MATH Research Publications
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The collection of peerreviewed research articles (co)authored by faculty of the Department of Mathematics and Statistics.
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Increasing stability in the inverse source problem with attenuation and many frequencies
(SIAM Publ., 2018)We study the interior inverse source problem for the Helmholtz equation from boundary Cauchy data of multiple wave numbers. The main goal of this paper is to understand the dependence of increasing stability on the ... 
Isotropic random fields with infinitely divisible marginal distributions
(Taylor & Francis, 2018)A simple but efficient approach is proposed in this paper to construct the isotropic random field in (d 2), whose univariate marginal distributions may be taken as any infinitely divisible distribution with finite variance. ... 
A generalization of ''Existence and behavior of the radial limits of a bounded capillary surface at a corner''
(Pacific Journal of Mathematics, 201802)The principal existence theorem (i.e., Theorem 1) of "Existence and behavior of the radial limits of a bounded capillary surface at a corner" (Pacific J. Math. 176:1 (1996), 165194) is extended to the case of a contact ... 
A capillary surface with no radial limits
(Pacific Journal of Mathematics, 2018)In 1996, Kirk Lancaster and David Siegel investigated the existence and behavior of radial limits at a corner of the boundary of the domain of solutions of capillary and other prescribed mean curvature problems with contact ... 
Preface  Inverse Problems for Partial Differential Equations First Edition Preface
(Springer, 2017)This book describes the contemporary state of the theory and some numerical aspects of inverse problems in partial differential equations. The topic is of substantial and growing interest for many scientists and engineers ... 
Preface  Inverse Problems for Partial Differential Equations Second Edition Preface
(Springer, 2017)In 8 years after the publication of the first version of this book, the rapidly progressing field of inverse problems witnessed changes and new developments. Parts of the book were used at several universities, and many ... 
Isotropic twodimensional pseudoRiemannian metrics uniquely constructed by a given curvature
(Elsevier, 201801)We prove a global uniqueness theorem of reconstruction of a twodimensional pseudo metric by a given Gaussian curvature. 
Appendix  Function Spaces
(Springer, 2017)We collect here definitions and known results about space Ck+λ of H¨older continuous functions and about Sobolev space Hkp that we used in the book 
Chapter 1  Inverse Problems
(Springer, 2017)In this chapter, we formulate basic inverse problems and indicate their applications. The choice of these problems is not random. We think that it represents their interconnections and some hierarchy. 
Chapter 3  Uniqueness and Stability in the Cauchy Problem
(Springer, 2017)In this chapter we formulate and in many cases prove results on the uniqueness and stability of solutions of the Cauchy problem for general partial differential equations. One of the basic tools is Carlemantype estimates. ... 
Preface  Inverse Problems for Partial Differential Equations Third Edition Preface
(Springer, 2017)In 10 years after the publication of the second edition of this book, the changing field of inverse problems witnessed further new developments. Parts of the book were used at several universities, and many colleagues and ... 
Chapter 2  IllPosed Problems and Regularization
(Springer, 2017)In this chapter, we consider the equation. 
Chapter 5  Elliptic Equations: Many Boundary Measurements
(Springer, 2017)We consider the Dirichlet problem (4.0.1), (4.0.2). At first we assume that for any Dirichlet data g0 we are given the Neumann data g1; in other words, we know the results of all possible boundary measurements, or the ... 
Chapter 7  Integral Geometry and Tomography
(Springer, 2017)The problems of integral geometry are to determine a function given (weighted) integrals of this function over a “rich” family of manifolds. These problems are of importance in medical applications (tomography), and they ... 
Chapter 4  Elliptic Equations: Single Boundary Measurements
(Springer, 2017)In this chapter we consider the elliptic secondorder differential equation Au=finΩ,f=f0−∑j=1n∂jfjAu=finΩ,f=f0−∑j=1n∂jfj with the Dirichlet boundary data u=g0on∂Ω.u=g0on∂Ω. We assume that A = div(−a∇) + b ⋅ ∇ + c ... 
Chapter 10  Some Numerical Methods
(Springer, 2017)In this chapter, we will briefly review some popular numerical methods widely used in practice. Of course it is not a comprehensive collection. We will demonstrate certain methods that are simple and widely used or, in our ... 
Chapter 9  Inverse Parabolic Problems
(Springer, 2017)In this chapter, we consider the secondorder parabolic equation (9.0.1) a0∂tu − div(a∇u) + b · ∇u + cu = f in Q = Ω × (0, T), where Ω is a bounded domain the space Rn with the C2smooth boundary ∂Ω. In Section 9.6, we ... 
Chapter 8  Hyperbolic Problems
(Springer, 2017)In this chapter, we are interested in finding coefficients of the secondorder hyperbolic operator. 
Chapter 6 Scattering Problems and Stationary Waves
(Springer, 2017)The stationary incoming wave u with the wave number k is a solution to the perturbed Helmholtz equation (scattering by medium) Au−k2u=0 in R3 
Time varying isotropic vector random fields on sphere
(Springer, 201712)For a vector random field that is isotropic and mean square continuous on a sphere and stationary on a temporal domain, this paper derives a general form of its covariance matrix function and provides a series representation ...