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Regularization-by-noise: Analysis of Numerics


Key facts

Type of research degree
Application deadline
Ongoing deadline
Country eligibility
International (open to all nationalities, including the UK)
Competition funded
Source of funding
University of Leeds
Dr Konstantinos Dareiotis
School of Mathematics
Research groups/institutes
Analysis, Probability and Financial Mathematics
<h2 class="heading hide-accessible">Summary</h2>

Regularisation-by-noise is a branch of stochastic analysis that studies the following phenomenon: Certain dynamical systems tend to behave better when a source of randomness is present. Let us give an example. One of the main concerns in the study of differential equations is the so-called well-posedness, that is, the existence and the uniqueness of solutions. However, there are many equations that suffer from lack of well-posedness, that is, they might have multiple (in fact infinitely many) solutions or might not have a solution at all. A remarkable result in mathematics states that for a large class of those equations, well-posedness can be retrieved provided that the system is perturbed by a random (stochastic), sufficiently rough force. <br /> <br /> Equations of this type, which need the presence of the noise in order to be well-posed, are very interesting from mathematical point of view. In addition, their importance goes beyond mathematics as they are increasingly used in the applied sciences. Among others, they are used in engineering in order to simulate transport-diffusion phenomena, in finance for modelling equity markets, and in neuroscience for modelling interacting neurons. As these equations are highly non-linear, their solutions cannot be given in a closed form and this is exactly the point where their numerical approximation becomes an important issue. While their qualitative properties have been studied satisfactorily in the last two decades, the numerical approximation of their solutions is still on a primary level. <br /> <br /> The aim of this project is to build a robust theory which will enable us to understand in depth and to quantify in an optimal way the performance of numerical algorithms for the approximation of this type of equations. Rather than brute-force calculations, the key point will be to develop and use analytical techniques that will allow us to capture and to exploit the regularising properties of the noise at a numerico-analytic level. <br /> It is expected that the applicants have a strong background in analysis, in particular, stochastic analysis. Some background in partial differential equations would also be beneficial. <br />

<h2 class="heading">How to apply</h2>

<p>Formal applications for research degree study should be made online through the&nbsp;<a href="">University&#39;s website</a>. Please state clearly in the Planned Course of Study section that you are applying for&nbsp;<em><strong>PhD Statistics FT&nbsp;</strong></em>and&nbsp;in the research information section&nbsp;that the research degree you wish to be considered for is<em><strong> Regularization-by-noise: Analysis of Numerics&nbsp;</strong></em>as well as <a href="">Dr. Konstantinos Dareiotis</a>&nbsp; as your proposed supervisor.</p> <p>If English is not your first language, you must provide evidence that you meet the University&#39;s minimum English language requirements (below).</p> <p style="margin-bottom:11px">As an international research-intensive university, we welcome students from all walks of life and from across the world. We foster an inclusive environment where all can flourish and prosper, and we are proud of our strong commitment to student education. Across all Faculties we are dedicated to diversifying our community and we welcome the unique contributions that individuals can bring, and particularly encourage applications from, but not limited to Black, Asian, people who belong to a minority ethnic community, people who identify as LGBT+ and people with disabilities. Applicants will always be selected based on merit and ability.</p> <p>&nbsp;</p>

<h2 class="heading heading--sm">Entry requirements</h2>

Applicants to research degree programmes should normally have at least a first class or an upper second class British Bachelors Honours degree (or equivalent) in an appropriate discipline. The criteria for entry for some research degrees may be higher, for example, several faculties, also require a Masters degree. Applicants are advised to check with the relevant School prior to making an application. Applicants who are uncertain about the requirements for a particular research degree are advised to contact the School or Graduate School prior to making an application.

<h2 class="heading heading--sm">English language requirements</h2>

The minimum English language entry requirement for research postgraduate research study is an IELTS of 6.0 overall with at least 5.5 in each component (reading, writing, listening and speaking) or equivalent. The test must be dated within two years of the start date of the course in order to be valid. Some schools and faculties have a higher requirement.

<h2 class="heading">Funding on offer</h2>

<p style="margin-bottom:12px"><strong>Self-Funded or externally sponsored students are welcome to apply.</strong></p> <p><strong>UK</strong>&nbsp;&ndash;&nbsp;The&nbsp;<a href="">Leeds Doctoral Scholarships</a> and <a href="">Leeds Opportunity Research Scholarship</a> are available to UK applicants. <a href="">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Non-UK</strong> &ndash;The&nbsp;<a href="">China Scholarship Council - University of Leeds Scholarship</a>&nbsp;is available to nationals of China. The&nbsp;<a href="">Leeds Marshall Scholarship</a>&nbsp;is available to support US citizens. <a href="">Alumni Bursary</a> is available to graduates of the University of Leeds.</p> <p><strong>Important:&nbsp;</strong> Any costs associated with your arrival at the University of Leeds to start your PhD including flights, immigration health surcharge/medical insurance and Visa costs are <strong>not</strong> covered under these studentships.</p> <p>Please refer to the <a href="">UKCISA</a> website for information regarding Fee Status for Non-UK Nationals.</p>

<h2 class="heading">Contact details</h2>

<p>For further information about your application, please contact Doctoral College Admissions: e:&nbsp;<a href=""></a></p> <p>For further information about this project, please contact Dr Konstantinos Dareiotis&nbsp;by email:&nbsp;<a href="">K</a><a href=""></a>.</p>