchten, Flux-averaged and volume-averaged concentrations in continuum approaches to solute transport, Water Resources Research, 20 (1984), 866-872. doi: 10.1029/WR020i007p00866.
[24]	I. Pavlyukevich and M. Riedle, Non-standard Skorokhod convergence of Lévy-driven convolution integrals in Hilbert spaces, Stochastic Analysis and Applications, 33 (2015), 271-305. doi: 10.1080/07362994.2014.988358.
[25]	I. Pavlyukevich and I. M. Sokolov, One-dimensional space-discrete transport subject to Lévy perturbations, The Journal of Statistical Physics, 133 (2008), 205-215. doi: 10.1007/s10955-008-9607-y.
[26]	A. Pazy, Semigroups of Linear Operators and Applications toPartial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[27]	S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, vol. 113 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.
[28]	A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton, FL, 2002.
[29]	M. Riedle, Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes, Potential Analysis, 42 (2015), 809-838. doi: 10.1007/s11118-014-9458-x.
[30]	A. V. Skorokhod, Limit theorems for stochastic processes, Theory of Probability and its Applications, 1 (1956), 289-319.
[31]	G. Tessitore and J. Zabczyk, Wong-Zakai approximations of stochastic evolution equations, Journal of Evolution Equations, 6 (2006), 621-655. doi: 10.1007/s00028-006-0280-9.
[32]	H. Triebel, Theory of Function Spaces II, Monographs in Mathematics, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0346-0419-2.
[33]	K. Twardowska, On the approximation theorem of the Wong-Zakai type for the functional stochastic differential equations, Probability and Mathematical Statistics, 12 (1991), 319-334.
[34]	P. P. Wang and C. Zheng, Contaminant transport models under random sources, Ground Water, 43 (2005), 423-433.
[35]	W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer, 2002.
[36]	E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, The Annals of Mathematical Statistics, 36 (1965), 1560-1564. doi: 10.1214/aoms/1177699916.
[37]	E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, International Journal of Engineering Science, 3 (1965), 213-229. doi: 10.1016/0020-7225(65)90045-5.

show all references

References:

