0000000001325837

AUTHOR

Shiqi Ma

showing 3 related works from this author

Determining a Random Schrödinger Operator : Both Potential and Source are Random

2020

We study an inverse scattering problem associated with a Schr\"odinger system where both the potential and source terms are random and unknown. The well-posedness of the forward scattering problem is first established in a proper sense. We then derive two unique recovery results in determining the rough strengths of the random source and the random potential, by using the corresponding far-field data. The first recovery result shows that a single realization of the passive scattering measurements uniquely recovers the rough strength of the random source. The second one shows that, by a single realization of the backscattering data, the rough strength of the random potential can be recovered…

Complex systemMicrolocal analysis01 natural sciencesinversio-ongelmatsähkömagneettinen säteilysymbols.namesakeOperator (computer programming)Mathematics - Analysis of PDEs0103 physical sciencessironta0101 mathematicsMathematical PhysicsMathematics35Q60 35J05 31B10 35R30 78A40osittaisdifferentiaaliyhtälötScattering010102 general mathematicsMathematical analysisErgodicityStatistical and Nonlinear PhysicsInverse scattering problemsymbols010307 mathematical physicsmatemaattiset mallitRealization (probability)Schrödinger's cat
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Fixed angle inverse scattering in the presence of a Riemannian metric

2020

We consider a fixed angle inverse scattering problem in the presence of a known Riemannian metric. First, assuming a no caustics condition, we study the direct problem by utilizing the progressing wave expansion. Under a symmetry assumption on the metric, we obtain uniqueness and stability results in the inverse scattering problem for a potential with data generated by two incident waves from opposite directions. Further, similar results are given using one measurement provided the potential also satisfies a symmetry assumption. This work extends the results of [23,24] from the Euclidean case to certain Riemannian metrics.

Mathematics - Differential GeometryWork (thermodynamics)01 natural sciencesinversio-ongelmatFixed angleMathematics - Analysis of PDEsIncident waveEuclidean geometryFOS: MathematicssirontaUniqueness0101 mathematicsinverse medium problemPhysicsosittaisdifferentiaaliyhtälöt35Q60 35J05 31B10 35R30 78A40Applied Mathematics010102 general mathematicsMathematical analysisCarleman estimatesRiemannian metricsSymmetry (physics)010101 applied mathematicsfixed angle scatteringDifferential Geometry (math.DG)Metric (mathematics)Inverse scattering problemAnalysis of PDEs (math.AP)
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Fixed angle inverse scattering for sound speeds close to constant

2021

We study the fixed angle inverse scattering problem of determining a sound speed from scattering measurements corresponding to a single incident wave. The main result shows that a sound speed close to constant can be stably determined by just one measurement. Our method is based on studying the linearized problem, which turns out to be related to the acoustic problem in photoacoustic imaging. We adapt the modified time-reversal method from [P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems 25 (2009), 075011] to solve the linearized problem in a stable way, and use this to give a local uniqueness result for the nonlinear inverse problem.

FOS: Mathematics35R30 35Q60 35J05 31B10 78A40Analysis of PDEs (math.AP)
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