Modern Characterization of Electromagnetic Systems and its Associated Metrology. Magdalena Salazar-Palma
and dedication for pursuing his visions for the advancement of science in our field and for building bridges between our technical know‐how and societally impactful applications. He made many enormous positive impacts not only on the technical societies in which he was member but also on so many of us as individuals. He was a good friend who would support you without any fear while also giving you his opinion even in a strong way when needed. His energy was contagious. He recognized that leadership is most effective when building by consensus: through talking, listening, and being willing to adjust course when called for. He was one of the key leaders of the AP‐S and promoted a global mindset, paving the way for substantial and unprecedented global reach for the AP‐S. He facilitated the creation of chapters, meetings, and conferences across the world; he visited and recruited members from all corners of the Earth. He was generous with his time and his ideas; he was also one of the most effective advocates (and implementers) of diversity, equity, and inclusion in our community. He was supportive of good ideas, seeking them wherever and from whomever they originated. He would energetically recruit you to implement those ideas regardless of whether you were a student or a decorated senior colleague. He will be missed, but his contributions will endure.
It is also proverbial among those who knew Professor Sarkar well his love for history, archeology, nature, botany, wild life, and animals in general. The world was too small for him. He travelled as much as he could, always learning from different cultures, making new friends, visiting old ones. Colleagues who had the chance to travel with him enjoyed immensely his enthusiasm, lively conversation, and eagerness to explore new places and people, and the warm welcome of his hosts.
He dearly loved his family enjoying immensely his trips back to Kolkata and other Indian cities. He had many pets (dogs, a variety of birds) and a terrace garden that he always enjoyed and personally contributed to improve.
Dr. Sarkar was the best of friends for his colleagues, coworkers, local Bengali community members, and students, always interested in any aspect of their life. At times, he did not speak much, but he will show his kindness and friendship in so many other ways.
It is incredibly difficult to accept Dr. Sarkar’s departure for those of us who knew him well and loved and respected him as a friend, a colleague, a mentor, and a leader.
These words are intended as a tribute to Dr. Sarkar and as a thank‐you note to him for everything he gave us: his invaluable scientific contributions, his mentorship, his leadership, and his friendship!!! He will be missed most dearly…
Magdalena Salazar Palma
Ming Da Zhu
Heng Chen
1 Mathematical Principles Related to Modern System Analysis
Summary
In the mathematical field of numerical analysis, model order reduction is the key to processing measured data. This also enables us to interpolate and extrapolate measured data. The philosophy of model order reduction is outlined in this chapter along with the concepts of total least squares and singular value decomposition.
1.1 Introduction
In mathematical physics many problems are characterized by a second order partial differential equation for a function as
and u(x, y) is the function to be solved for a given excitation f (x, y), where
(1.2)
When Β2 − AC < 0 and assuming uxy = uyx, then (1.1) is called an elliptic partial differential equation. These classes of problems arise in the solution of boundary value problems. In this case, the solution u(x, y) is known only over a boundary {or equivalently a contour B(x, y)} and the goal is to continue the given solution u(x, y) from the boundary to the entire region of the real plane ℜ(x, y).
When B2 − AC = 0 we obtain a parabolic partial differential equation for (1.1), which arises in the solution of the diffusion equation or an acoustic propagation in the ocean. Such applications are characterized by the term initial value problems. The solution is given for the initial condition u(x, y = 0) and the goal is to find the solution u(x, y) for all values of x and y.
Finally when Β2 − AC > 0, we obtain a hyperbolic partial differential equation. This type of equation arises from the solution of the wave equation. The characteristic of the wave equation is that if a disturbance is made in the initial data, then not every point of space feels the disturbance at once. The disturbance has a finite propagation speed. This feature makes it distinct from the elliptic and parabolic partial differential equations when a disturbance of the initial data is felt at once by all points in the domain. Even though these equations have significantly different mathematical properties, the solution methodology, just like for every numerical method in solution of an operator equation, is essentially the same, by exploiting the principle of analytic continuation.
The solution u of these equations is made in a straight forward fashion by assuming: it to be of the form
where ϕi(x, y) are some known basis functions; and the final solution is to be composed of these functions multiplied by some constants αi which are the unknowns to be determined using the specific given boundary conditions. The solution procedure then translates the solution of a functional equation to the solution of a matrix equation, the solution of these unknown constants is much easier to address. The methodology starts by substituting (1.3) into (1.1) and then solving for the unknown coefficients αi from the boundary conditions for the problem if the equations are in the differential form or by integrating if it is an integral equation. Then once the unknown coefficients αi are determined, the general solution for the problem can be obtained using (1.3).
A question that is now raised is: what is the optimum way to choose the known basis functions ϕi as the quality of the final solution depends on the choice of ϕi? It is well known in the numerical community that the best choices of the basis functions are the eigenfunctions of the operator that characterizes the system. Since in most examples one is dealing with a real life system, then the operators, in general, are linear time invariant (LTI) and have a bounded input and bounded output (BIBO) response resulting in a second‐order differential equation, which is the case for Maxwell’s equations. In the general case, the eigenfunctions of these operators are the complex exponentials, and in the transformed domain, they form a ratio of two rational polynomials. Therefore, our goal is to fit the given data for a LTI system either by a sum of complex exponentials or in the transformed domain approximate it by a ratio of polynomials. Next, it is illustrated how the eigenfunctions are used through a bias‐variance tradeoff in reduced rank modelling [1, 2].
1.2