[1]	E. Alòs and S. Bonaccorsi, Stability for stochastic partial differential equations with Dirichlet white-noise boundary conditions, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 5 (2002), 465-481. doi: 10.1142/S0219025702000948.
[2]	E. Alòs and S. Bonaccorsi, Stochastic partial differential equations with Dirichlet white-noise boundary conditions, Ann. Inst. H. Poincaré Probab. Statist, 38 (2002), 125-154. doi: 10.1016/S0246-0203(01)01097-4.
[3]	A. V. Balakrishnan, Applied Functional Analysis, vol. 3 of Applications of Mathematics, 2nd edition, Springer, New York, 1981.
[4]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.
[5]	P. Brune, J. Duan and B. Schmalfuss, Random dynamics of the Boussinesq system with dynamical boundary conditions, Stochastic Analysis and Applications, 27 (2009), 1096-1116. doi: 10.1080/07362990902976546.
[6]	Z. Brzeźniak and F. Flandoli, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations, Stochastic Processes and Their Applications, 55 (1995), 329-358. doi: 10.1016/0304-4149(94)00037-T.
[7]	Z. Brzeźniak, B. Goldys, S. Peszat and F. Russo, Second order PDEs with Dirichlet white noise boundary conditions, Journal of Evolution Equations, 15 (2015), 1-26. doi: 10.1007/s00028-014-0246-2.
[8]	Z. Brzeźniak and S. Peszat, Hyperbolic equations with random boundary conditions, in Recent Development in Stochastic Dynamics and Stochastic Analysis (eds. J. Duan, S. Luo and C. Wang), vol. 8 of Interdisciplinary Mathematical Sciences, World Scientific, Singapore, 2010, 1-21. doi: 10.1142/9789814277266_0001.
[9]	H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solid, The Clarendon Press, Oxford University Press, New York, 1988.
[10]	A. Chaudhuri and M. Sekhar, Stochastic modeling of solute transport in 3-D heterogeneous porous media with random source condition, Stochastic Environmental Research and Risk Assessment, 21 (2006), 159-173. doi: 10.1007/s00477-006-0053-6.
[11]	A. Chojnowska-Michalik, On processes of Ornstein-Uhlenbeck type in Hilbert space, Stochastics, 21 (1987), 251-286. doi: 10.1080/17442508708833459.
[12]	I. Chueshov and B. Schmalfuss, Parabolic stochastic partial differential equations with dynamical boundary conditions, Differential and Integral Equations, 17 (2004), 751-780.
[13]	G. Da Prato and J. Zabczyk, Evolution equations with white-noise boundary conditions, Stochastics and Stochastics Reports, 42 (1993), 167-182. doi: 10.1080/17442509308833817.
[14]	G. Fabbri and B. Goldys, An LQ problem for the heat equation on the halfline with Dirichlet boundary control and noise, SIAM Journal on Control and Optimization, 48 (2009), 1473-1488. doi: 10.1137/070711529.
[15]	D. D. Haroske and H. Triebel, Distributions, Sobolev Spaces, Elliptic Equations, EMS Textbooks in Mathematics, European Mathematical Society, Zürich, 2008.
[16]	E. Hausenblas and P. A. Razafimandimby, Controllability and qualitative properties of the solutions to SPDEs driven by boundary Lévy noise, Stochastic Partial Differential Equations: Analysis and Computations, 3 (2015), 221-271. doi: 10.1007/s40072-015-0047-9.
[17]	W. A. Jury and H. Flühler, Transport of chemicals through soil: Mechanisms, models, and field applications, Advances in agronomy, 47 (1992), 141-201. doi: 10.1016/S0065-2113(08)60490-3.
[18]	A. Kreft and A. Zuber, On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions, Chemical Engineering Science, 33 (1978), 1471-1480. doi: 10.1016/0009-2509(78)85196-3.
[19]	J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, vol. 181 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1972.
[20]	C. Man and C. W. Tsai, Stochastic partial differential equation-based model for suspended sediment transport in surface water flows, Journal of Engineering Mechanics, 133 (2007), 422-430. doi: 10.1061/(ASCE)0733-9399(2007)133:4(422).
[21]	F. Masiero, A stochastic optimal control problem for the heat equation on the halfline with Dirichlet boundary-noise and boundary-control, Applied Mathematics & Optimization, 62 (2010), 253-294. doi: 10.1007/s00245-010-9103-z.
[22]	S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation on the haf-line, Transactions of the American Mathematical Society, 353 (2000), 1635-1659. doi: 10.1090/S0002-9947-00-02665-9.
[23]	J. C. Parker and M. T. van Genuchten, Flux-averaged and volume-averaged concentrations in continuum approaches to solute transport, Water Resources Research, 20 (1984), 866-872. doi: 10.1029/WR020i007p00866.
[24]	I. Pavlyukevich and M. Riedle, Non-standard Skorokhod convergence of Lévy-driven convolution integrals in Hilbert spaces, Stochastic Analysis and Applications, 33 (2015), 271-305. doi: 10.1080/07362994.2014.988358.
[25]	I. Pavlyukevich and I. M. Sokolov, One-dimensional space-discrete transport subject to Lévy perturbations, The Journal of Statistical Physics, 133 (2008), 205-215. doi: 10.1007/s10955-008-9607-y.
[26]	A. Pazy, Semigroups of Linear Operators and Applications toPartial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[27]	S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, vol. 113 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511721373.
[28]	A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton, FL, 2002.
[29]	M. Riedle, Ornstein-Uhlenbeck processes driven by cylindrical Lévy processes, Potential Analysis, 42 (2015), 809-838. doi: 10.1007/s11118-014-9458-x.
[30]	A. V. Skorokhod, Limit theorems for stochastic processes, Theory of Probability and its Applications, 1 (1956), 289-319.
[31]	G. Tessitore and J. Zabczyk, Wong-Zakai approximations of stochastic evolution equations, Journal of Evolution Equations, 6 (2006), 621-655. doi: 10.1007/s00028-006-0280-9.
[32]	H. Triebel, Theory of Function Spaces II, Monographs in Mathematics, Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0346-0419-2.
[33]	K. Twardowska, On the approximation theorem of the Wong-Zakai type for the functional stochastic differential equations, Probability and Mathematical Statistics, 12 (1991), 319-334.
[34]	P. P. Wang and C. Zheng, Contaminant transport models under random sources, Ground Water, 43 (2005), 423-433.
[35]	W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer, 2002.
[36]	E. Wong and M. Zakai, On the convergence of ordinary integrals to stochastic integrals, The Annals of Mathematical Statistics, 36 (1965), 1560-1564. doi: 10.1214/aoms/1177699916.
[37]	E. Wong and M. Zakai, On the relation between ordinary and stochastic differential equations, International Journal of Engineering Science, 3 (1965), 213-229. doi: 10.1016/0020-7225(65)90045-5.

