# Bath Numerical Analysis seminar - current

Fridays at **12.15** (**online**)

** Everyone is welcome at these talks. **

Date | Speaker | Title |

8 Oct 2021 | Tobias Hartung (Bath) |
Zoom
## Dimensional Expressivity Analysis for Parametric Quantum Circuits
A standard tool in quantum computing are Variational Quantum Simulations (VQS) which form a class of hybrid quantum-classical algorithms for solving optimization problems. For example, the objective may be to find the ground state of a Hamiltonian by minimizing the energy. As such, VQS use parametric quantum circuit designs to generate a family of quantum states (e.g., states obeying physical symmetries) and efficiently evaluate a cost function for the given set of variational parameters (e.g., energy of the current quantum state) on a quantum device. The optimization is then performed using a classical feedback loop based on the measurement outcomes of the quantum device. In the case of energy minimization, the optimal parameter set therefore encodes the ground state corresponding to the given Hamiltonian provided that the parametric quantum circuit is able to encode the ground state. Hence, the design of parametric quantum circuits is subject to two competing drivers. On one hand, the set of states, that can be generated by the parametric quantum circuit, has to be large enough to contain the ground state. On the other hand, the circuit should contain as few parametric quantum gates as possible to minimize noise from the quantum device. In other words, when designing a parametric quantum circuit we want to ensure that there are no redundant parameters. Thus, in this talk, I will introduce the dimensional expressivity analysis as a means of analyzing a given parametric design in order to remove redundant parameters as well as any unwanted symmetries. Time permitting, we may also discuss best-approximation errors for non-maximally expressive parametric quantum circuits or how to custom design parametric quantum circuits for specific physical applications in which physical states are restricted by a class of symmetries. |

15 Oct 2021 | Kristian Bredies (University of Graz, Austria) |
Zoom
## Dynamic inverse problems in spaces of measures with optimal-transport regularization
We discuss the solution of dynamic inverse problems in which for each time point, a time-dependent linear forward operator mapping the space of measures to a time-dependent Hilbert space has to be inverted. These problems are regularized with dynamic optimal-transport energies that base on the continuity equation as well as convex functionals of Benamou-Brenier-type. Well-posedness of respective Tikhonov minimization is discussed in detail. Further, for the purpose of deriving properties of the solutions as well as numerical algorithms, we present sparsity results for general inverse problems that are connected with the extremal points of the Benamou-Brenier energy subject to the continuity equation. For the latter, it is proven that the extremal points are realized by point masses moving along curves with Sobolev regularity. This result will be employed in numerical optimization algorithms of generalized conditional gradient type. We present instances of this algorithm that are tailored towards dynamic inverse problems associated with point tracking. Finally, the application and numerical performance of the method is demonstrated for sparse dynamic superresolution. This is joint work with Marcello Carioni, Silvio Fanzon and Francisco Romero. References: [1] Kristian Bredies, Silvio Fanzon. An optimal transport approach for solving dynamic inverse problems in spaces of measures. ESAIM: Mathematical Modelling and Numerical Analysis, 54(6):2351-2382, 2020. [2] Kristian Bredies, Marcello Carioni. Sparsity of solutions for variational inverse problems with finite-dimensional data. Calculus of Variations and Partial Differential Equations, 59:14, 2020. [3] Kristian Bredies, Marcello Carioni, Silvio Fanzon, Francisco Romero. On the extremal points of the ball of the Benamou-Brenier energy. Bulletin of the London Mathematical Society, 2021. [4] Kristian Bredies, Marcello Carioni, Silvio Fanzon and Francisco Romero. A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization. arXiv:2012.11706, 2020. |

22 Oct 2021 | Jingwei Liang (Shanghai Jiao Tong University, China) |
Zoom
## A framework for analyzing variance reduced stochastic gradient methods and a new one
Over the past years, variance reduced stochastic gradient methods have become increasingly popular, not only in the machine learning community, but also other areas including inverse problems and mathematical imaging to name a few. However, despite the varieties of variance reduced stochastic gradient descent methods, their analysis varies from each other. In this talk, I will first present a unified framework, under which we manage to abstract different variance reduced stochastic gradient methods into one. Then I will introduce a new stochastic method for composed optimization problems, and illustrate its performance via several imaging problems. |

29 Oct 2021 | Sebastian Banert (Lund, Sweden) |
Zoom
## TBC
TBC |

5 Nov 2021 | Jemma Shipton (Exeter) |
Zoom
## TBC
TBC |

12 Nov 2021 | Tony Shardlow (Bath) |
Zoom
## TBC
TBC |

19 Nov 2021 | Alex Bespalov (Birmingham) |
Zoom
## TBC
TBC |

26 Nov 2021 | Lisa Maria Kreusser (Bath) |
Zoom
## TBC
TBC |

3 Dec 2021 | Matthew Griffith (Bath) |
Zoom
## TBC
TBC |

10 Dec 2021 | Jacob Byrne, Sam Cook, Alexandros Gonos, Danny Goodacre, Tom Ryan |
Zoom
## Year-Long-Project student presentatations
TBC |

17 Dec 2021 | no seminar |
Zoom
## n/a
n/a |

## Seminar calendar

You can subscribe to the NA calendar directly from your calendar client, including Outlook, Apple’s iCalendar or Google calendar. The web address of the calendar is this ICS link which you will need to copy.

To subscribe to a calendar in Outlook:

- In Calendar view, select “Add Calendar” (large green +)
- Select “From Internet”
- Copy paste the ICS link, click OK, and click Yes to subscribe.

To subscribe to a calendar in iCalendar, please follow these instructions. Copy paste the ICS link in “web address”.

To subscribe to a calendar in Google Calendar:

- Go to link.
- On the left side go to "Other Calendars" and click on the dropdown.
- Choose "Add by URL".
- Copy paste the ICS link in the URL of the calendar.
- Click on "Add Calendar" and wait for Google to import your events. This creates a calendar with a somewhat unreadable name.
- To give a readable name to the calendar, click on the three vertical dots sign next to the newly created calendar and select Settings.
- Choose a name for the calendar, eg. Numerical Analysis @ Bath, and click back button on top left.

## How to get to Bath

See here for instructions how to get to Bath. Please email Matthias Ehrhardt (m.ehrhardt@bath.ac.uk) if you intend to come by car and require a parking permit for Bath University Campus for the day.## Tips for giving talks

#### Tips for new students on giving talks

Since the audience of the NA seminar contains both PhD students and staff with quite wide interests and backgrounds, the following are some guidelines/hints to make sure people don't give you evil looks at lunch afterwards.

Before too much time passes in your talk, ideally the audience should know the answers to the following 4 questions:

- What is the problem you're considering?
- Why do you find this interesting?
- What has been done before on this problem/what's the background?
- What is your approach/what are you going to talk about?

There are lots of different ways to communicate this information. One way, if you're doing a slide show, could be for the first 4 slides to cover these 4 questions; although in this case you may want to revisit these points later on in the talk (e.g. to give more detail).

Remember:

- "vertebrate style" (structure hidden inside - like the skeleton of a vertebrate) = good for detective stories, bad for maths talks.
- "crustacean style" (structure visible from outside - like the skeleton of a crustacean) = bad for detective stories, good for maths talks.