Figure 1. A sample path of an $\alpha$-stable Lévy subordinator $Z$ with ${\bf E} \text{e}^{-\lambda Z_1} = \text{e}^{-\lambda^\alpha}$ for $\alpha = 0.9$ (a); solutions $t\mapsto u_D(t, x)$ of equation (2.2) with Dirichlet boundary noise for $\nu = -1$, $x = 1$ (b) and $\nu = 1$, $x = 1$ (d); the concentration curve $x\mapsto u_D(t, x)$ for $\nu = 1$, $t = 55$ (c)

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. A sample path of a symmetric $\alpha$-stable Lévy process $Z$ with ${\bf{E}} \text{e}^{-\text{i} \lambda Z_1} = \text{e}^{-|\lambda|^\alpha}$ for $\alpha = 1.75$ (a); the solution $t\mapsto u_D(t, x)$ of equation (2.2) with Dirichlet boundary noise for $\nu = 1$, $x = 1$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. The scales $c(x)$ of the limiting distribution in the Dirichlet case for $\nu = \pm1, 0$ (left), and the Neumann case for $\nu = -1$ (right); $\alpha = 0.9$, $c = 1$

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Anhui Gu. Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-31. doi: 10.3934/dcdsb.2019104
[2]	Yeping Li, Jie Liao. Stability and $ L^{p}$ convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1281-1302. doi: 10.3934/cpaa.2019062
[3]	Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068
[4]	Yupeng Li, Wuchen Li, Guo Cao. Image segmentation via $ L_1 $ Monge-Kantorovich problem. Inverse Problems & Imaging, 2019, 13 (4) : 805-826. doi: 10.3934/ipi.2019037
[5]	Lidan Li, Hongwei Zhang, Liwei Zhang. Inverse quadratic programming problem with $ l_1 $ norm measure. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2019061
[6]	Tuan Anh Dao, Michael Reissig. $ L^1 $ estimates for oscillating integrals and their applications to semi-linear models with $ \sigma $-evolution like structural damping. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5431-5463. doi: 10.3934/dcds.2019222
[7]	Anhui Gu, Kening Lu, Bixiang Wang. Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 185-218. doi: 10.3934/dcds.2019008
[8]	Zalman Balanov, Yakov Krasnov. On good deformations of $ A_m $-singularities. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1851-1866. doi: 10.3934/dcdss.2019122
[9]	Ziheng Chen, Siqing Gan, Xiaojie Wang. Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-33. doi: 10.3934/dcdsb.2019154
[10]	Harbir Antil, Mahamadi Warma. Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence. Mathematical Control & Related Fields, 2019, 9 (1) : 1-38. doi: 10.3934/mcrf.2019001
[11]	Connor Mooney, Ovidiu Savin. Regularity results for the equation $ u_{11}u_{22} = 1 $. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-12. doi: 10.3934/dcds.2019235
[12]	Teresa Alberico, Costantino Capozzoli, Luigi D'Onofrio, Roberta Schiattarella. $G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 129-137. doi: 10.3934/dcdss.2019009
[13]	Yonglin Cao, Yuan Cao, Hai Q. Dinh, Fang-Wei Fu, Jian Gao, Songsak Sriboonchitta. Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. Advances in Mathematics of Communications, 2018, 12 (2) : 231-262. doi: 10.3934/amc.2018016
[14]	Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $ L^2 $-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122
[15]	Wenqiang Zhao. Random dynamics of non-autonomous semi-linear degenerate parabolic equations on $\mathbb{R}^N$ driven by an unbounded additive noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2499-2526. doi: 10.3934/dcdsb.2018065
[16]	Jun Wang, Xing Tao Wang. Sparse signal reconstruction via the approximations of $ \ell_{0} $ quasinorm. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2019035
[17]	Hideaki Takagi. Times until service completion and abandonment in an M/M/$ m$ preemptive-resume LCFS queue with impatient customers. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1701-1726. doi: 10.3934/jimo.2018028
[18]	Koya Nishimura. Global existence for the Boltzmann equation in $ L^r_v L^\infty_t L^\infty_x $ spaces. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1769-1782. doi: 10.3934/cpaa.2019083
[19]	Linglong Du. Long time behavior for the visco-elastic damped wave equation in $\mathbb{R}^n_+$ and the boundary effect. Networks & Heterogeneous Media, 2018, 13 (4) : 549-565. doi: 10.3934/nhm.2018025
[20]	Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319

2018 Impact Factor: 1.008

Tools

Metrics

Tools

Article outline

Figures and Tables

2018 Impact Factor: 1.008

Tools

Figures and Tables

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences